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Beyond Advanced · Laws of Motion

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1
A wedge of mass M=2 kgM=2\,\text{kg} with base angles 30∘30^\circ and 60∘60^\circ stands on a rough horizontal floor; the coefficient of static friction between wedge and floor is μ\mu. Both inclined faces are smooth. A block of mass 1 kg1\,\text{kg} lies on the 30∘30^\circ face and a block of mass 4 kg4\,\text{kg} on the 60∘60^\circ face. They are joined by a light inextensible string that runs parallel to the faces over a light frictionless pulley fixed at the apex of the wedge. The blocks are released from rest and the wedge stays at rest. Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the smallest value of μ\mu for which the wedge does not slip while the blocks move?
Not attempted
1 kg4 kg30°60°M = 2 kg (on rough floor)
A
0.2430.243
B
0.3240.324
C
0.1860.186
D
0.1940.194
2
A symmetric wedge of mass M=10 kgM=10\,\text{kg} has its apex pointing down; each of its two smooth faces makes an angle of 30∘30^\circ with the vertical. The apex is pushed between two identical blocks of mass m=4 kgm=4\,\text{kg} that rest on a rough horizontal floor with coefficient of friction μ=0.25\mu=0.25 (static and kinetic equal), each block having a smooth sloping face in contact with the corresponding face of the wedge, as in the figure. The wedge is released from rest, moves vertically (guided so that it cannot tilt or rotate) and drives the blocks apart. Treat the blocks as sliding without tipping. Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the downward acceleration of the wedge?
Not attempted
M = 10 kg4 kg4 kgreleased30°
A
7.89 m s−27.89\,\text{m s}^{-2}
B
6.98 m s−26.98\,\text{m s}^{-2}
C
4.47 m s−24.47\,\text{m s}^{-2}
D
6.60 m s−26.60\,\text{m s}^{-2}
3
A wedge of mass M=6 kgM=6\,\text{kg} with a smooth face inclined at θ\theta (sin⁡θ=0.6\sin\theta=0.6, cos⁡θ=0.8\cos\theta=0.8) rests on a horizontal floor; the coefficient of static friction between wedge and floor is μ=0.3\mu=0.3. A block of mass m=2 kgm=2\,\text{kg} is on the face. A horizontal force FF is applied to the block alone, directed towards the wedge, as shown, and the block is observed to slide up the face. Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the least value of FF for which the wedge itself begins to slip on the floor?
Not attempted
37°mFM
A
56.7 N56.7\,\text{N}
B
23.3 N23.3\,\text{N}
C
24.0 N24.0\,\text{N}
D
9.0 N9.0\,\text{N}
4
An incline of mass 12 kg12\,\text{kg}, with its face at 37∘37^\circ to the horizontal (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8), stands on a platform scale; horizontal forces on the incline are taken by a stop, so the scale records only the vertical force. A block of mass 2 kg2\,\text{kg} lies on the rough face (coefficient of friction 0.30.3). A light string tied to it runs up the face, over a light fixed pulley at the top of the incline, then down under a light movable pulley that carries a hanging block of mass 6 kg6\,\text{kg}, and up again to a hook fixed to the top of the incline, as shown (all string segments except the first are vertical). The pulleys and hook are parts of the incline. The system is released from rest and the blocks start moving. Take g=10 m s−2g=10\,\text{m s}^{-2}. What does the scale read at that instant?
Not attempted
37°2 kg6 kgplatform scaleMᵢ = 12 kg
A
200.0 N200.0\,\text{N}
B
188.7 N188.7\,\text{N}
C
193.2 N193.2\,\text{N}
D
184.2 N184.2\,\text{N}
5
A smooth vertical circular wire of radius RR carries a small bead of mass m=1.0 kgm=1.0\,\text{kg}. A tiny smooth pulley is fixed at the highest point of the wire. A light inextensible string of length 2R2R is tied to the bead, runs straight along a chord to the pulley, passes over it, and carries a block of mass M=1.2 kgM=1.2\,\text{kg} hanging freely along the vertical diameter. Besides the lowest point of the wire, the bead has one other position of equilibrium on the wire. Which diagram shows the bead (black dot) in that other equilibrium position? The diagrams are drawn to the same scale and differ only in the position of the bead.
Not attempted
A
Diagram 1
B
Diagram 2
C
Diagram 3
D
Diagram 4
6
A bead AA of mass 2 kg2\,\text{kg} slides on a smooth fixed vertical wire, and a bead BB of mass 3 kg3\,\text{kg} slides on a smooth fixed horizontal wire that meets it at the base. The beads are joined by a light rigid rod that makes 37∘37^\circ with the horizontal (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8), as in the figure. A constant horizontal force P=40 NP=40\,\text{N} acts on BB towards the vertical wire. The system is released from rest with the rod in this position. Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the vertical acceleration of AA immediately after release?
Not attempted
A 2 kgB 3 kgP37°light rod
A
5.42 m s−25.42\,\text{m s}^{-2} downward
B
12.35 m s−212.35\,\text{m s}^{-2} upward
C
1.53 m s−21.53\,\text{m s}^{-2} upward
D
2.71 m s−22.71\,\text{m s}^{-2} upward
7
Three blocks are stacked on a rough horizontal floor: CC (1 kg1\,\text{kg}) on the floor, a long plank-like block BB (2 kg2\,\text{kg}) on CC, and AA (1 kg1\,\text{kg}) on BB, as in the figure. A horizontal force FF, applied to BB only, is increased slowly from zero. The coefficients of friction (static equal to kinetic) are 0.40.4 between AA and BB, 0.30.3 between BB and CC and 0.10.1 between CC and the floor. Take g=10 m s−2g=10\,\text{m s}^{-2}. Which graph shows the acceleration aa of block BB as a function of FF for 0≤F≤30 N0\le F\le 30\,\text{N}? The graphs share the same scales.
Not attempted
CBAF
A
Graph 1
F (N)a (m/s²)1020302468
B
Graph 2
F (N)a (m/s²)1020302468
C
Graph 3
F (N)a (m/s²)1020302468
D
Graph 4
F (N)a (m/s²)1020302468
8
A uniform rope of mass per unit length 0.40 kg m−10.40\,\text{kg m}^{-1} hangs over a light smooth fixed pulley. At the instant shown, a 1.5 m1.5\,\text{m} length hangs on the left and carries a 1 kg1\,\text{kg} block on its end, and a 2.5 m2.5\,\text{m} length hangs on the right and carries a 3 kg3\,\text{kg} block (the length of rope in contact with the pulley is negligible). The system is released from rest, and g=10 m s−2g=10\,\text{m s}^{-2}. Immediately after release, which graph shows the tension TT in the rope against the distance ss measured along the rope from the 1 kg1\,\text{kg} block (the rope is 4.0 m4.0\,\text{m} long, so s=4s=4 is at the 3 kg3\,\text{kg} block)? The numbers beside the corner points of each graph give the values of TT in newton there.
Not attempted
1 kg3 kgrope 1.5 mrope 2.5 m
A
Graph 1
s (m)T (N)15.024.015.01234102030
B
Graph 2
s (m)T (N)14.322.917.11234102030
C
Graph 3
s (m)T (N)14.320.317.11234102030
D
Graph 4
s (m)T (N)14.327.117.11234102030
9
Two smooth inclines of angles 30∘30^\circ and 60∘60^\circ face each other, with a light smooth fixed pulley at the top of each, the two pulleys being at the same height and 1.2 m1.2\,\text{m} apart. A single light inextensible string joins a 2 kg2\,\text{kg} block on the 30∘30^\circ incline to a 1 kg1\,\text{kg} block on the 60∘60^\circ incline: it runs up each incline parallel to it, over the pulley, then down to a light smooth movable pulley and under it, as shown. A 3 kg3\,\text{kg} block hangs from the axle of the movable pulley. The system is released from rest when the movable pulley is exactly midway between the fixed pulleys and each of the two straight segments of string between a fixed pulley and the movable pulley is 1.0 m1.0\,\text{m} long. Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the acceleration of the 3 kg3\,\text{kg} block immediately after release?
Not attempted
2 kg1 kg3 kg1.2 m30°60°
A
3.28 m s−23.28\,\text{m s}^{-2}
B
2.52 m s−22.52\,\text{m s}^{-2}
C
3.77 m s−23.77\,\text{m s}^{-2}
D
3.06 m s−23.06\,\text{m s}^{-2}
10
A smooth wedge of mass 3 kg3\,\text{kg}, with a face inclined at 37∘37^\circ (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8), lies on a smooth horizontal floor and is free to slide. A smooth block of mass 2 kg2\,\text{kg} rests on the face and is also held by a light inextensible string tied to a fixed point far above and to the right, the string making 53∘53^\circ with the horizontal at the instant shown (it passes clear above the wedge). The whole system is released from rest with the string taut. Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the acceleration of the wedge immediately after release?
Not attempted
37°2 kg3 kg53°
A
2.58 m s−22.58\,\text{m s}^{-2}
B
1.98 m s−21.98\,\text{m s}^{-2}
C
1.88 m s−21.88\,\text{m s}^{-2}
D
2.11 m s−22.11\,\text{m s}^{-2}
11
A small bead of mass m=1 kgm=1\,\text{kg} is threaded on a smooth fixed horizontal rod. A light inextensible string tied to the bead passes over a tiny smooth pulley at height h=0.6 mh=0.6\,\text{m} above the rod and carries a hanging block of mass M=2 kgM=2\,\text{kg} on its free end, which hangs vertically (the hanging part passes the rod without touching it), as shown. The bead is released from rest when the straight part of the string between the bead and the pulley is 1.0 m1.0\,\text{m} long, so that it makes an angle θ\theta with the rod (cos⁡θ=0.8\cos\theta=0.8, sin⁡θ=0.6\sin\theta=0.6). Take g=10 m s−2g=10\,\text{m s}^{-2}. Which of the following statements about the situation immediately after release is/are correct?
Not attempted
mMhθsmooth rod
A
The initial acceleration of the bead is 7.02 m s−27.02\,\text{m s}^{-2}.
B
The hanging block initially accelerates downward at 7.02 m s−27.02\,\text{m s}^{-2}, the same as the bead.
C
Immediately after release the force exerted by the rod on the bead is 4.74 N4.74\,\text{N} directed upward.
D
However large the mass MM of the hanging block is made, the initial acceleration of the bead cannot exceed gg.
12
A cart of mass 4 kg4\,\text{kg} on a smooth horizontal floor is pushed by a horizontal force FF. On its rough horizontal top (coefficient of friction 0.50.5 between the top and the block, static and kinetic equal) lies a block of mass 2 kg2\,\text{kg}. A light string joins this block, over a light smooth pulley at the front edge of the cart, to a block of mass 3 kg3\,\text{kg} that hangs against the smooth vertical front face of the cart, as shown. Take g=10 m s−2g=10\,\text{m s}^{-2}. Which of the following statements is/are correct? (Both blocks are small, and the string segments are straight, horizontal and vertical.)
Not attempted
2 kg3 kgcart 4 kgF
A
The blocks can remain at rest relative to the cart for a force FF as small as 90 N90\,\text{N}.
B
If F=135 NF=135\,\text{N}, there is no friction force between the top block and the cart.
C
At the largest force for which the blocks remain at rest relative to the cart, the cart's vertical front face pushes the hanging block with a horizontal force of 60 N60\,\text{N}.
D
If F=100 NF=100\,\text{N} is applied, the top block slides towards the pulley relative to the cart.
13
A wedge of mass 3 kg3\,\text{kg} with a face inclined at 37∘37^\circ (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8) rests on a smooth horizontal floor and is free to move. A block of mass 2 kg2\,\text{kg} is released from rest on the face; the coefficient of friction between block and face is 0.500.50 (static equal to kinetic). Take g=10 m s−2g=10\,\text{m s}^{-2} and consider the situation just after release. Which of the following statements is/are correct?
Not attempted
37°2 kg3 kgsmooth floor
A
The initial acceleration of the wedge has magnitude 0.99 m s−20.99\,\text{m s}^{-2}.
B
The block slides down the face with acceleration 2.0 m s−22.0\,\text{m s}^{-2} relative to the wedge.
C
If the coefficient of friction were 0.800.80 (static and kinetic), the block would still start sliding.
D
While the block slides, the normal reaction of the floor on the wedge equals 50 N50\,\text{N}.
14
A block of mass 5 kg5\,\text{kg} rests on a light (massless) plank 2, which rests on another light (massless) plank 1, which rests on rough horizontal ground, as in the figure. The coefficients of friction (static, kinetic) are: block-plank 2: (0.25, 0.18)(0.25,\,0.18); plank 2-plank 1: (0.30, 0.22)(0.30,\,0.22); plank 1-ground: (0.40, 0.36)(0.40,\,0.36). A horizontal force FF is applied to the block and increased slowly from zero. Take g=10 m s−2g=10\,\text{m s}^{-2}, sin⁡37∘=0.60\sin37^\circ=0.60 and cos⁡37∘=0.80\cos37^\circ=0.80. Which of the following statements is/are correct?
Not attempted
plank 1plank 25 kgFground
A
The block begins to slide over the upper plank when FF reaches 12.5 N12.5\,\text{N}.
B
When F=30 NF=30\,\text{N} the block's acceleration is 4.2 m s−24.2\,\text{m s}^{-2}.
C
When F=30 NF=30\,\text{N} both planks stay at rest and the friction force at the ground is 9 N9\,\text{N}.
D
If instead F=30 NF=30\,\text{N} is applied at 37∘37^\circ above the horizontal, the block's acceleration is 3.65 m s−23.65\,\text{m s}^{-2}.
15
Three smooth uniform cylinders of the same radius rr lie in a vertical plane. Two of them, each of mass 2 kg2\,\text{kg}, rest on a smooth horizontal floor with their axes joined by a light horizontal string; the third, of mass 3 kg3\,\text{kg}, rests on both of them. The distance between the axes of the lower two is 2.4 r2.4\,r (so each line of centres from the upper cylinder to a lower one makes an angle φ\varphi with the vertical, with sin⁡φ=0.6\sin\varphi=0.6), as in the figure. Take g=10 m s−2g=10\,\text{m s}^{-2}. At a certain instant the string is cut. Which of the following statements is/are correct?
Not attempted
3 kg2 kg2 kgstring
A
Before the string is cut, its tension is 11.25 N11.25\,\text{N}.
B
Immediately after the string is cut, the force between the upper cylinder and each lower cylinder is still 18.75 N18.75\,\text{N}.
C
Immediately after the string is cut, the upper cylinder has a downward acceleration of 2.97 m s−22.97\,\text{m s}^{-2}.
D
Immediately after the string is cut, the normal reaction of the floor on each lower cylinder is still 35 N35\,\text{N}.
16
A lift (elevator) accelerates downward at 4 m s−24\,\text{m s}^{-2}. A light fixed smooth pulley hangs from the lift's ceiling by a spring balance. Over it passes a light inextensible string carrying a 3 kg3\,\text{kg} block (m1m_1) on one side and, on the other, the axle of a light smooth movable pulley PP. A second string over PP carries blocks m2=2 kgm_2=2\,\text{kg} and m3=6 kgm_3=6\,\text{kg}. All string segments are vertical and taut, and the blocks are released from rest relative to the lift. Take g=10 m s−2g=10\,\text{m s}^{-2}. Which of the following statements is/are correct, immediately after release?
Not attempted
lift accelerating downward at 4 m/s²spring balance3 kgP (light)2 kg6 kg
A
The block m2m_2 remains at rest relative to the lift.
B
The spring balance from which the fixed pulley hangs reads 48 N48\,\text{N} (taking the reading as the force it exerts on the pulley).
C
Relative to the ground, the block m3m_3 accelerates downward at 4 m s−24\,\text{m s}^{-2}.
D
If the lift were instead in free fall, the spring balance would still read 48 N48\,\text{N}.
17
In each of the four arrangements (all pulleys light and smooth, strings light and inextensible, tables and the 30∘30^\circ incline smooth, block AA of mass mm) the system is released from rest. In P, the hanging block of mass 2m2m is tied directly to AA over the edge pulley. In Q, the string from AA goes over the edge pulley, under a light movable pulley carrying a block of mass 3m3m, and up to a fixed support above the table edge. In R, the string is tied to a hook at the table edge, runs back round a light pulley fixed to AA, and then over the edge pulley to a hanging block of mass mm. In S, AA is on the incline and a block of mass mm hangs over a fixed pulley at the top. Match each arrangement in List-I with the magnitude of the acceleration of AA in List-II. List-IList-IIP) Arrangement P1) 2g/5Q) Arrangement Q2) 6g/7R) Arrangement R3) g/4S) Arrangement S4) g/25) 2g/3\begin{array}{ll}\textbf{List-I} & \textbf{List-II} \\ \text{P)}\ \text{Arrangement P} & \text{1)}\ 2g/5 \\ \text{Q)}\ \text{Arrangement Q} & \text{2)}\ 6g/7 \\ \text{R)}\ \text{Arrangement R} & \text{3)}\ g/4 \\ \text{S)}\ \text{Arrangement S} & \text{4)}\ g/2 \\ & \text{5)}\ 2g/3 \end{array}
Not attempted
A2mPA3mQAmRAm30°S
A
P-5, Q-1, R-2, S-3
B
P-4, Q-2, R-1, S-5
C
P-5, Q-2, R-1, S-3
D
P-5, Q-4, R-3, S-1
18
A smooth wedge of mass M=2 kgM=2\,\text{kg} with a face inclined at 37∘37^\circ (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8) lies on a smooth horizontal floor. A smooth block of mass m=1 kgm=1\,\text{kg} is placed on the face, which slopes upward towards the right, as shown. A horizontal force FF pulls the wedge to the right through a string tied to its vertical side. The system is released from rest with the block touching the face. Take g=10 m s−2g=10\,\text{m s}^{-2}. Match each value of FF in List-I with the acceleration of the wedge just after release in List-II (in m s−2\text{m s}^{-2}, to the nearest tenth). List-IList-IIP) F=01) 12.2Q) F=10 N2) 20.0R) F=24 N3) 2.0S) F=40 N4) 19.05) 6.3\begin{array}{ll}\textbf{List-I} & \textbf{List-II} \\ \text{P)}\ F=0 & \text{1)}\ 12.2 \\ \text{Q)}\ F=10\,\text{N} & \text{2)}\ 20.0 \\ \text{R)}\ F=24\,\text{N} & \text{3)}\ 2.0 \\ \text{S)}\ F=40\,\text{N} & \text{4)}\ 19.0 \\ & \text{5)}\ 6.3 \end{array}
Not attempted
37°mMF
A
P-3, Q-5, R-2, S-4
B
P-5, Q-3, R-1, S-2
C
P-3, Q-1, R-5, S-4
D
P-3, Q-5, R-1, S-2
19
A light inextensible string of length 2.00 m2.00\,\text{m} has its two ends tied to two hooks at the same height, 1.20 m1.20\,\text{m} apart. A smooth ring of mass 1.00 kg1.00\,\text{kg} is threaded on the string. A horizontal force of 10.0 N10.0\,\text{N}, directed parallel to the line joining the hooks, is applied to the ring, and the ring is held in stable equilibrium (the string stays taut and in one vertical plane). Take g=10 m s−2g=10\,\text{m s}^{-2}. Find the tension in the string, in newton, correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 7.81
20
A cart of mass M=2 kgM=2\,\text{kg} stands on a rough horizontal floor (the friction force on the cart equals μ=0.1\mu=0.1 times the normal reaction of the floor on the cart, for kinetic and static friction alike). Its top is smooth and horizontal, with a block of mass m1=4 kgm_1=4\,\text{kg} on it. A light inextensible string joins this block, over a light smooth pulley at the front edge of the cart, to a block of mass m2=3 kgm_2=3\,\text{kg} that hangs inside a smooth vertical shaft cut in the cart, so that it can move up and down relative to the cart but must share its horizontal motion. The system is released from rest. Take g=10 m s−2g=10\,\text{m s}^{-2}. Find the magnitude of the cart's acceleration (in m s−2\text{m s}^{-2}) just after release, correct to two decimal places.
Not attempted
3 kg4 kgcart, M = 2 kg
Your answer: —
Correct answer: 1.44
21
A smooth wedge of mass M=2 kgM=2\,\text{kg} with an inclined face at 37∘37^\circ (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8) lies on a smooth horizontal table. A smooth block of mass m=1 kgm=1\,\text{kg} rests on the inclined face, which slopes upward towards the pulley, as in the figure. A light string tied to the vertical side of the wedge runs horizontally over a light smooth pulley at the edge of the table and carries a hanging block of mass m3=6 kgm_3=6\,\text{kg}. The system is released from rest. Take g=10 m s−2g=10\,\text{m s}^{-2} and assume the small block stays in contact with the face. What is the magnitude of the acceleration of the small block relative to the ground, in m s−2\text{m s}^{-2}, correct to two decimal places?
Not attempted
mMm₃a
Your answer: —
Correct answer: 7.59
22
In the arrangement shown, a light string carries a block m1m_1 on one side of a fixed smooth pulley and, on the other side, supports the axle of a movable smooth pulley P1P_1 of mass 1.5 kg1.5\,\text{kg}. A second light string runs over P1P_1; it carries a 4 kg4\,\text{kg} block m2m_2 on one end, and the axle of a light movable smooth pulley P2P_2 on the other end. A third string over P2P_2 carries blocks m4=2 kgm_4=2\,\text{kg} and m5=7 kgm_5=7\,\text{kg}. All strings are light and inextensible, and all segments are vertical. The system is released from rest. For what mass m1m_1 (in kg) does m1m_1 remain at rest while all the other blocks move? Give the answer correct to two decimal places.
Not attempted
m1P1m2P2m4m5
Your answer: —
Correct answer: 11.24
23
A light inextensible string has one end tied to a block AA of mass 2 kg2\,\text{kg}; it passes over a fixed smooth pulley, runs down under a light smooth movable pulley PP, and its other end is tied to the ceiling. A block BB of mass 7 kg7\,\text{kg} is attached to the axle of PP and rests on the floor, and both strands going down from the fixed pulley and up to the ceiling are vertical. Initially the string is slack by an amount such that AA, released from rest, falls freely through 0.80 m0.80\,\text{m} before the string becomes taut. When the string becomes taut it jerks (the tension acts impulsively and its impulse is much larger than the impulse of the weights during the jerk), after which both blocks move with the string taut. Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the greatest vertical distance, in metres, that AA descends from its release point? Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 1.37
24
A wedge of mass 3 kg3\,\text{kg} with smooth faces inclined at 30∘30^\circ and 60∘60^\circ stands on a smooth horizontal table. A block of mass 2 kg2\,\text{kg} lies on the 30∘30^\circ face and a block of mass 4 kg4\,\text{kg} on the 60∘60^\circ face; a light inextensible string joins them over a light smooth pulley at the apex, parallel to both faces. A second light string is tied horizontally to the bottom of the wedge on the 60∘60^\circ side, runs over a light smooth pulley at the edge of the table and carries a hanging block of mass 5 kg5\,\text{kg}, as shown. The system is released from rest, and the blocks on the faces slide with the string taut and in contact with the faces. Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the acceleration of the wedge (in m s−2\text{m s}^{-2}), correct to two decimal places?
Not attempted
2 kg4 kg30°60°M = 3 kg5 kg
Your answer: —
Correct answer: 2.97
25
A uniform chain of length L=1.5 mL=1.5\,\text{m} and mass M=2 kgM=2\,\text{kg} lies straight on a horizontal table with its end links 0.25L0.25L hanging over the smooth edge (the hanging part is vertical). The table is smooth except for a rough strip of width 0.30L0.30L adjacent to the edge, where the coefficient of friction (static and kinetic) between the chain and the table is μ=0.50\mu=0.50. The chain is released from rest and slides off the table with its links staying in a straight line on the table and a straight vertical line below the edge, so that its whole length remains taut. Take g=10 m s−2g=10\,\text{m s}^{-2}. Among all points of the chain and all instants before the last link leaves the table, what is the largest tension in the chain?
Not attempted
chain: length L, mass 2 kgrough 0.3Lsmooth0.25L
A
7.50 N7.50\,\text{N}
B
4.50 N4.50\,\text{N}
C
6.61 N6.61\,\text{N}
D
6.30 N6.30\,\text{N}
26
A small ball is thrown vertically upward with speed u=30 m s−1u=30\,\text{m s}^{-1} in air. The air exerts a drag force of magnitude cv2cv^2 directed opposite to the velocity, where vv is the speed and cc is a constant for which the terminal speed of the falling ball would be vT=10 m s−1v_T=10\,\text{m s}^{-1}. Take g=10 m s−2g=10\,\text{m s}^{-2} (constant) and ignore buoyancy. What is the speed of the ball when, on its way DOWN, it passes through the point that is half of the maximum height above the launch point?
Not attempted
A
8.27 m s−18.27\,\text{m s}^{-1}
B
9.49 m s−19.49\,\text{m s}^{-1}
C
7.07 m s−17.07\,\text{m s}^{-1}
D
6.71 m s−16.71\,\text{m s}^{-1}
27
A uniform flexible chain of linear density λ=0.80 kg m−1\lambda=0.80\,\text{kg m}^{-1} is lowered vertically by a hand that moves down at the constant speed v=1.5 m s−1v=1.5\,\text{m s}^{-1}. Below it a small table is raised vertically from rest with constant acceleration 2.0 m s−22.0\,\text{m s}^{-2}. At t=0t=0 the lowest link of the chain just touches the table, and from then on the links come to rest on the table one after another and form a compact heap on it (no rebound, heap height negligible). The chain above the heap hangs straight and moves at the constant speed vv. Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the force exerted by the table on the chain at t=1.0 st=1.0\,\text{s}?
Not attempted
A
29.80 N29.80\,\text{N}
B
25.80 N25.80\,\text{N}
C
24.20 N24.20\,\text{N}
D
33.80 N33.80\,\text{N}
28
A uniform flexible chain of length 1.00 m1.00\,\text{m} and mass 2.0 kg2.0\,\text{kg} hangs vertically from a hook, with its lowest link 0.80 m0.80\,\text{m} above the pan of a weighing scale. At t=0t=0 the chain is released and falls freely, links staying in a vertical line. As each link reaches the pan it comes to rest at once and stays there (no rebound, negligible heap height). Take g=10 m s−2g=10\,\text{m s}^{-2}. Which graph correctly shows the force NN exerted by the pan on the chain as a function of time tt from release? (Each graph is drawn to the same scales.)
Not attempted
A
Graph 1
t (s)N (N)O0.20.40.60.820406080100
B
Graph 2
t (s)N (N)O0.20.40.60.820406080100
C
Graph 3
t (s)N (N)O0.20.40.60.820406080100
D
Graph 4
t (s)N (N)O0.20.40.60.820406080100
29
A block of mass 2.0 kg2.0\,\text{kg} lies at rest on a rough horizontal floor with coefficients of friction μs=0.50\mu_s=0.50 (static) and μk=0.30\mu_k=0.30 (kinetic). A horizontal force F(t)F(t) acts on it: it rises linearly from 00 to 20 N20\,\text{N} in 5 s5\,\text{s}, then falls linearly back to 00 at t=10 st=10\,\text{s}, and stays zero afterwards, as shown. Take g=10 m s−2g=10\,\text{m s}^{-2}. Which graph correctly shows the speed vv of the block against time tt?
Not attempted
t (s)F (N)O51015201020
A
Graph 1
t (s)v (m/s)O5101520102030
B
Graph 2
t (s)v (m/s)O5101520102030
C
Graph 3
t (s)v (m/s)O5101520102030
D
Graph 4
t (s)v (m/s)O5101520102030
30
A rope of length L=1.00 mL=1.00\,\text{m} lies straight on a smooth horizontal table. One end is pivoted at the fixed point O, and a small bead of mass 0.50 kg0.50\,\text{kg} is tied to the other end. The rope has linear density λ(r)=λ0(1+r/L)\lambda(r)=\lambda_0(1+r/L) with λ0=1.00 kg m−1\lambda_0=1.00\,\text{kg m}^{-1}, where rr is the distance from O. The whole arrangement rotates uniformly about O with ω=4.0 rad s−1\omega=4.0\,\text{rad s}^{-1}, the rope staying straight. Which graph correctly shows the tension TT in the rope as a function of the distance rr from O? (Each graph has the same scales; the vertical axis starts at 6 N6\,\text{N}.)
Not attempted
A
Graph 1
r (m)T (N)0.250.50.751.0812162024
B
Graph 2
r (m)T (N)0.250.50.751.0812162024
C
Graph 3
r (m)T (N)0.250.50.751.0812162024
D
Graph 4
r (m)T (N)0.250.50.751.0812162024
31
A beaker containing water stands on the platform of a spring scale fixed to the floor of a lift. The beaker plus water (not counting the ball described next) has mass 3.0 kg3.0\,\text{kg}. A small ball of mass 0.40 kg0.40\,\text{kg} and volume 1.0×10−3 m31.0\times10^{-3}\,\text{m}^3 is fully submerged and is rising inside the water; the density of water is 1000 kg m−31000\,\text{kg m}^{-3} and the water level in the beaker stays fixed. At a certain instant the lift is accelerating upward at 2.0 m s−22.0\,\text{m s}^{-2} and the ball is accelerating upward relative to the beaker at 4.0 m s−24.0\,\text{m s}^{-2}. Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the reading of the scale (in newtons) at that instant?
Not attempted
A
40.8 N40.8\,\text{N}
B
38.4 N38.4\,\text{N}
C
43.2 N43.2\,\text{N}
D
42.4 N42.4\,\text{N}
32
A horizontal jet of water of constant cross-section A=5.0 cm2A=5.0\,\text{cm}^2 leaves a nozzle with speed 20 m s−120\,\text{m s}^{-1} (density 1000 kg m−31000\,\text{kg m}^{-3}). A cart of mass 12 kg12\,\text{kg} runs on smooth rails parallel to the jet and carries a smooth flat plate inclined at 37∘37^\circ to the direction of the jet (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8), as shown. The cart is moving away from the nozzle with speed 4.0 m s−14.0\,\text{m s}^{-1} at the instant considered, and the water that strikes the plate spreads out and leaves along the plate's surface without losing the component of its relative velocity parallel to the plate (no splash); ignore gravity acting on the water and the weight of water on the plate. The rails take up any force perpendicular to them. What is the acceleration of the cart at that instant?
Not attempted
12 kgjet, speed 20 m/su = 4 m/s37°
A
3.84 m s−23.84\,\text{m s}^{-2}
B
6.00 m s−26.00\,\text{m s}^{-2}
C
6.40 m s−26.40\,\text{m s}^{-2}
D
4.80 m s−24.80\,\text{m s}^{-2}
33
A fixed horizontal rough cylinder (coefficient of friction μ=0.50\mu=0.50 between rope and cylinder) sits at the top edge of an incline of angle 37∘37^\circ (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8). A light rope runs from a block AA of mass 2.0 kg2.0\,\text{kg} on the incline, parallel to the incline, up to the top of the cylinder, over it, and then hangs vertically supporting a block BB of mass mm, as in the figure. The coefficient of friction between block AA and the incline is 0.250.25 (static and kinetic equal). The system is at rest. Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the largest value of mm for which the system can stay at rest?
Not attempted
A 2 kgBfixed rough cylinderμ = 0.5037°incline: μ = 0.25 under A
A
7.70 kg7.70\,\text{kg}
B
2.54 kg2.54\,\text{kg}
C
4.85 kg4.85\,\text{kg}
D
3.63 kg3.63\,\text{kg}
34
A chain of length L=2.00 mL=2.00\,\text{m} passes over a very small, light, smooth pulley fixed to a spring balance, which hangs from the ceiling of a lift accelerating upward at 2.0 m s−22.0\,\text{m s}^{-2}. The chain is not uniform: its linear density is λ(s)=λ0(1+s/L)\lambda(s)=\lambda_0(1+s/L) with λ0=1.00 kg m−1\lambda_0=1.00\,\text{kg m}^{-1}, where ss is the distance along the chain measured from end PP (so end PP is the lighter end). The chain is released from rest relative to the lift with 1.40 m1.40\,\text{m} hanging on the PP side, and slides over the pulley. At the instant when 1.70 m1.70\,\text{m} hangs on the PP side (the other side hangs 0.30 m0.30\,\text{m}), what does the spring balance read? Take g=10 m s−2g=10\,\text{m s}^{-2} and assume both hanging parts are vertical.
Not attempted
A
33.57 N33.57\,\text{N}
B
36.00 N36.00\,\text{N}
C
22.38 N22.38\,\text{N}
D
10.80 N10.80\,\text{N}
35
A closed box rests on the platform of a scale. Inside, near the top, a hopper holds sand which begins to pour out at t=0t=0 at a constant rate μ=0.80 kg s−1\mu=0.80\,\text{kg s}^{-1} and continues for 6.0 s6.0\,\text{s} before the hopper is empty. The sand falls freely through 1.25 m1.25\,\text{m} onto the floor of the box and stops there at once (no bounce). The box, hopper and the sand together weigh WW at all times (the sand in the air counts as part of WW). Take g=10 m s−2g=10\,\text{m s}^{-2} and treat the stream as continuous. Which of the following statements about the scale reading N(t)N(t) are correct?
Not attempted
A
At t=0.30 st=0.30\,\text{s} the reading is lower than WW by 2.4 N2.4\,\text{N}.
B
For 0.50 s<t<6.0 s0.50\,\text{s}<t<6.0\,\text{s} the reading is exactly WW.
C
Immediately after the hopper empties at t=6.0 st=6.0\,\text{s} the reading jumps to W+4.0 NW+4.0\,\text{N}.
D
The largest reading during the whole process is W+4.0 NW+4.0\,\text{N}, reached at t=6.5 st=6.5\,\text{s}.
36
A uniform spring of mass 1.0 kg1.0\,\text{kg} and force constant k=200 N m−1k=200\,\text{N m}^{-1} lies on a horizontal table (the table is smooth for the spring). One end is attached to a block of mass 3.0 kg3.0\,\text{kg} which rests on a rough part of the table with coefficient of friction μ=0.25\mu=0.25 (static and kinetic equal), and the free end of the spring is pulled with a constant horizontal force F=16 NF=16\,\text{N} along the spring. After a short time the spring and block move together with a steady acceleration and the spring has a constant extension (the extension is measured as the total change in length). Take g=10 m s−2g=10\,\text{m s}^{-2}. Which of the following statements are correct?
Not attempted
A
The force exerted by the spring on the block is about 13.9 N13.9\,\text{N}.
B
The extension of the spring is 8.0 cm8.0\,\text{cm}.
C
The tension at the midpoint of the spring is about 14.9 N14.9\,\text{N}.
D
The acceleration of the block is (F−μMg)/M≈2.83 m s−2(F-\mu Mg)/M\approx 2.83\,\text{m s}^{-2}.
37
A block of mass 0.50 kg0.50\,\text{kg} lies on a rough horizontal table and is attached to a fixed wall by a light spring of force constant 50 N m−150\,\text{N m}^{-1} lying along the table. The coefficient of friction between block and table is 0.200.20 (static and kinetic equal). The spring is stretched by 0.31 m0.31\,\text{m} from its natural length and the block is released from rest. Take g=10 m s−2g=10\,\text{m s}^{-2}. Which of the following statements are correct?
Not attempted
A
The block comes to rest permanently after exactly 77 half-oscillations.
B
The total distance travelled by the block is 2.40 m2.40\,\text{m}.
C
The maximum speed of the block during the motion is 3.1 m s−13.1\,\text{m s}^{-1}.
D
The block comes to rest 1.0 cm1.0\,\text{cm} from the natural-length position, on the same side of it as the release point.
38
A block AA of mass 2.0 kg2.0\,\text{kg} lies on a long block BB of mass 4.0 kg4.0\,\text{kg} which rests on a smooth horizontal floor. The coefficient of friction between AA and BB is 0.300.30 (static and kinetic equal). Starting from rest at t=0t=0, a horizontal force F=(4.0 N s−1) tF=(4.0\,\text{N s}^{-1})\,t is applied to block AA (not to BB). Block AA stays on BB throughout. Take g=10 m s−2g=10\,\text{m s}^{-2}. Which of the following statements are correct?
Not attempted
A
Block AA begins to slide on block BB at t=2.25 st=2.25\,\text{s}.
B
At t=4.0 st=4.0\,\text{s} the speed of AA is 7.375 m s−17.375\,\text{m s}^{-1}.
C
At t=4.0 st=4.0\,\text{s} the speed of BB is about 4.31 m s−14.31\,\text{m s}^{-1}.
D
Between the beginning of slipping and t=4.0 st=4.0\,\text{s}, AA slides forward on BB through about 1.79 m1.79\,\text{m}.
39
Sand falls vertically from a hopper at the constant rate 5.0 kg s−15.0\,\text{kg s}^{-1} through 1.8 m1.8\,\text{m} onto a long horizontal conveyor belt moving at the constant speed 2.0 m s−12.0\,\text{m s}^{-1}. The sand lands with no horizontal velocity, slips on the belt for a short time and is then carried along at the belt's speed (the coefficient of friction between sand and belt is unknown but large enough for the slipping time to be short). Take g=10 m s−2g=10\,\text{m s}^{-2}. Consider the steady state. Which of the following statements are correct?
Not attempted
A
The extra horizontal force that the motor must supply to hold the belt speed constant is 10 N10\,\text{N}.
B
The extra power that the motor must supply is 10 W10\,\text{W}.
C
The horizontal kinetic energy of the sand increases at the rate of 10 W10\,\text{W}.
D
Heat is generated by the sliding of the sand on the belt at the rate of 10 W10\,\text{W}.
40
A tank on smooth wheels carries water that is kept at a constant depth h=1.25 mh=1.25\,\text{m} above a small side hole of area 2.0 cm22.0\,\text{cm}^2 (the tank is continuously refilled from above by a vertical pipe that adds no horizontal momentum). Water leaves the hole horizontally with the Torricelli speed. Take g=10 m s−2g=10\,\text{m s}^{-2} and the density of water 1000 kg m−31000\,\text{kg m}^{-3}. Ignore air resistance and the motion of the cart itself when computing relative speeds. Which of the following statements are correct?
Not attempted
A
With nothing in front of the hole, the horizontal force on the tank has magnitude 5.0 N5.0\,\text{N}, opposite to the direction of the jet.
B
If a rigid plate fixed to the cart is placed in front of the hole so that the water hits it and runs straight down, the net horizontal force on the cart plus plate is zero.
C
If a plate fixed to the ground is placed in front of the hole so that the jet is stopped by it, the horizontal force on the tank becomes zero.
D
If a plate fixed to the cart sends the jet back along its original line with the same speed, the force on the cart becomes 10 N10\,\text{N} in the same direction as in the no-plate case.
41
In each situation a block of mass 2.0 kg2.0\,\text{kg} is at rest on a rough horizontal table (μs=0.50\mu_s=0.50, μk=0.40\mu_k=0.40, g=10 m s−2g=10\,\text{m s}^{-2}, sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8), held in equilibrium by a spring (stretched, pulling the block one way, horizontal) and a string (pulling the other way) as described. The string is then cut. Match each situation in List-I with the magnitude of the block's acceleration just after the cut in List-II (the spring force is unchanged at that instant). List-IList-IIP) Spring pulls with 12 N; a horizontal string pulls back with 12 N1) 0Q) Spring pulls with 9 N; a horizontal string pulls back with 9 N2) 1.5 m s−2R) Spring pulls with 18 N; a horizontal string pulls back with 26 N3) 2.0 m s−2S) Spring pulls with 11 N; a string at 37∘ above the horizontal pulls back with 10 N4) 5.0 m s−25) 2.7 m s−2\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P) Spring pulls with } 12\,\text{N; a horizontal string pulls back with } 12\,\text{N} & \text{1) } 0 \\ \text{Q) Spring pulls with } 9\,\text{N; a horizontal string pulls back with } 9\,\text{N} & \text{2) } 1.5\,\text{m s}^{-2} \\ \text{R) Spring pulls with } 18\,\text{N; a horizontal string pulls back with } 26\,\text{N} & \text{3) } 2.0\,\text{m s}^{-2} \\ \text{S) Spring pulls with } 11\,\text{N; a string at } 37^\circ \text{ above the horizontal pulls back with } 10\,\text{N} & \text{4) } 5.0\,\text{m s}^{-2} \\ & \text{5) } 2.7\,\text{m s}^{-2} \end{array}
Not attempted
A
P-3, Q-1, R-4, S-2
B
P-1, Q-3, R-4, S-2
C
P-3, Q-1, R-2, S-4
D
P-4, Q-1, R-3, S-5
42
Take g=10 m s−2g=10\,\text{m s}^{-2} and the density of water 1000 kg m−31000\,\text{kg m}^{-3}. In rows R and S the water jet has cross-section 2.0 cm22.0\,\text{cm}^2 and speed 10 m s−110\,\text{m s}^{-1}; in S the water runs off along the plate without losing the tangential component of its velocity. In rows P and Q, the pan is that of a scale, the sand and the chain stop at once on reaching it and do not rebound. Match each situation in List-I with the correct force in List-II. List-IList-IIP) Chain of mass 2 kg and length 1 m released from rest with its lowest link touching a pan: maximum pan force1) 5 NQ) Sand poured at 2.0 kg s−1 from 0.80 m above a pan: steady reading minus the weight of sand lying on the pan2) 8 NR) Water jet hitting a fixed smooth vertical wall normally and rebounding with its speed unchanged: force on the wall3) 10 NS) The same jet hitting a fixed smooth plate inclined at 30∘ to the jet: component of the force on the plate along the jet4) 40 N5) 60 N\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P) Chain of mass } 2\,\text{kg and length } 1\,\text{m released from rest with its lowest link touching a pan: maximum pan force} & \text{1) } 5\,\text{N} \\ \text{Q) Sand poured at } 2.0\,\text{kg s}^{-1}\text{ from } 0.80\,\text{m above a pan: steady reading minus the weight of sand lying on the pan} & \text{2) } 8\,\text{N} \\ \text{R) Water jet hitting a fixed smooth vertical wall normally and rebounding with its speed unchanged: force on the wall} & \text{3) } 10\,\text{N} \\ \text{S) The same jet hitting a fixed smooth plate inclined at } 30^\circ \text{ to the jet: component of the force on the plate along the jet} & \text{4) } 40\,\text{N} \\ & \text{5) } 60\,\text{N} \end{array}
Not attempted
A
P-5, Q-2, R-1, S-4
B
P-5, Q-2, R-4, S-1
C
P-5, Q-3, R-4, S-2
D
P-1, Q-2, R-3, S-5
43
A uniform rope of length 2.00 m2.00\,\text{m} and linear density 0.50 kg m−10.50\,\text{kg m}^{-1} hangs over a very small, smooth, fixed pulley. A small ball of mass 0.20 kg0.20\,\text{kg} is tied to one end of the rope. The system is released from rest with equal lengths of rope on both sides of the pulley (the ball on the left). The ball is heavy enough to descend. Take g=10 m s−2g=10\,\text{m s}^{-2}. How long (in seconds) after the release does the free end of the rope on the right leave the pulley? Assume the ball does not reach the pulley, the rope never slips off before this and both hanging parts stay vertical.
Not attempted
Your answer: —
Correct answer: 0.86
44
A block of mass 2.0 kg2.0\,\text{kg} lies at rest on a smooth horizontal surface. A horizontal force F(t)F(t) along a fixed line acts on it, as shown (straight segments between the points (0,0)(0,0), (4 s,28 N)(4\,\text{s},28\,\text{N}), (6 s,28 N)(6\,\text{s},28\,\text{N}), (9 s,0)(9\,\text{s},0), (10 s,−10 N)(10\,\text{s},-10\,\text{N}), (12 s,−10 N)(12\,\text{s},-10\,\text{N}), (13 s,0)(13\,\text{s},0), and F=0F=0 afterwards). In addition the block experiences a resistive force −bv-bv opposing its velocity, with b=3.5 N s m−1b=3.5\,\text{N s m}^{-1}. After the force has stopped acting the block eventually comes to rest. What is its net displacement (in metres) from its starting point, in the direction of positive FF? Give the answer to two decimal places.
Not attempted
t (s)F (N)O24681012-10102028
Your answer: —
Correct answer: 35.43
45
A uniform chain of length 1.40 m1.40\,\text{m} and linear density 1.00 kg m−11.00\,\text{kg m}^{-1} lies in a smooth V-shaped trough formed by two fixed planes inclined at 37∘37^\circ and 53∘53^\circ to the horizontal (sin⁡37∘=0.60\sin37^\circ=0.60, sin⁡53∘=0.80\sin53^\circ=0.80), the chain lying along the line of greatest slope and passing smoothly round the bottom of the V. A small bead of mass 0.35 kg0.35\,\text{kg} is fixed to the chain's end that lies on the 53∘53^\circ plane. The chain is released from rest with its whole length on the 37∘37^\circ plane (the bead just at the bottom). Take g=10 m s−2g=10\,\text{m s}^{-2}. The chain then oscillates, always with a portion on each plane except at the turning point. Find the period of the oscillation in seconds, to two decimal places.
Not attempted
Your answer: —
Correct answer: 2.22
46
A cart on a horizontal track is pulled by a constant horizontal force of 40 N40\,\text{N}. Resistance on the track exerts a constant horizontal force of 10 N10\,\text{N} opposing the motion. The cart (including the sand it holds) has constant mass 10 kg10\,\text{kg} because sand pours into it from a fixed hopper above at 2.0 kg s−12.0\,\text{kg s}^{-1}, arriving vertically with no horizontal velocity, while sand leaks out through a hole in the floor of the cart at exactly the same rate (leaving the cart with the cart's own horizontal velocity at that instant). The cart starts from rest at t=0t=0. Take e−1=0.3679e^{-1}=0.3679 if needed. What is the speed of the cart (in m s−1\text{m s}^{-1}) at t=5.0 st=5.0\,\text{s}? Give the answer to two decimal places.
Not attempted
Your answer: —
Correct answer: 9.48
47
A flexible uniform chain of length 3.0 m3.0\,\text{m} and linear density 0.60 kg m−10.60\,\text{kg m}^{-1} lies in a small compact heap on a table. Its upper end is lifted by a constant vertical force F=9.0 NF=9.0\,\text{N} applied from the instant t=0t=0. Links leave the heap one at a time, each being jerked into motion with the speed of the part already rising (a perfectly inelastic pickup, no friction with the heap, negligible heap size). Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the maximum height (in metres) above the table reached by the upper end of the chain? Give the answer to two decimal places.
Not attempted
Your answer: —
Correct answer: 2.25
48
A block of mass 2.0 kg2.0\,\text{kg} rests on a rough horizontal floor (coefficient of friction μ=0.50\mu=0.50, static and kinetic equal). A light string attached to it makes a fixed angle of 37∘37^\circ with the horizontal (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8), and its tension is increased steadily as T=(10 N s−1) tT=(10\,\text{N s}^{-1})\,t from t=0t=0. Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the speed of the block (in m s−1\text{m s}^{-1}) at the instant it just leaves the floor? Give the answer to two decimal places.
Not attempted
Your answer: —
Correct answer: 16.16
49
A small cart (treat as a particle) runs on the inner, concave side of a smooth rigid track whose shape is the upper half of the ellipse x2a2+y2b2=1\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1, with a=3 ma=3\,\text{m} and b=2 mb=2\,\text{m}. It enters at the left end (−a,0)(-a,0) of the horizontal diameter, moving vertically upward with speed uu, and rides on the underside of the arc over the crest (0,b)(0,b) down to the right end (a,0)(a,0). Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the least uu for which the cart stays in contact with the track along the whole arc?
Not attempted
entryb = 2 m2a = 6 mu
A
8.37 m s−18.37\,\text{m s}^{-1}
B
7.75 m s−17.75\,\text{m s}^{-1}
C
9.22 m s−19.22\,\text{m s}^{-1}
D
7.30 m s−17.30\,\text{m s}^{-1}
50
A smooth hemispherical dome of radius RR is fixed on a cart that moves on a horizontal road with constant acceleration a=3g4a=\dfrac{3g}{4} (the cart's velocity is increasing in the forward direction). A small block is placed at rest, relative to the cart, at the very top of the dome and gets a negligible nudge toward the rear. It slides along the dome and leaves the surface at a certain point. What is the angle between the radius to that point and the vertical?
Not attempted
A
20.9∘20.9^\circ
B
48.2∘48.2^\circ
C
57.8∘57.8^\circ
D
11.3∘11.3^\circ
51
A small bob of mass m=0.2 kgm=0.2\,\text{kg} is tied to a vertical rod by two light strings, each of length 0.5 m0.5\,\text{m}, fixed at points PP (upper) and QQ (lower) of the rod, 0.6 m0.6\,\text{m} apart. The rod is spun about its own axis and its angular speed ω\omega is increased very slowly from zero, the bob always moving in a horizontal circle. Take g=10 m s−2g=10\,\text{m s}^{-2}. Let T1T_1 and T2T_2 be the tensions in the upper and lower strings. Which graph correctly shows T1T_1 (solid) and T2T_2 (dashed) against ω2\omega^2?
Not attempted
PQrPQ = 0.6 m0.5 m0.5 mωrotating rod
A
Graph 1
ω² (rad²/s²)T (N)020406080100246solid: T₁ (upper)dashed: T₂ (lower)
B
Graph 2
ω² (rad²/s²)T (N)020406080100246solid: T₁ (upper)dashed: T₂ (lower)
C
Graph 3
ω² (rad²/s²)T (N)020406080100246solid: T₁ (upper)dashed: T₂ (lower)
D
Graph 4
ω² (rad²/s²)T (N)020406080100246solid: T₁ (upper)dashed: T₂ (lower)
52
A vertical circular hoop of radius R=0.5 mR=0.5\,\text{m} is spun with constant angular speed ω\omega about its vertical diameter. A small bead is threaded on the hoop; the coefficient of static friction between bead and hoop is μ=1\mu=1. The bead is to remain at rest relative to the hoop at the point where the radius to the bead makes 53∘53^\circ with the upward vertical (that is, above the centre; sin⁡53∘=0.8\sin53^\circ=0.8, cos⁡53∘=0.6\cos53^\circ=0.6). Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the smallest ω\omega for which this is possible?
Not attempted
R53°axis (rotates at ω)bead (rough)
A
10.00 rad s−110.00\,\text{rad s}^{-1}
B
4.33 rad s−14.33\,\text{rad s}^{-1}
C
5.00 rad s−15.00\,\text{rad s}^{-1}
D
13.23 rad s−113.23\,\text{rad s}^{-1}
53
A small block slides on the inner surface of a smooth fixed vertical circular track of radius R=1.0 mR=1.0\,\text{m}. At the lowest point it is given a horizontal speed v0v_0 toward the right. It climbs the right half, leaves the track in the upper right quadrant, flies freely and next touches the track exactly at the point AA, the left end of the horizontal diameter. Take g=10 m s−2g=10\,\text{m s}^{-2}. Find v0v_0.
Not attempted
Av₀Otrack radius R = 1.0 m (full circle, smooth)
A
6.11 m s−16.11\,\text{m s}^{-1}
B
6.78 m s−16.78\,\text{m s}^{-1}
C
6.42 m s−16.42\,\text{m s}^{-1}
D
7.07 m s−17.07\,\text{m s}^{-1}
54
A truck moves along a straight level road with constant forward acceleration a=3g4a=\dfrac{3g}{4}. A light inextensible string of length L=0.8 mL=0.8\,\text{m} is fixed to the ceiling of the truck and carries a small bob. At some instant the string is vertical and the bob is given a horizontal speed uu relative to the truck, along the length of the truck (in either sense). Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the smallest uu for which the bob goes right round a complete circle (string never slack)?
Not attempted
A
6.32 m s−16.32\,\text{m s}^{-1}
B
7.07 m s−17.07\,\text{m s}^{-1}
C
6.48 m s−16.48\,\text{m s}^{-1}
D
6.78 m s−16.78\,\text{m s}^{-1}
55
A car (treated as a particle, no air drag, no engine thrust) is on a circular banked road of radius RR with banking angle θ\theta, where sin⁡θ=0.6\sin\theta=0.6 and cos⁡θ=0.8\cos\theta=0.8 (i.e. tan⁡θ=34\tan\theta=\tfrac34). The coefficient of friction between tyres and road is μ=0.5\mu=0.5 in every direction. At some instant the car is moving along the horizontal circle of radius RR with speed vv (inside the range of speeds for which it can do so) and the brakes are applied so that it slows down along its path while staying on the same circle at that instant. Let at,max⁡a_{t,\max} be the largest possible rate of decrease of its speed at that instant without skidding. Which graph shows at,max⁡/ga_{t,\max}/g against v2/(Rg)v^2/(Rg)?
Not attempted
A
Graph 1
v²/(Rg)aₜ,max/g00.51.01.52.00.20.40.6
B
Graph 2
v²/(Rg)aₜ,max/g00.51.01.52.00.20.40.6
C
Graph 3
v²/(Rg)aₜ,max/g00.51.01.52.00.20.40.6
D
Graph 4
v²/(Rg)aₜ,max/g00.51.01.52.00.20.40.6
56
A small bob of mass mm moves as a conical pendulum on a light string of length L1=1.00 mL_1=1.00\,\text{m}, the string making 37∘37^\circ with the vertical (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8). The string passes freely through a small smooth ring at the point of suspension, and the free end is pulled slowly through the ring until the string below the ring makes 60∘60^\circ with the vertical (sin⁡60∘=32\sin60^\circ=\tfrac{\sqrt3}{2}, cos⁡60∘=12\cos60^\circ=\tfrac12). The pulling is slow enough that the bob moves in an almost exact horizontal circle at every stage. Take g=10 m s−2g=10\,\text{m s}^{-2}. In units of mgL1mgL_1, how much work does the person pulling the string do?
Not attempted
A
0.706 mgL10.706\,mgL_1
B
0.650 mgL10.650\,mgL_1
C
0.538 mgL10.538\,mgL_1
D
0.168 mgL10.168\,mgL_1
57
A car rounds a circular banked bend (banking angle unknown, radius unknown) with uniform speed, and a small bob hangs at rest from the roof by a light string. The coefficient of friction between tyres and road is μ\mu. A passenger notes that when the speed is the largest at which the car does not slide up the bank, the string makes 45∘45^\circ with the vertical, and when the speed is the smallest at which it does not slide down the bank, the string makes 15∘15^\circ with the vertical; in both cases the bob is displaced toward the outside of the bend. What is μ\mu?
Not attempted
A
0.270.27
B
0.580.58
C
0.730.73
D
0.130.13
58
A bob of mass mm on a light inextensible string of length LL hangs from a fixed point OO. The bob is held at rest with the string just straight at 37∘37^\circ above the horizontal through OO (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8) and then released. It first falls vertically with the string slack until the string becomes taut again, then swings. Let θ\theta be the angle of the string from the downward vertical, taken positive on the side of the release point. Immediately after the string becomes taut the motion continues along the circle. Which graph shows the tension TT (in units of mgmg) against θ\theta from that moment until the string next goes slack or the bob first comes to rest?
Not attempted
Orelease (string slack)string taut againθ = 037°
A
Graph 1
θ (deg)T/mg−90−450451234
B
Graph 2
θ (deg)T/mg−90−450451234
C
Graph 3
θ (deg)T/mg−90−450451234
D
Graph 4
θ (deg)T/mg−90−450451234
59
A simple pendulum (bob of mass mm, light inextensible string of length LL) is released from rest with the string horizontal. Let θ\theta be the angle of the string from the downward vertical, and a⃗\vec a the acceleration of the bob while the string is taut during the first swing down to the lowest point. Which of the following are correct?
Not attempted
A
The acceleration a⃗\vec a is exactly horizontal at the instant at which the downward component of the bob's velocity is greatest.
B
At the instant described in the first statement the magnitude of a⃗\vec a is 2 g\sqrt2\,g.
C
When a⃗\vec a makes an angle of 45∘45^\circ with the string, the string itself is at θ=45∘\theta=45^\circ to the vertical.
D
The magnitude of a⃗\vec a is 1.5 g1.5\,g when the string makes 60∘60^\circ with the vertical.
60
A particle of mass mm is given horizontal speed v0=3gRv_0=\sqrt{3gR} at the lowest point of a smooth vertical circular track of radius RR and centre OO, and moves along the inner surface of the track (it can be pushed by the track toward OO but not pulled away from it). Separately, a bead of the same mass is given the same speed at the lowest point of a smooth circular tube of radius RR (the tube can push it either toward or away from OO). Which of the following statements are correct?
Not attempted
A
The particle on the track loses contact when it is at height R/3R/3 above OO, where its speed is gR/3\sqrt{gR/3}.
B
After losing contact, the highest point reached by the particle is at height 4027R\dfrac{40}{27}R above the lowest point of the track.
C
After losing contact, the particle next strikes the track at a point 2327R\dfrac{23}{27}R below OO.
D
The bead in the tube comes momentarily to rest at height 32R\dfrac32R above the lowest point, where the tube pushes it toward OO with a force mg2\dfrac{mg}{2}.
61
A small bob of mass mm hangs from a point OO on the ceiling of a lift by a light inextensible string of length L=1.0 mL=1.0\,\text{m} and moves as a conical pendulum, the string making 37∘37^\circ with the vertical (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8). At the instant the string is at its usual cone, the lift's cable snaps and the lift falls freely (air resistance and rotation of the lift are ignored). Take g=10 m s−2g=10\,\text{m s}^{-2}. As observed from the lift, which of the following are correct?
Not attempted
A
Just after the cable snaps the tension in the string is 0.45 mg0.45\,mg, compared with 1.25 mg1.25\,mg just before.
B
The bob rises to a maximum height of 0.8 m0.8\,\text{m} above OO.
C
The string stays taut and the bob moves in a single plane that contains OO.
D
The bob reaches its highest point 1.48 s1.48\,\text{s} after the cable snaps.
62
A small coin rests on a horizontal turntable at distance r=2.0 mr=2.0\,\text{m} from the axis. The coefficient of static friction is μ=0.5\mu=0.5. Starting from rest, the turntable is made to rotate with constant angular acceleration α=1.5 rad s−2\alpha=1.5\,\text{rad s}^{-2}. Take g=10 m s−2g=10\,\text{m s}^{-2}. Which of the following statements about the motion up to the instant at which the coin begins to slip are correct?
Not attempted
A
The turntable has turned through 23\dfrac23 rad when the coin starts to slip.
B
At the instant of slipping the static friction force makes an angle β\beta with the inward radial direction, with tan⁡β=43\tan\beta=\dfrac43.
C
The coin starts to slip at time t=1αμg/r≈1.05 st=\dfrac{1}{\alpha}\sqrt{\mu g/r}\approx1.05\,\text{s}.
D
At the instant of slipping the friction force has magnitude μmg\mu mg and does no work on the coin at that instant.
63
A rigid funnel in the shape of an inverted cone (apex down) rotates about its vertical axis with constant angular speed ω\omega. Its inner wall is rough, with coefficient of static friction μ\mu with a small block, and makes an angle of 53∘53^\circ with the horizontal (sin⁡53∘=0.8\sin53^\circ=0.8, cos⁡53∘=0.6\cos53^\circ=0.6). The block is placed on the wall at a fixed horizontal distance r=0.5 mr=0.5\,\text{m} from the axis and is required to stay at rest relative to the funnel. Take g=10 m s−2g=10\,\text{m s}^{-2}. Which of the following are correct?
Not attempted
A
For μ=0.25\mu=0.25 the block stays at rest only if 4.0 rad s−1≤ω≤6.9 rad s−14.0\,\text{rad s}^{-1}\le\omega\le6.9\,\text{rad s}^{-1}.
B
For μ=0.75\mu=0.75 the block stays at rest for every ω≥0\omega\ge0 up to some finite maximum.
C
For μ=0.75\mu=0.75 and ω=8 rad s−1\omega=8\,\text{rad s}^{-1} the static friction on the block is directed down the wall and has magnitude 1.12 mg1.12\,mg.
D
For μ=1.2\mu=1.2 the block stays at rest for every value of ω≥0\omega\ge0.
64
A block of mass m=1 kgm=1\,\text{kg} moves in a circle of radius r0=1 mr_0=1\,\text{m} on a smooth horizontal table, held by a light string that passes through a small smooth hole in the table and carries a hanging block of mass M=1 kgM=1\,\text{kg} which hangs at rest. A second block, also of mass 1 kg1\,\text{kg}, is now suddenly attached to the hanging block (so the hanging mass becomes 2 kg2\,\text{kg}) without any jerk. The string stays taut; take g=10 m s−2g=10\,\text{m s}^{-2}. Which of the following are correct?
Not attempted
A
The smallest distance of the circling block from the hole is r0/2r_0/2.
B
At the instant the circling block is closest to the hole, the 2 kg2\,\text{kg} hanging mass accelerates upward at about 6.0 m s−26.0\,\text{m s}^{-2}.
C
At the instant the circling block is closest to the hole, the tension in the string is about 32 N32\,\text{N}.
D
Immediately after the second block is attached, the tension in the string is 43×10 N\dfrac43\times10\,\text{N}.
65
A smooth vertical hoop of radius RR rotates with constant angular speed ω\omega about its vertical diameter, where ω2R=2g\omega^2R=2g. A small bead of mass mm is threaded on the hoop. The bead has a stable relative equilibrium position at polar angle θ0\theta_0 from the lowest point of the hoop. Match each quantity in List-I with its value in List-II (all in terms of gg and RR where needed). List-IList-IIP) Height of the equilibrium position above the lowest point, in units of R1)    1.00Q) Magnitude of the hoop’s reaction on the bead at θ0, in units of mg2)    1.22R) Angular frequency of small oscillations about θ0 along the hoop, in units of g/R3)    0.50S) Exponential growth rate of a tiny displacement from the lowest point, in units of g/R4)    2.005)    1.50\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P) Height of the equilibrium position above the lowest point, in units of } R & \text{1)} \;\; 1.00 \\ \text{Q) Magnitude of the hoop's reaction on the bead at } \theta_0, \text{ in units of } mg & \text{2)} \;\; 1.22 \\ \text{R) Angular frequency of small oscillations about } \theta_0 \text{ along the hoop, in units of } \sqrt{g/R} & \text{3)} \;\; 0.50 \\ \text{S) Exponential growth rate of a tiny displacement from the lowest point, in units of } \sqrt{g/R} & \text{4)} \;\; 2.00 \\ & \text{5)} \;\; 1.50 \end{array}
Not attempted
axis, ωsmooth hoop, radius Rθ₀
A
P-3, Q-1, R-5, S-2
B
P-3, Q-4, R-2, S-1
C
P-3, Q-4, R-5, S-2
D
P-1, Q-4, R-2, S-3
66
A small body of mass mm is given a horizontal speed v0v_0 at the lowest point of a smooth vertical circular path of radius RR with centre OO. Four situations are listed in List-I. In each, the greatest height above the lowest point reached by the body is HH. Match each situation with H/RH/R in List-II. List-IList-IIP) Body tied to a string of length R from O,  v02=2gR1)    1.00Q) Body on the inside of a track (cannot be pulled away), v02=3gR2)    1.33R) Bead in a smooth tube, v02=3gR3)    1.48S) Body tied to a string of length R from O,  v02=4gR4)    1.505)    1.85\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P) Body tied to a string of length } R \text{ from } O, \; v_0^2=2gR & \text{1)} \;\; 1.00 \\ \text{Q) Body on the inside of a track (cannot be pulled away), } v_0^2=3gR & \text{2)} \;\; 1.33 \\ \text{R) Bead in a smooth tube, } v_0^2=3gR & \text{3)} \;\; 1.48 \\ \text{S) Body tied to a string of length } R \text{ from } O, \; v_0^2=4gR & \text{4)} \;\; 1.50 \\ & \text{5)} \;\; 1.85 \end{array}
Not attempted
A
P-1, Q-2, R-4, S-3
B
P-1, Q-4, R-3, S-5
C
P-1, Q-3, R-4, S-5
D
P-3, Q-1, R-4, S-5
67
A particle of mass mm moves on the inside of a smooth fixed cone whose axis is vertical and whose apex is at the bottom; the surface makes an angle of 30∘30^\circ with the axis. At the horizontal level h0=0.40 mh_0=0.40\,\text{m} above the apex the particle is projected horizontally along the surface with speed v0v_0 such that v02=2gh0v_0^2=2gh_0, with g=10 m s−2g=10\,\text{m s}^{-2}. It spirals up to a highest level and then returns. At the highest level reached, what is the normal reaction of the cone on the particle in units of mgmg? Give the value to two decimal places.
Not attempted
Your answer: —
Correct answer: 1.21
68
A bead of mass m=0.50 kgm=0.50\,\text{kg} is threaded on a smooth horizontal rod which rotates in a horizontal plane about a vertical axis through one end of the rod with constant angular speed ω=50 rad s−1\omega=\sqrt{50}\,\text{rad s}^{-1}. The bead is connected to the axis by a light spring of force constant k=50 N m−1k=50\,\text{N m}^{-1} and natural length l0=0.20 ml_0=0.20\,\text{m} lying along the rod. The bead is released from rest relative to the rod with the spring at its natural length. At the instant at which the bead's speed relative to the rod is greatest, what is the magnitude of the total force exerted by the rod on the bead? Take g=10 m s−2g=10\,\text{m s}^{-2}. Give the answer in newtons to two decimal places.
Not attempted
axiskbead msmooth horizontal rod, rotating about the vertical axisspring natural length l₀
Your answer: —
Correct answer: 11.18
69
A thin smooth straight rod is fixed to a vertical axis at its lower end OO and slopes upward and outward, making an angle of 37∘37^\circ with the upward vertical (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8). The whole arrangement rotates with constant angular speed ω=5.0 rad s−1\omega=5.0\,\text{rad s}^{-1} about the axis. A bead of mass 0.50 kg0.50\,\text{kg} threaded on the rod is released from rest relative to the rod at a distance 1.2 m1.2\,\text{m} from OO (measured along the rod) and slides outward. When the bead is 2.0 m2.0\,\text{m} from OO along the rod, what is the magnitude of the total force exerted on it by the rod? Take g=10 m s−2g=10\,\text{m s}^{-2} and give the answer in newtons to two decimal places.
Not attempted
Oaxis (angular speed ω)bead37°
Your answer: —
Correct answer: 17.81
70
A small block of mass mm is released from rest at the top end (at the level of the centre) of a fixed quarter-circular track of radius RR whose concave surface faces up and inward; the block slides along the inside surface down to the lowest point. The coefficient of kinetic friction between block and track is μ=0.30\mu=0.30 and the track is rough throughout. Find the normal reaction of the track on the block just as it reaches the lowest point, in units of mgmg (two decimal places). Use e−0.3π=0.3897e^{-0.3\pi}=0.3897.
Not attempted
start (rest)lowest pointcentreRrough track, μ = 0.30
Your answer: —
Correct answer: 1.69
71
A small puck of mass mm slides in a narrow horizontal circular groove of radius R=5.0 mR=5.0\,\text{m} cut in a flat table (so the puck is in contact with the floor of the groove and with its outer wall). The coefficient of kinetic friction is 0.200.20 between the puck and the floor of the groove and 0.100.10 between the puck and the wall of the groove. The puck is set moving along the groove with initial speed 24 m s−124\,\text{m s}^{-1}. Take g=10 m s−2g=10\,\text{m s}^{-2}. How many complete revolutions (a fractional number, to two decimal places) does the puck make before it stops?
Not attempted
Your answer: —
Correct answer: 1.52
72
A horizontal turntable, initially at rest, is made to rotate about its vertical axis with constant angular acceleration α=2.0 rad s−2\alpha=2.0\,\text{rad s}^{-2}. Two small blocks lie on it along the same radius: block 1 of mass 1.0 kg1.0\,\text{kg} at r1=0.50 mr_1=0.50\,\text{m} and block 2 of mass 2.0 kg2.0\,\text{kg} at r2=1.00 mr_2=1.00\,\text{m}. They are joined by a light inextensible string stretched along that radius (so it is always radial and can only pull). The coefficients of static friction between the blocks and the turntable are μ1=0.60\mu_1=0.60 and μ2=0.30\mu_2=0.30, respectively. Take g=10 m s−2g=10\,\text{m s}^{-2}. Through what angle (in radians, to two decimal places) has the turntable turned when the blocks first begin to slip?
Not attempted
Your answer: —
Correct answer: 1.04
73
A wedge of mass M=3 kgM=3\,\text{kg} rests on a rough horizontal floor (coefficient of static friction μf\mu_f). Its inclined face makes 37∘37^\circ with the horizontal (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8) and the coefficient of friction between a block of mass m=2 kgm=2\,\text{kg} and this face is 0.250.25 (take static equal to kinetic). The block is projected up the face along the line of greatest slope with a speed large enough that it travels a considerable distance up, stops momentarily, and then slides back down. Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the smallest μf\mu_f for which the wedge stays at rest during the ENTIRE motion (upward leg and return leg)?
Not attempted
m37°Mvrough floor (μf)face: μ = 0.25
A
0.2560.256
B
0.3170.317
C
0.2830.283
D
0.1420.142
74
A block of mass m=2 kgm=2\,\text{kg} lies at rest on a rough horizontal floor. A light string attached to it makes an angle of 37∘37^\circ above the horizontal (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8), and its tension is increased steadily from zero as T=ktT=kt with k=10 N s−1k=10\,\text{N s}^{-1}. The coefficients of friction between block and floor are μs=0.6\mu_s=0.6 and μk=0.4\mu_k=0.4; g=10 m s−2g=10\,\text{m s}^{-2}. The string stays taut and at 37∘37^\circ throughout. Which graph correctly shows the magnitude ff of the friction force on the block against time tt from t=0t=0 to t=4 st=4\,\text{s} (after the block leaves the floor the friction force is zero)?
Not attempted
A
Graph 1
t (s)f (N)O12344812
B
Graph 2
t (s)f (N)O12344812
C
Graph 3
t (s)f (N)O12344812
D
Graph 4
t (s)f (N)O12344812
75
A plank of mass M=3 kgM=3\,\text{kg} lies on a rough horizontal floor with coefficient of friction 0.10.1 (static equal to kinetic). A block of mass m=2 kgm=2\,\text{kg} rests on the plank; the coefficient of friction between block and plank is 0.50.5 (static equal to kinetic). A force of magnitude FF pushes the block forward and downward, at 37∘37^\circ below the horizontal (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8), and FF is slowly increased from 00 to 60 N60\,\text{N}. g=10 m s−2g=10\,\text{m s}^{-2}. Which graph shows the acceleration of the PLANK against FF?
Not attempted
A
Graph 1
F (N)a (m s⁻²)O102030405060246810
B
Graph 2
F (N)a (m s⁻²)O102030405060246810
C
Graph 3
F (N)a (m s⁻²)O102030405060246810
D
Graph 4
F (N)a (m s⁻²)O102030405060246810
76
A wedge of mass M=4 kgM=4\,\text{kg} stands on a rough horizontal floor (coefficient of friction 0.10.1, static equal to kinetic). Its inclined face makes 37∘37^\circ with the horizontal (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8) and is rough, with coefficient of friction 0.50.5 (static equal to kinetic) with a block of mass m=1 kgm=1\,\text{kg} resting on it. A horizontal force FF is applied to the vertical side of the wedge, pushing the wedge to the right as shown (the slope descends to the right). The block is not otherwise held. g=10 m s−2g=10\,\text{m s}^{-2}. For which range of FF does the block remain at rest relative to the wedge?
Not attempted
mM37°Ffloor: μf = 0.1face: μ = 0.5
A
9.1 N to 100.0 N9.1\,\text{N}\ \text{to}\ 100.0\,\text{N}
B
13.1 N to 104.0 N13.1\,\text{N}\ \text{to}\ 104.0\,\text{N}
C
15.0 N to 55.0 N15.0\,\text{N}\ \text{to}\ 55.0\,\text{N}
D
14.1 N to 105.0 N14.1\,\text{N}\ \text{to}\ 105.0\,\text{N}
77
A block m1=2 kgm_1=2\,\text{kg} rests on a rough incline of angle 37∘37^\circ (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8); a second block m3=1 kgm_3=1\,\text{kg} rests on top of m1m_1. A light inextensible string tied to m1m_1 runs parallel to the incline over a light smooth pulley at the top and carries a hanging mass M=5 kgM=5\,\text{kg}. The friction coefficient between m1m_1 and the incline is 0.30.3 and between m3m_3 and m1m_1 is 0.90.9 (static equal to kinetic in both). The system is released from rest with the string taut. Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the tension in the string just after release?
Not attempted
m₁m₃M37°incline: μ(m₁) = 0.3, μ(m₃ on m₁) = 0.9
A
34.5 N34.5\,\text{N}
B
28.0 N28.0\,\text{N}
C
33.1 N33.1\,\text{N}
D
26.3 N26.3\,\text{N}
78
A carrier of mass M=3 kgM=3\,\text{kg} has the shape of a right triangle with its top face horizontal. It lies on a rough incline of angle 37∘37^\circ (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8) with coefficient of friction μ2\mu_2 (static equal to kinetic), and is free to slide down the incline. A block of mass m=1 kgm=1\,\text{kg} is placed on the horizontal top face, where the coefficient of friction is μ1=0.5\mu_1=0.5 (static equal to kinetic). The carrier is released from rest and g=10 m s−2g=10\,\text{m s}^{-2}. Find the smallest μ2\mu_2 for which the block does not slide on the carrier.
Not attempted
mM37°top of carrier: μ₁ = 0.5; incline: μ₂a
A
0.1820.182
B
00 (the block never slips, whatever the incline friction)
C
0.2690.269
D
0.1450.145
79
A block of mass m=2 kgm=2\,\text{kg} lies in a long V-shaped groove whose two plane faces meet at a right angle. Looking along the groove (axis horizontal, perpendicular to the page in the figure) one face is inclined at 30∘30^\circ and the other at 60∘60^\circ to the horizontal; the block touches both faces and the coefficient of kinetic friction with each face is 0.40.4. A horizontal force along the axis of the groove drags the block at constant speed along it. g=10 m s−2g=10\,\text{m s}^{-2}. What is the magnitude of this force?
Not attempted
m30°60°cross-section of the groove(its axis is horizontal, into the page)each face has μ = 0.4
A
8.00 N8.00\,\text{N}
B
11.31 N11.31\,\text{N}
C
12.62 N12.62\,\text{N}
D
10.93 N10.93\,\text{N}
80
A flat rough plane is inclined at θ=37∘\theta=37^\circ to the horizontal (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8) and the coefficient of kinetic friction between a small block and the plane is exactly μ=tan⁡θ=0.75\mu=\tan\theta=0.75. The block is projected with speed v0=10 m s−1v_0=10\,\text{m s}^{-1} along the plane in the horizontal direction (along a contour line), as sketched in plan view. Friction is the only force besides gravity and the normal reaction. The block never comes to rest. At the instant when its velocity makes an angle whose sine is 0.60.6 with the original horizontal direction, what is its speed?
Not attempted
lower edge of the planev₀down theslopeplan view of the inclined plane
A
8.00 m s−18.00\,\text{m s}^{-1}
B
6.25 m s−16.25\,\text{m s}^{-1}
C
5.00 m s−15.00\,\text{m s}^{-1}
D
5.56 m s−15.56\,\text{m s}^{-1}
81
Three blocks are stacked on a rough horizontal floor: CC (3 kg3\,\text{kg}) on the floor, BB (2 kg2\,\text{kg}) on CC and AA (1 kg1\,\text{kg}) on BB. The coefficients of friction (static equal to kinetic) are 0.10.1 between CC and the floor, 0.40.4 between BB and CC, and 0.10.1 between AA and BB. A horizontal force F=ktF=kt with k=4 N s−1k=4\,\text{N s}^{-1} acts on BB only, starting at t=0t=0 from rest; g=10 m s−2g=10\,\text{m s}^{-2}. Which graph shows the acceleration of block BB against time for 0≤t≤6 s0\le t\le6\,\text{s}?
Not attempted
A
Graph 1
t (s)a (m s⁻²)O123456123456
B
Graph 2
t (s)a (m s⁻²)O123456123456
C
Graph 3
t (s)a (m s⁻²)O123456123456
D
Graph 4
t (s)a (m s⁻²)O123456123456
82
A block of mass mm is confined between smooth vertical guides so that it can only move vertically. It rests on the inclined face of a wedge of negligible mass whose face makes an angle θ\theta with the horizontal, sin⁡θ=0.6\sin\theta=0.6 and cos⁡θ=0.8\cos\theta=0.8; the wedge lies on a horizontal floor. The coefficient of friction (static) has the same value μ\mu between the wedge and the block as between the wedge and the floor. Nobody pushes the wedge. What is the smallest μ\mu for which the wedge stays in equilibrium (does not get squeezed out) for every value of mm?
Not attempted
mmassless wedgeθsmooth vertical guides on the block;same μ at both rubbing surfaces
A
0.3750.375
B
0.7500.750
C
0.3330.333
D
0.5000.500
83
Block m1=3 kgm_1=3\,\text{kg} rests on a rough horizontal table (coefficient of friction 0.20.2) and carries a block m3=1 kgm_3=1\,\text{kg} on top; the coefficient of friction between m3m_3 and m1m_1 is 0.50.5 (static equal to kinetic for both pairs). A light string tied to m1m_1 runs horizontally over a light smooth pulley at the table edge and carries a hanging block of mass MM; the system is released from rest with the string taut. Take g=10 m s−2g=10\,\text{m s}^{-2}. Which of the following statements is/are correct?
Not attempted
m₁m₃Mm₁–table μ = 0.2; m₃–m₁ μ = 0.5
A
For M=3 kgM=3\,\text{kg} the tension in the string is about 20.6 N20.6\,\text{N}.
B
The block m3m_3 starts slipping on m1m_1 as soon as MM exceeds 4.6 kg4.6\,\text{kg}.
C
For M=8 kgM=8\,\text{kg}, m3m_3 accelerates at 5.0 m s−25.0\,\text{m s}^{-2} while m1m_1 accelerates at about 6.09 m s−26.09\,\text{m s}^{-2}, and the friction exerted by m3m_3 on m1m_1 is 5 N5\,\text{N} directed opposite to the motion of m1m_1.
D
For M=0.75 kgM=0.75\,\text{kg} the system stays at rest, the table exerts a friction force of 7.5 N7.5\,\text{N} on m1m_1, and no friction acts between m1m_1 and m3m_3.
84
A cart moves forward on a horizontal road with constant acceleration aa and pushes two blocks that are in front of it. Block BB (2 kg2\,\text{kg}) touches the vertical front face of the cart and block AA (1 kg1\,\text{kg}) touches the front of BB; the blocks are in the air and are supported only by friction (nothing underneath, no contact with anything else), and the cart face is large enough that both blocks stay on it. The coefficient of friction (static equal to kinetic) is 0.50.5 between the cart and BB and 0.40.4 between BB and AA; g=10 m s−2g=10\,\text{m s}^{-2}. The blocks start at rest relative to the cart. Which of the following statements is/are correct?
Not attempted
BAablocks in the air, held by friction onlycart–B: μ = 0.5; B–A: μ = 0.4
A
At a=15 m s−2a=15\,\text{m s}^{-2} block BB still does not slip, and block AA slides down at 4.0 m s−24.0\,\text{m s}^{-2}.
B
The two blocks can both be kept from sliding only if the cart's acceleration is at least 25 m s−225\,\text{m s}^{-2}, and this limiting value would be the same for any masses of the two blocks.
C
At a=15 m s−2a=15\,\text{m s}^{-2} block BB slides down with an acceleration of 5.5 m s−25.5\,\text{m s}^{-2}.
D
At a=22 m s−2a=22\,\text{m s}^{-2} block AA slides down with an acceleration of 1.2 m s−21.2\,\text{m s}^{-2} while block BB does not move relative to the cart; the friction force the cart exerts on BB is then 28.8 N28.8\,\text{N}.
85
A block of mass m=4 kgm=4\,\text{kg} lies on a rough horizontal floor; the coefficient of friction (static equal to kinetic) is μ=0.5\mu=0.5. g=10 m s−2g=10\,\text{m s}^{-2}. A force of magnitude FF can be applied to the block either as a pull (directed forward and upward) or as a push (directed forward and downward), at a chosen angle to the horizontal. Which of the following statements is/are correct?
Not attempted
A
The smallest force with which the block can be set moving by pulling is about 17.9 N17.9\,\text{N}, and at that force the normal reaction of the floor on the block is 32 N32\,\text{N}.
B
If the block is pushed by a force directed 37∘37^\circ below the horizontal (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8), it starts to move only when the force reaches 40 N40\,\text{N}.
C
If the block is pushed by a force directed below the horizontal at an angle α\alpha with tan⁡α=2.5\tan\alpha=2.5 (sin⁡α≈0.928\sin\alpha\approx0.928, cos⁡α≈0.371\cos\alpha\approx0.371), no magnitude of the force, however large, can set the block in motion.
D
A pull of 20 N20\,\text{N} directed 37∘37^\circ above the horizontal gives the block an acceleration of 0.5 m s−20.5\,\text{m s}^{-2}.
86
Two small blocks AA (2 kg2\,\text{kg}, coefficient of friction 0.20.2 with the incline) and BB (3 kg3\,\text{kg}, coefficient 0.60.6) are placed on the same rough incline of angle θ\theta one directly behind the other along the line of greatest slope, initially touching and at rest (static equal to kinetic). Use sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8 and g=10 m s−2g=10\,\text{m s}^{-2}; for the second angle tan⁡θ=0.5\tan\theta=0.5, i.e. sin⁡θ=1/5\sin\theta=1/\sqrt5, cos⁡θ=2/5\cos\theta=2/\sqrt5. They are released. Which of the following statements is/are correct?
Not attempted
A
At 37∘37^\circ, with AA placed higher up the incline and touching BB, the contact force between them is 7.0 N7.0\,\text{N}.
B
On an incline with tan⁡θ=0.5\tan\theta=0.5 the two blocks stay at rest whichever of them is placed higher up, because BB alone could not slide at that angle.
C
At 37∘37^\circ, with BB placed higher up the incline and touching AA, the two blocks move down together with an acceleration of 2.48 m s−22.48\,\text{m s}^{-2}.
D
On an incline with tan⁡θ=0.5\tan\theta=0.5, if BB is placed higher up and touches AA, then BB stays at rest while AA slides away down the slope with an acceleration of about 2.7 m s−22.7\,\text{m s}^{-2}.
87
Block AA (2 kg2\,\text{kg}) rests on block BB (3 kg3\,\text{kg}), which rests on a rough floor. A light inextensible string tied to AA runs horizontally to a light smooth pulley fixed to a wall and returns horizontally to the end of BB nearest the wall, as sketched. A horizontal force FF pulls BB directly away from the wall. The coefficient of friction (static equal to kinetic) is 0.30.3 between AA and BB and 0.20.2 between BB and the floor; g=10 m s−2g=10\,\text{m s}^{-2}. The system starts from rest. Which of the following statements is/are correct?
Not attempted
B (3 kg)A (2 kg)FB–floor μ = 0.2; A–B μ = 0.3
A
The smallest force FF that makes BB start moving is 16 N16\,\text{N}.
B
For F=40 NF=40\,\text{N} the two blocks move with equal accelerations of 3.6 m s−23.6\,\text{m s}^{-2} in opposite directions and the tension in the string is 13.2 N13.2\,\text{N}.
C
If the string is cut while F=40 NF=40\,\text{N} is applied, AA and BB move together with an acceleration of 6 m s−26\,\text{m s}^{-2}.
D
If the string is cut while F=40 NF=40\,\text{N} is applied, AA accelerates at 3 m s−23\,\text{m s}^{-2} in the direction of FF and BB at 8 m s−28\,\text{m s}^{-2}.
88
A truck on a level road brakes with a constant deceleration ata_t. On its horizontal bed rests crate BB (4 kg4\,\text{kg}), and crate AA (1 kg1\,\text{kg}) rests on top of BB. Before braking all three move together. The coefficient of friction (static equal to kinetic) is 0.50.5 between BB and the bed and 0.30.3 between AA and BB; g=10 m s−2g=10\,\text{m s}^{-2}. The crates are far from the edges of the bed. Which of the following statements is/are correct?
Not attempted
A
Crate BB cannot remain at rest relative to the truck once the truck's deceleration exceeds μbed g=5 m s−2\mu_{bed}\,g=5\,\text{m s}^{-2}.
B
At a truck deceleration of 4 m s−24\,\text{m s}^{-2}, crate AA slides forward on BB while BB stays at rest relative to the truck; the acceleration of AA relative to BB is 1 m s−21\,\text{m s}^{-2} (forward).
C
At a truck deceleration of 6 m s−26\,\text{m s}^{-2}, crate BB slides forward relative to the truck with an acceleration of 0.5 m s−20.5\,\text{m s}^{-2} and crate AA with 3 m s−23\,\text{m s}^{-2} (both relative to the truck).
D
At a truck deceleration of 5 m s−25\,\text{m s}^{-2}, crate BB does not slip on the bed, and the friction force the bed exerts on BB is 23 N23\,\text{N} (backward).
89
A block of mass 2 kg2\,\text{kg} rests on a plank of mass 3 kg3\,\text{kg} lying on a horizontal floor. The coefficient of friction (static equal to kinetic) between block and plank is 0.50.5; the floor-plank value is given in each row. A horizontal force acts as stated, applied from rest and kept constant; g=10 m s−2g=10\,\text{m s}^{-2}. Match each situation in List-I with the resulting acceleration of the BLOCK in List-II. List-IList-IIP) Force 12 N on the block, floor μ=0.11) 0Q) Force 36 N on the plank, floor μ=0.12) 1.0 m s−2R) Force 12 N on the block, floor μ=0.33) 1.4 m s−2S) Force 15 N on the block, floor μ=0.254) 2.5 m s−25) 5.0 m s−2\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \text{ Force } 12\,\text{N} \text{ on the block, floor } \mu=0.1 & \text{1)} \ 0 \\ \text{Q)} \text{ Force } 36\,\text{N} \text{ on the plank, floor } \mu=0.1 & \text{2)} \ 1.0\,\text{m s}^{-2} \\ \text{R)} \text{ Force } 12\,\text{N} \text{ on the block, floor } \mu=0.3 & \text{3)} \ 1.4\,\text{m s}^{-2} \\ \text{S)} \text{ Force } 15\,\text{N} \text{ on the block, floor } \mu=0.25 & \text{4)} \ 2.5\,\text{m s}^{-2} \\ & \text{5)} \ 5.0\,\text{m s}^{-2} \end{array}
Not attempted
A
P-2, Q-5, R-1, S-4
B
P-3, Q-5, R-2, S-4
C
P-3, Q-4, R-5, S-1
D
P-5, Q-3, R-2, S-4
90
A block of mass 2 kg2\,\text{kg} rests on a rough plane inclined at 37∘37^\circ to the horizontal (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8); the coefficients of friction are μs=0.9\mu_s=0.9 and μk=0.8\mu_k=0.8. g=10 m s−2g=10\,\text{m s}^{-2}. A force HH lying in the plane of the incline and directed horizontally (along a contour line, perpendicular to the line of greatest slope) is applied to the block and increased slowly from zero to the value stated in List-I. Match each value of HH with the magnitude of the friction force acting on the block at the moment HH has reached that value. List-IList-IIP) H=01) 12.0 NQ) H=5 N2) 12.8 NR) H=7 N3) 13.0 NS) H=10 N4) 13.9 N5) 14.4 N\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \ H=0 & \text{1)} \ 12.0\,\text{N} \\ \text{Q)} \ H=5\,\text{N} & \text{2)} \ 12.8\,\text{N} \\ \text{R)} \ H=7\,\text{N} & \text{3)} \ 13.0\,\text{N} \\ \text{S)} \ H=10\,\text{N} & \text{4)} \ 13.9\,\text{N} \\ & \text{5)} \ 14.4\,\text{N} \end{array}
Not attempted
A
P-1, Q-3, R-4, S-2
B
P-1, Q-3, R-5, S-4
C
P-3, Q-1, R-4, S-2
D
P-1, Q-4, R-2, S-5
91
A long straight horizontal conveyor belt moves at a steady speed u=2.5 m s−1u=2.5\,\text{m s}^{-1} along its length. A small block is slid onto the belt from the side, with velocity v0=6.0 m s−1v_0=6.0\,\text{m s}^{-1} (measured relative to the ground) perpendicular to the direction of the belt, as seen in the plan view. The coefficient of kinetic friction between block and belt is 0.350.35 and g=10 m s−2g=10\,\text{m s}^{-2}. The belt is wide enough that the block never reaches its edges, and friction is the only horizontal force. Find the minimum speed (relative to the ground) that the block has during the whole motion, in m s−1\text{m s}^{-1}, correct to two decimal places.
Not attempted
belt (speed u)v₀blockplan view
Your answer: —
Correct answer: 2.31
92
A uniform rope of length 1.5 m1.5\,\text{m} and mass per unit length 2.0 kg m−12.0\,\text{kg m}^{-1} lies along a rough horizontal table, perpendicular to a smooth straight edge, with a block of mass 1.5 kg1.5\,\text{kg} tied to its rear end; the front part of the rope hangs vertically over the edge. The coefficients of friction of the rope and of the block with the table are μs=0.5\mu_s=0.5 and μk=0.35\mu_k=0.35, and g=10 m s−2g=10\,\text{m s}^{-2}. The hanging length is set to the smallest value for which the rope can just begin to slide, and the system then slides from rest. Treat the rope as flexible, inextensible, with a smooth turn over the edge. What is the speed of the system, in m s−1\text{m s}^{-1} (two decimal places), at the instant the block reaches the edge of the table?
Not attempted
mrope on the tablehanging part, length xrope: 2 kg/m, total length 1.5 mrope and block with table: μs = 0.5, μk = 0.35
Your answer: —
Correct answer: 2.21
93
A row of identical books, each of mass 2 kg2\,\text{kg}, is held vertically between two parallel vertical plates pressed against the two ends of the row by horizontal forces of 100 N100\,\text{N} each (the plates themselves are massless and held fixed so that they do not move). The coefficient of friction between a plate and a book is 0.650.65, and between two neighbouring books it is 0.350.35 (static in both cases). The books rest on nothing else, and g=10 m s−2g=10\,\text{m s}^{-2}. What is the largest number of books that can be held in this way without slipping?
Not attempted
Your answer: —
Correct answer: 5
94
A small block of mass 1 kg1\,\text{kg} lies on a plank of mass 4 kg4\,\text{kg} that rests on a rough horizontal floor. Friction coefficients (static equal to kinetic): 0.20.2 between block and plank, 0.050.05 between plank and floor. g=10 m s−2g=10\,\text{m s}^{-2}. At t=0t=0 both are at rest and a horizontal force of 14 N14\,\text{N} in the forward direction is applied to the plank. At t=2 st=2\,\text{s} the force is suddenly reversed to 9 N9\,\text{N} in the backward direction and then stays constant for the rest of the motion. The block is a point-like body that may be placed anywhere on the plank at t=0t=0. What is the minimum length of the plank, in metres (two decimal places), for which the block never leaves the plank?
Not attempted
Your answer: —
Correct answer: 0.87
95
A block of mass 2 kg2\,\text{kg} rests on a rough incline of angle 37∘37^\circ (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8) with coefficients of friction μs=0.8\mu_s=0.8 and μk=0.6\mu_k=0.6. Starting at t=0t=0, a horizontal force F=ktF=kt with k=10 N s−1k=10\,\text{N s}^{-1} pushes the block toward the incline, in the vertical plane containing the line of greatest slope, as shown. g=10 m s−2g=10\,\text{m s}^{-2}. What is the speed of the block at t=10 st=10\,\text{s}, in m s−1\text{m s}^{-1} (two decimal places)? The block stays on the incline and the incline is long enough.
Not attempted
m37°F = ktμs = 0.8, μk = 0.6
Your answer: —
Correct answer: 19.63
96
A horizontal turntable rotates about a vertical axis through its centre with an angular speed that is increased very slowly from zero. Two small blocks sit on it on the same radius: AA (1 kg1\,\text{kg}) at 0.20 m0.20\,\text{m} from the axis and BB (2 kg2\,\text{kg}) at 0.50 m0.50\,\text{m}. They are joined by a light inextensible string lying along the radius, just taut at the start. The coefficient of static friction with the turntable is 0.50.5 for AA and 0.350.35 for BB; g=10 m s−2g=10\,\text{m s}^{-2}. At what angular speed, in rad s−1\text{rad s}^{-1} (two decimal places), do the blocks first start to slide?
Not attempted
ABplan view of the turntable (axis at the dot)rA = 0.2 m, rB = 0.5 m; string along the radius
Your answer: —
Correct answer: 3.16
97
A rigid wedge of angle θ\theta with sin⁡θ=0.6\sin\theta=0.6, cos⁡θ=0.8\cos\theta=0.8 is fixed on a trolley so that its inclined face rises toward the rear of the trolley. The trolley accelerates forward with a constant acceleration aa. Two blocks are stacked on the face: a lower block of mass 2 kg2\,\text{kg} with friction coefficient μ1=0.90\mu_1=0.90 against the face, and an upper block of mass 1 kg1\,\text{kg} with friction coefficient μ2=0.50\mu_2=0.50 against the lower block (static equal to kinetic in both cases). Everything is at rest relative to the trolley at the start of each situation below. Take g=10 m s−2g=10\,\text{m s}^{-2}. Which of the following statements are correct?
Not attempted
A
Both blocks remain at rest relative to the trolley for every constant aa in the range 1.82 m s−2≤a≤20.0 m s−21.82\,\text{m s}^{-2}\le a\le20.0\,\text{m s}^{-2} (and only in this range), whatever the masses of the two blocks.
B
If the trolley's acceleration is zero, the upper block slides down the lower block while the lower block stays at rest on the face.
C
If a=25 m s−2a=25\,\text{m s}^{-2}, the upper block slides up the face of the lower block, while the lower block stays at rest relative to the trolley.
D
If μ1\mu_1 were reduced to 0.300.30 (everything else unchanged), the range of aa for which both blocks stay at rest would be unchanged.
98
A cart on a horizontal road accelerates forward with a constant acceleration a=7.5 m s−2a=7.5\,\text{m s}^{-2}. A pendulum of length L=0.80 mL=0.80\,\text{m}, whose bob has mass m=0.20 kgm=0.20\,\text{kg} on a light inextensible string, hangs from a point on the cart's ceiling. The bob is held with the string taut and horizontal, the bob lying on the front side of the pivot, at rest relative to the cart, and is then released. Whenever the string becomes taut after being slack, the component of the bob's velocity along the string is lost (inelastic jerk), the perpendicular component being retained. Take g=10 m s−2g=10\,\text{m s}^{-2}. Ignoring the impulsive tension of that jerk, find the largest tension in the string (in newtons) during the motion that follows.
Not attempted
bob (released here)Lapivot
Your answer: —
Correct answer: 8.34
99
A long hall-shaped compartment of height 2.5 m2.5\,\text{m} is fixed on a truck that accelerates forward with a constant a=7.5 m s−2a=7.5\,\text{m s}^{-2}. A ball is launched from the rear-bottom corner of the compartment, in the vertical plane along the direction of motion, with speed uu relative to the truck. It must reach the front-top corner, which is 6.0 m6.0\,\text{m} ahead of the launch point horizontally and 2.5 m2.5\,\text{m} above it. The ball must not strike the ceiling or any wall before that: touching the ceiling exactly at the corner is allowed. Take g=10 m s−2g=10\,\text{m s}^{-2} and neglect the size of the ball. Find the minimum possible uu (in m s−1\text{m s}^{-1}) to two decimal places.
Not attempted
launchtarget6.0 m2.5 ma
Your answer: —
Correct answer: 13.19
100
A trolley is driven so that its acceleration is exactly a=3.0 m s−2a=3.0\,\text{m s}^{-2} forward from rest. On its horizontal floor lie two blocks joined by a light spring of force constant k=80 N m−1k=80\,\text{N m}^{-1} (natural length at the start, blocks at rest relative to the trolley). Block AA (mA=1.0 kgm_A=1.0\,\text{kg}) is behind block BB (mB=2.0 kgm_B=2.0\,\text{kg}). The friction coefficients between the floor and the blocks are μA=0.10\mu_A=0.10 and μB=0.45\mu_B=0.45 (static equal to kinetic), the motion is along a line, and g=10 m s−2g=10\,\text{m s}^{-2}. Find the maximum extension of the spring during the motion, in centimetres, to two decimal places.
Not attempted
ABak
Your answer: —
Correct answer: 4.87
101
A smooth double wedge is fixed on a trolley: its left face rises at 37∘37^\circ (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8) to a rounded smooth apex, and its right face descends from the apex at 53∘53^\circ (sin⁡53∘=0.8\sin53^\circ=0.8, cos⁡53∘=0.6\cos53^\circ=0.6); both faces are long. A uniform flexible chain of length 1.6 m1.6\,\text{m} is draped over the apex, lying along both faces. The trolley moves on a straight horizontal road with a constant acceleration of 10 m s−210\,\text{m s}^{-2} directed from the left face toward the right face. The chain is released at rest relative to the trolley with 0.35 m0.35\,\text{m} of it on the left face (the rest on the right face). Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the speed of the chain relative to the trolley at the instant its last element passes the apex?
Not attempted
37°53°a
A
2.13 m s−12.13\,\text{m s}^{-1}
B
2.98 m s−12.98\,\text{m s}^{-1}
C
4.40 m s−14.40\,\text{m s}^{-1}
D
2.53 m s−12.53\,\text{m s}^{-1}
102
A wedge of mass 3.0 kg3.0\,\text{kg} has a horizontal upper face and slides down a fixed smooth incline of angle θ\theta (sin⁡θ=0.6\sin\theta=0.6, cos⁡θ=0.8\cos\theta=0.8). A block of mass 1.0 kg1.0\,\text{kg} rests on the horizontal upper face, the coefficient of friction between them being 0.500.50 (static equal to kinetic). The system is released from rest. Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the magnitude of the acceleration of the block relative to the wedge?
Not attempted
A
2.05 m s−22.05\,\text{m s}^{-2}
B
1.60 m s−21.60\,\text{m s}^{-2}
C
5.00 m s−25.00\,\text{m s}^{-2}
D
5.13 m s−25.13\,\text{m s}^{-2}
103
A trolley of mass M=2.0 kgM=2.0\,\text{kg} (including a light, smooth pulley fixed at the end of a boom that sticks out behind its rear end) is pulled forward on a smooth horizontal floor by a horizontal force FF so that it moves with a constant acceleration a=7.5 m s−2a=7.5\,\text{m s}^{-2}. A light inextensible string over the pulley carries masses m1=3.0 kgm_1=3.0\,\text{kg} and m2=1.0 kgm_2=1.0\,\text{kg} on its two sides; both hang freely in air behind the trolley. At the instant considered the system is in the steady state in which both strings are straight with fixed directions and the masses move along them (the size of the pulley is negligible). Take g=10 m s−2g=10\,\text{m s}^{-2}. What is FF at the instant considered?
Not attempted
31aMmasses in kg
A
45.0 N45.0\,\text{N}
B
37.5 N37.5\,\text{N}
C
39.0 N39.0\,\text{N}
D
35.0 N35.0\,\text{N}
104
A lift is descending with an increasing speed, its downward acceleration being 4.0 m s−24.0\,\text{m s}^{-2}. Inside, a light spring S1S_1 hangs from the ceiling and carries a block m1=1 kgm_1=1\,\text{kg}; from m1m_1 a light string hangs and carries m2=2 kgm_2=2\,\text{kg}; from m2m_2 a second light spring S2S_2 hangs and carries m3=3 kgm_3=3\,\text{kg}. Everything is at rest relative to the lift. At one instant the lift's supporting cable snaps (so the lift is thereafter in free fall) and, at that very same instant, the string between m1m_1 and m2m_2 breaks. Take g=10 m s−2g=10\,\text{m s}^{-2}. What are the accelerations of m1m_1, m2m_2, m3m_3 relative to the lift immediately afterwards?
Not attempted
A
30 m s−230\,\text{m s}^{-2} up, 15 m s−215\,\text{m s}^{-2} down, 00
B
50 m s−250\,\text{m s}^{-2} up, 25 m s−225\,\text{m s}^{-2} down, 00
C
26 m s−226\,\text{m s}^{-2} up, 19 m s−219\,\text{m s}^{-2} down, 4 m s−24\,\text{m s}^{-2} down
D
36 m s−236\,\text{m s}^{-2} up, 9 m s−29\,\text{m s}^{-2} down, 6 m s−26\,\text{m s}^{-2} up
105
A smooth ramp of length 2.5 m2.5\,\text{m}, inclined at 30∘30^\circ to the horizontal, is fixed on the floor of a lift. The lift starts from rest at t=0t=0 and moves upward with an acceleration that grows linearly, aL=kta_L=kt with k=6.0 m s−3k=6.0\,\text{m s}^{-3}. A small block is released at the top of the ramp, at rest relative to it, at t=0t=0. Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the speed of the block relative to the ground at the instant it reaches the bottom of the ramp?
Not attempted
A
8.40 m s−18.40\,\text{m s}^{-1}
B
5.09 m s−15.09\,\text{m s}^{-1}
C
5.86 m s−15.86\,\text{m s}^{-1}
D
7.46 m s−17.46\,\text{m s}^{-1}
106
A plank of mass M=40 kgM=40\,\text{kg} lies on a smooth horizontal floor and is attached to a fixed wall by a light horizontal spring of force constant k=500 N m−1k=500\,\text{N m}^{-1} (at its natural length, plank at rest). A man of mass m=60 kgm=60\,\text{kg} standing on the plank starts to run along it, away from the wall, with a constant acceleration b=0.80 m s−2b=0.80\,\text{m s}^{-2} relative to the plank, and keeps this relative acceleration until the plank first comes momentarily to rest again. Treat the man as a particle whose shoes grip the plank with a coefficient of static friction μ\mu. Take g=10 m s−2g=10\,\text{m s}^{-2}. What is the minimum μ\mu that lets him do this?
Not attempted
A
0.0320.032
B
0.0800.080
C
0.1280.128
D
0.3200.320
107
A pendulum, a bob of mass 0.50 kg0.50\,\text{kg} on a light inextensible string of length ℓ\ell, hangs at rest from the ceiling of a cart that is at rest on a horizontal road. At t=0t=0 the cart starts to accelerate forward with a constant acceleration equal to gg, and it keeps this acceleration only until the instant at which the string first becomes horizontal (the bob then being behind the pivot); from that instant the cart moves with constant velocity. Take g=10 m s−2g=10\,\text{m s}^{-2} and 2=1.414\sqrt2=1.414. What is the largest tension in the string at any time during the whole motion?
Not attempted
bob (at rest at the start)a = gpivot
A
11.2 N11.2\,\text{N}
B
21.2 N21.2\,\text{N}
C
10.0 N10.0\,\text{N}
D
15.0 N15.0\,\text{N}
108
A block of mass 1.0 kg1.0\,\text{kg} hangs at rest from a light spring (k=100 N m−1k=100\,\text{N m}^{-1}) attached to the ceiling of a lift at rest; the spring force is read on a fast recorder. At t=0t=0 the lift begins to accelerate upward at 2.0 m s−22.0\,\text{m s}^{-2}, and it keeps this acceleration for exactly a time τ=πm/k\tau=\pi\sqrt{m/k} (half the natural period of the block on the spring), after which the lift moves with constant velocity. Take g=10 m s−2g=10\,\text{m s}^{-2} and no damping. Which graph correctly shows the spring force FF against time for 0<t<4τ0<t<4\tau?
Not attempted
A
Graph 1
t (s)F (N)τ3τ61014
B
Graph 2
t (s)F (N)τ3τ61014
C
Graph 3
t (s)F (N)τ3τ61014
D
Graph 4
t (s)F (N)τ3τ61014
109
A trolley (a box with a small pendulum hung from the centre of its roof, parallel to the floor of the box) slides freely down a fixed incline of angle 53∘53^\circ (sin⁡53∘=0.8\sin53^\circ=0.8, cos⁡53∘=0.6\cos53^\circ=0.6), its floor in contact with the incline. The kinetic friction coefficient between the trolley and the incline is 0.500.50, and the trolley is moving down the incline with a steady, constant acceleration. The bob has come to rest relative to the trolley. Which figure shows the equilibrium direction of the string? (In each figure the incline rises toward the right; only the direction of the string matters.)
Not attempted
A
String 1
53°
B
String 2
53°
C
String 3
53°
D
String 4
53°
110
A lift is moving upward with an upward acceleration 5.0 m s−25.0\,\text{m s}^{-2}. At t=0t=0 a ball is thrown horizontally with speed 3.0 m s−13.0\,\text{m s}^{-1} (relative to the lift) from a point 4.0 m4.0\,\text{m} above the lift floor (the lift is tall and wide). At t=0.30 st=0.30\,\text{s} the cable snaps and the lift thereafter falls freely. Neglect air resistance; g=10 m s−2g=10\,\text{m s}^{-2}. Which figure shows the path of the ball as seen by an observer inside the lift? In each figure the horizontal axis (xx) and the vertical axis (yy, height above the floor) are in metres and the origin is directly below the point of projection on the floor.
Not attempted
A
Path 1
x (m)y (m)O123424
B
Path 2
x (m)y (m)O123424
C
Path 3
x (m)y (m)O123424
D
Path 4
x (m)y (m)O123424
111
A trolley of mass 3.0 kg3.0\,\text{kg} stands on a smooth horizontal table. A light string tied to its front passes over a light smooth pulley at the table's edge and carries a hanging mass of 4.0 kg4.0\,\text{kg}. A small bob of mass 1.0 kg1.0\,\text{kg} hangs by a light string from the ceiling of the trolley, and a block of mass 2.0 kg2.0\,\text{kg} rests on the floor of the trolley with friction coefficient 0.300.30 (static equal to kinetic). The trolley is released from rest, the bob has settled at a steady inclination, and the block slides along the floor. Take g=10 m s−2g=10\,\text{m s}^{-2}. Which of the following statements are correct once the motion is steady (the hanging mass is still falling and the block is still on the floor)?
Not attempted
A
The acceleration of the trolley is 4.25 m s−24.25\,\text{m s}^{-2}.
B
The string holding the bob makes an angle with the vertical whose tangent is 0.400.40.
C
The tension in the string over the pulley is 24 N24\,\text{N}.
D
The block slides backward relative to the trolley with an acceleration of 1.25 m s−21.25\,\text{m s}^{-2}.
112
A block of mass 0.50 kg0.50\,\text{kg} lies on the smooth floor of a trolley, joined to the trolley's rear wall by a light horizontal spring of force constant 50 N m−150\,\text{N m}^{-1}, which is at its natural length. The block and trolley are at rest on a horizontal road when, at t=0t=0, the trolley begins to move forward with a constant acceleration 2.0 m s−22.0\,\text{m s}^{-2}; the trolley floor is smooth, so only the spring acts on the block horizontally. Which of the following statements are correct?
Not attempted
A
The maximum compression of the spring is 2.0 cm2.0\,\text{cm}.
B
The maximum speed of the block relative to the trolley is 0.40 m s−10.40\,\text{m s}^{-1}.
C
At the instant of maximum compression, the kinetic energy of the block in the ground frame is equal to the energy stored in the spring.
D
The first time the spring returns to its natural length (after t=0t=0), the speed of the block relative to the ground is 1.26 m s−11.26\,\text{m s}^{-1}.
113
A smooth wedge, whose inclined face makes an angle θ\theta with the horizontal (sin⁡θ=0.6\sin\theta=0.6, cos⁡θ=0.8\cos\theta=0.8) and rises toward the front, is fixed on a trolley. A block of mass 0.50 kg0.50\,\text{kg} lies on the face, connected to the top edge of the face by a light spring of force constant 100 N m−1100\,\text{N m}^{-1} and natural length 0.30 m0.30\,\text{m}, the spring being free to turn about its upper end and lying in the vertical plane through the face's line of greatest slope. The trolley accelerates forward with constant acceleration aa, and the block is at rest relative to the trolley. Take g=10 m s−2g=10\,\text{m s}^{-2} and 1000=31.6\sqrt{1000}=31.6. Which of the following statements are correct?
Not attempted
37°am
A
For a=0a=0 the spring is stretched by 3.0 cm3.0\,\text{cm}.
B
For a=10 m s−2a=10\,\text{m s}^{-2} the block still presses on the face, with a normal force of 1.0 N1.0\,\text{N}, and the spring is stretched by 7.0 cm7.0\,\text{cm}.
C
For a=30 m s−2a=30\,\text{m s}^{-2} the spring is stretched by 15.0 cm15.0\,\text{cm}.
D
For a=30 m s−2a=30\,\text{m s}^{-2} the block is not in contact with the face, and the spring makes an angle of about 18.4∘18.4^\circ with the face.
114
A lift is moving upward at 2.0 m s−12.0\,\text{m s}^{-1} and, at that instant (t=0t=0), its upward acceleration becomes 5.0 m s−25.0\,\text{m s}^{-2}, which it keeps for a long time. At t=0t=0 a small ball is released from rest relative to the lift, at a height of 0.60 m0.60\,\text{m} above the lift floor. The collisions of the ball with the floor are such that the speed of separation is 0.500.50 times the speed of approach (both measured relative to the floor). Take g=10 m s−2g=10\,\text{m s}^{-2}. Which of the following statements are correct?
Not attempted
A
The ball first strikes the floor at t=0.283 st=0.283\,\text{s}.
B
Just before the first impact the ball moves downward with a speed of 0.83 m s−10.83\,\text{m s}^{-1} relative to the ground.
C
Just after the first impact the ball moves upward with a speed of 5.54 m s−15.54\,\text{m s}^{-1} relative to the ground.
D
The ball comes to rest relative to the lift (after infinitely many bounces) at t=0.85 st=0.85\,\text{s}, having travelled a total distance of 1.0 m1.0\,\text{m} relative to the lift.
115
In a lift whose upward acceleration is 2.0 m s−22.0\,\text{m s}^{-2}, a light rope passes over a smooth light pulley fixed to the ceiling. One end of the rope is tied to a platform of mass 20 kg20\,\text{kg}; the other end is held by a man of mass 60 kg60\,\text{kg} who stands on the platform and pulls the rope vertically downward. The platform carries a weighing scale (of negligible mass) under the man's feet, which reads the force between his feet and the platform. The man pulls so that he and the platform together accelerate upward at 1.0 m s−21.0\,\text{m s}^{-2} relative to the lift. Take g=10 m s−2g=10\,\text{m s}^{-2}. Which of the following statements are correct?
Not attempted
A
The tension in the rope is 480 N480\,\text{N}.
B
The scale reads 260 N260\,\text{N}.
C
If the lift were in free fall and the man pulled with exactly the same force as above, he and the platform would accelerate upward relative to the lift at 13 m s−213\,\text{m s}^{-2}.
D
The pulley exerts a force of 520 N520\,\text{N} on the ceiling.
116
A simple pendulum of length ℓ\ell has period T0T_0 when hung in a laboratory at rest. It is now used in four different non-inertial settings, with its period TT. Match each setting in List-I with the value of T4/T04T^4/T_0^4 in List-II. In the incline cases the pendulum hangs from the roof of a trolley that is released from rest on a fixed incline of angle 37∘37^\circ (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8), the friction coefficient between trolley and incline (static equal to kinetic) is given, and the oscillations are small. List-IList-IIP) Lift accelerating upward at g/31)  1Q) Trolley on a smooth incline2)  9/16R) Trolley on the incline, friction coefficient 0.503)  5/4S) Trolley on the incline, friction coefficient 0.754)  25/165)  16/25\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \text{ Lift accelerating upward at } g/3 & \text{1)} \; 1 \\ \text{Q)} \text{ Trolley on a smooth incline} & \text{2)} \; 9/16 \\ \text{R)} \text{ Trolley on the incline, friction coefficient } 0.50 & \text{3)} \; 5/4 \\ \text{S)} \text{ Trolley on the incline, friction coefficient } 0.75 & \text{4)} \; 25/16 \\ & \text{5)} \; 16/25 \end{array}
Not attempted
A
P-5, Q-4, R-1, S-3
B
P-2, Q-3, R-4, S-1
C
P-1, Q-5, R-3, S-2
D
P-2, Q-4, R-3, S-1
117
In each situation of List-I a small block is released from rest, at a point on a fixed inclined face of angle 37∘37^\circ (sin⁡37∘=0.6\sin37^\circ=0.6, cos⁡37∘=0.8\cos37^\circ=0.8), which is rigidly attached to a frame (a lift or a trolley). Match each situation with the magnitude of the block's acceleration relative to the frame (in m s−2\text{m s}^{-2}, taking g=10 m s−2g=10\,\text{m s}^{-2}) immediately after release, from List-II. If the block loses contact with the face, give its acceleration relative to the frame (it then moves freely under the effective gravity). List-IList-IIP) Smooth face in a lift accelerating upward at 5 m s−21)  0Q) Smooth face in a lift whose cable has snapped (free fall)2)  2.5R) Smooth face in a lift accelerating downward at 20 m s−23)  9.0S) Face with friction coefficient 0.5 (rising toward the rear) on a trolley accelerating forward at 25 m s−24)  105)  14\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \text{ Smooth face in a lift accelerating upward at } 5\,\text{m s}^{-2} & \text{1)} \; 0 \\ \text{Q)} \text{ Smooth face in a lift whose cable has snapped (free fall)} & \text{2)} \; 2.5 \\ \text{R)} \text{ Smooth face in a lift accelerating downward at } 20\,\text{m s}^{-2} & \text{3)} \; 9.0 \\ \text{S)} \text{ Face with friction coefficient } 0.5 \text{ (rising toward the rear) on a trolley accelerating forward at } 25\,\text{m s}^{-2} & \text{4)} \; 10 \\ & \text{5)} \; 14 \end{array}
Not attempted
A
P-3, Q-1, R-4, S-2
B
P-3, Q-1, R-2, S-5
C
P-5, Q-1, R-4, S-3
D
P-3, Q-4, R-1, S-5
118
A smooth solid hemisphere of radius RR is fixed with its flat face down on the horizontal roof of a cart, the curved surface upward. The cart moves forward with a constant acceleration a=7.5 m s−2a=7.5\,\text{m s}^{-2}. A small particle of mass mm, resting on the topmost point of the hemisphere (at rest relative to the cart), is given an infinitesimal push, so that it starts sliding down the surface. Take g=10 m s−2g=10\,\text{m s}^{-2}. The particle leaves the surface when the radius to it makes an angle θ\theta with the vertical. Find θ\theta in degrees, to two decimal places. (Use cos⁡−1(8/15)=57.77∘\cos^{-1}(8/15)=57.77^\circ if needed, and tan⁡−1(0.75)=36.87∘\tan^{-1}(0.75)=36.87^\circ.)
Not attempted
ma
Your answer: —
Correct answer: 20.9
119
A lift starts from rest at t=0t=0. Its acceleration aLa_L, taken positive upward, varies with time as in the graph: +5.0 m s−2+5.0\,\text{m s}^{-2} for 0<t<0.4 s0<t<0.4\,\text{s}, −10 m s−2-10\,\text{m s}^{-2} for 0.4<t<1.0 s0.4<t<1.0\,\text{s}, −15 m s−2-15\,\text{m s}^{-2} for 1.0<t<2.0 s1.0<t<2.0\,\text{s} and 00 for t>2.0 st>2.0\,\text{s} (the lift is very tall, so it is not limited by any building). At t=0t=0 a small ball is released from rest relative to the lift, at a height 9.0 m9.0\,\text{m} above the lift floor and clear of the ceiling. There is no air resistance, g=10 m s−2g=10\,\text{m s}^{-2}. Find the time tt (in seconds, measured from t=0t=0) at which the ball hits the floor of the lift, to two decimal places.
Not attempted
t (s)a (m/s²)0.41.02.03.05-10-15
Your answer: —
Correct answer: 2.29
120
A plank of mass M=2.0 kgM=2.0\,\text{kg} lies on a rough horizontal floor (coefficient of friction 0.100.10 with the floor, static equal to kinetic). A small block of mass m=1.0 kgm=1.0\,\text{kg} rests on it, 0.40 m0.40\,\text{m} from the rear end of the plank, the coefficient of friction between block and plank being 0.300.30 (static equal to kinetic). A horizontal force F=ktF=kt with k=6.0 N s−1k=6.0\,\text{N s}^{-1} is applied to the plank from t=0t=0, directed forward. Take g=10 m s−2g=10\,\text{m s}^{-2}. At what time tt (in seconds from t=0t=0) does the block slide off the rear end of the plank? Give two decimal places and use (0.8)1/3=0.9283(0.8)^{1/3}=0.9283 if needed.
Not attempted
plank MmF = ktrough floor
Your answer: —
Correct answer: 2.93
Three regimes in sequence: nothing moves, plank and block move together, then the block slips. In the last regime, find how the relative acceleration depends on time.
Nothing moves until ktkt exceeds the floor friction limit 0.10(M+m)g=3 N0.10(M+m)g=3\,\text{N}, i.e. t=0.5 st=0.5\,\text{s}. Then the plank and block move together with a=kt−33a=\frac{kt-3}{3}. The block needs friction mama; slipping begins when a=μg=3 m s−2a=\mu g=3\,\text{m s}^{-2}, i.e. kt−3=9kt-3=9, t0=2.0 st_0=2.0\,\text{s}. For t>t0t>t_0 the block has constant acceleration μg=3 m s−2\mu g=3\,\text{m s}^{-2} forward, while the plank has ap=kt−μmg−3M=6t−62a_p=\frac{kt-\mu mg-3}{M}=\frac{6t-6}{2}. In the block's frame the plank's relative acceleration is 6t−62−3=3(t−2)\frac{6t-6}{2}-3=3(t-2), zero at t0t_0, so the relative speed is 32τ2\tfrac32\tau^2 and the relative displacement is 12τ3\tfrac12\tau^3 with τ=t−t0\tau=t-t_0. The block slides off the rear when the plank has moved forward 0.40 m0.40\,\text{m} relative to it: 12τ3=0.40\tfrac12\tau^3=0.40, τ=(0.8)1/3=0.9283 s\tau=(0.8)^{1/3}=0.9283\,\text{s}. So t=2.00+0.9283=2.93 st=2.00+0.9283=2.93\,\text{s}. Using the constant relative acceleration found at the instant of slipping, or ignoring the floor friction, gives different times.