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Beyond Advanced · Rotational Motion

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1
A uniform thin rod AB of mass MM and length LL carries a small bead of mass MM fixed at end B. Let I(x)I(x) be the moment of inertia of the whole system about an axis perpendicular to the rod passing through the point of line AB at distance xx from A (xx is measured along AB; it may be negative or exceed LL). Which curve correctly shows I/(ML2)I/(ML^2) against x/Lx/L?
Not attempted
A
Curve 1
x/LI/(ML²)O00.510.51.01.5
B
Curve 2
x/LI/(ML²)O00.510.51.01.5
C
Curve 3
x/LI/(ML²)O00.510.51.01.5
D
Curve 4
x/LI/(ML²)O00.510.51.01.5
2
Four uniform square plates have the same side 12 cm, thickness and material. Each has material removed as described in the options, with coordinates in cm measured from the geometric centre of the square (marked with a cross in each figure) (xx to the right, yy upward). The holes lie fully inside the square; the slot is open at the edge. Which plate has its centre of mass farthest from the centre of the square?
Not attempted
A
One square hole of side 3.5, centre at (4, 4)
B
One circular hole: radius 3, centre at (3, 0)
C
One rectangular slot 3 wide and 6 tall, centre at (-4.5, 0), open at the left edge
D
Two circular holes: radius 2.4 at (3.5, 0) and radius 2.2 at (-3, 0)
3
A thin uniform-cross-section wire is bent into a circular ring of radius RR lying in the xyxy-plane with its centre at the origin. Its linear mass density depends on the polar angle θ\theta (measured from the xx-axis) as λ=λ0(1+sin⁡θ)\lambda=\lambda_0(1+\sin\theta). Let I1I_1 be the moment of inertia about the line in the plane of the ring parallel to the xx-axis and tangent to the ring at its lightest point, and I2I_2 the moment of inertia about the line in the plane parallel to the xx-axis passing through the centre of mass of the ring. Find I1/I2I_1/I_2.
Not attempted
A
10
B
5
C
3
D
6.25
4
List-I describes four equilibrium situations; List-II gives values. Match each situation with the value of the quantity asked (dimensionless). List-IList-IIP) Smallest pull, over all directions, on the axle of a wheel (weight W, radius 1.0 m) to start lifting it over a step 0.4 m high (in units of W)1) 0.577Q) Smallest floor friction coefficient for a ladder at 60∘ on a smooth wall, ladder weight W with a man of weight W three-quarters of the way up2) 0.361R) Tension (in units of W) in a string making 37∘ with a horizontal hinged uniform rod of weight W, tied to its free end3) 0.800S) Force of the wall on a sphere (in units of its weight W) when two identical smooth spheres are stacked in a vertical smooth tube of inner radius 1.5 times the sphere radius4) 0.8315) 1.333\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P) Smallest pull, over all directions, on the axle of a wheel (weight } W\text{, radius } 1.0\text{ m) to start lifting it over a step } 0.4\text{ m high (in units of } W\text{)} & \text{1) } 0.577 \\ \text{Q) Smallest floor friction coefficient for a ladder at } 60^\circ \text{ on a smooth wall, ladder weight } W \text{ with a man of weight } W \text{ three-quarters of the way up} & \text{2) } 0.361 \\ \text{R) Tension (in units of } W\text{) in a string making } 37^\circ \text{ with a horizontal hinged uniform rod of weight } W\text{, tied to its free end} & \text{3) } 0.800 \\ \text{S) Force of the wall on a sphere (in units of its weight } W\text{) when two identical smooth spheres are stacked in a vertical smooth tube of inner radius } 1.5\text{ times the sphere radius} & \text{4) } 0.831 \\ & \text{5) } 1.333 \end{array}
Not attempted
A
P-3, Q-2, R-4, S-1
B
P-5, Q-2, R-1, S-4
C
P-3, Q-4, R-2, S-1
D
P-1, Q-3, R-4, S-2
5
A uniform wire of total mass MM is shaped into a plane frame consisting of a regular hexagon of side aa together with its three long diagonals (all joined at the centre O), as in the figure. The wire has the same mass per unit length everywhere. Let IdI_d be the moment of inertia about a long diagonal and IℓI_\ell that about the line ℓ\ell in the plane, along one side of the hexagon. Find Iℓ/IdI_\ell/I_d.
Not attempted
ℓOhexagon with its three long diagonals
A
257\dfrac{25}{7}
B
145\dfrac{14}{5}
C
167\dfrac{16}{7}
D
317\dfrac{31}{7}
6
A uniform thin plate has the shape of a T with a displaced stem. In cm, with the origin at the centre of its 7×67\times6 bounding rectangle (xx to the right, yy upward), the plate occupies the union of the bar −3.5≤x≤3.5, 1.5≤y≤3-3.5\le x\le3.5,\ 1.5\le y\le3 and the stem 0.5≤x≤2.5, −3≤y≤1.50.5\le x\le2.5,\ -3\le y\le1.5. Four lines in the plane of the plate are described in the options (each is drawn dashed on a copy of the plate). About which line is the moment of inertia of the plate the least?
Not attempted
A
Horizontal line through the centre of the 7 cm by 6 cm bounding rectangle
B
Horizontal line through the centre of mass of the plate
C
Vertical line through the centre of mass of the plate
D
Vertical line through the centre of the 7 cm by 6 cm bounding rectangle
7
A uniform rod AB of length 0.80 m0.80\ \text{m} and mass mm is hung from a fixed hook O by two light inextensible strings tied to its ends: OA=0.50 m\text{OA}=0.50\ \text{m} and OB=0.70 m\text{OB}=0.70\ \text{m} (figure). The rod hangs at rest. Find the tension in the shorter string OA.
Not attempted
OAB0.50 m0.70 mrod AB = 0.80 m, uniform, mass m (figure to scale)
A
0.764 mgmg
B
0.625 mgmg
C
0.546 mgmg
D
0.500 mgmg
8
A uniform crate in the shape of a cube of side 1.0 m1.0\ \text{m} has a rectangular channel (void) running through its whole width, cut at the top of the front side: in a side view, the void occupies the part of the front half 0.5 m0.5\ \text{m} long (from the front face) that lies within 0.4 m0.4\ \text{m} of the top, as shown. The crate stands on the horizontal bed of a truck, with coefficient of static friction 1.21.2 between crate and bed, and does not slide or tip when the truck moves uniformly. The truck now moves in a straight line on level ground. Let afa_f be the largest forward acceleration and aba_b the largest braking deceleration (acceleration directed backwards) for which the crate stays at rest relative to the truck. Find afa_f and aba_b.
Not attempted
voidfronttruck bed, crate side view: 1.0 m × 1.0 m0.5 m0.4 m
A
1.20gg and 1.20gg
B
1.03gg and 1.20gg
C
1.03gg and 1.32gg
D
1.20gg and 1.03gg
9
A uniform square plate of side aa and mass MM is folded along a median line into a V shape, so that two identical rectangular halves of size a×a/2a\times a/2 meet along the crease. The halves make equal angles with the vertical symmetry plane, as in the figure. Let O be the midpoint of the crease. Take the xx-axis along the crease, the zz-axis vertical through O (in the symmetry plane) and the yy-axis horizontal and perpendicular to the crease. The moments of inertia about the zz-axis and the yy-axis are measured to be in the ratio Iz/Iy=0.60I_z/I_y=0.60. Find the angle between the two halves.
Not attempted
crease: x-axis, out of the pagezyend view; angle between the halves not to scale
A
20.7°
B
82.8°
C
138.6°
D
41.4°
10
Each body has mass MM and radius RR. List-I describes a body and an axis; List-II gives the moment of inertia about that axis in units of MR2MR^2. Match each entry of List-I with its value. List-IList-IIP) Uniform solid hemisphere: axis parallel to the flat face through the centre of mass1) 0.259Q) Thin hemispherical shell (open): axis parallel to the rim plane, tangent to the dome at its pole2) 0.320R) Uniform semicircular lamina: axis perpendicular to its plane through its centre of mass3) 0.595S) Uniform semicircular wire: axis perpendicular to its plane through its centre of mass4) 0.6675) 0.400\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P) Uniform solid hemisphere: axis parallel to the flat face through the centre of mass} & \text{1) } 0.259 \\ \text{Q) Thin hemispherical shell (open): axis parallel to the rim plane, tangent to the dome at its pole} & \text{2) } 0.320 \\ \text{R) Uniform semicircular lamina: axis perpendicular to its plane through its centre of mass} & \text{3) } 0.595 \\ \text{S) Uniform semicircular wire: axis perpendicular to its plane through its centre of mass} & \text{4) } 0.667 \\ & \text{5) } 0.400 \end{array}
Not attempted
A
P-5, Q-4, R-3, S-2
B
P-1, Q-2, R-4, S-3
C
P-1, Q-4, R-2, S-3
D
P-5, Q-3, R-2, S-4
11
A shell of mass 4 kg4\ \text{kg} is fired from level ground with velocity components 30 m s−130\ \text{m s}^{-1} horizontally and 40 m s−140\ \text{m s}^{-1} vertically (g=10 m s−2g=10\ \text{m s}^{-2}, no air resistance). Exactly 1.0 s1.0\ \text{s} after firing it bursts into two fragments of masses 3 kg3\ \text{kg} and 1 kg1\ \text{kg}. Just after the burst the heavier fragment has velocity 10 m s−110\ \text{m s}^{-1} vertically upward (no horizontal component) and both fragments stay in the original vertical plane. The ground is smooth and horizontal: when a fragment reaches it, its vertical velocity is destroyed at once and it then slides on frictionlessly with its horizontal velocity unchanged. The centre of mass of the pair means the mass-weighted mean position of both fragments, wherever they are. Which of the following statements are correct?
Not attempted
A
When the heavier fragment lands, the centre of mass of the two fragments is at a height of 76.6 m above the ground.
B
When the lighter fragment lands, the horizontal coordinate of the centre of mass is 581.4 m from the firing point.
C
At the instant the lighter fragment lands, the two fragments are 2.07 km apart.
D
The centre of mass of the two fragments keeps following the original parabola until the lighter fragment lands.
12
Two blocks of masses 1.0 kg1.0\ \text{kg} and 3.0 kg3.0\ \text{kg} are joined by a light spring of constant 400 N m−1400\ \text{N m}^{-1} and rest on a smooth horizontal floor, the lighter block touching a fixed vertical wall. The spring is compressed by 0.10 m0.10\ \text{m} by pushing the heavier block towards the wall, and the system is released from rest. Afterwards the lighter block leaves the wall and the pair moves away from it along the line of the spring. Which of the following statements are correct? (Velocities are relative to the floor.)
Not attempted
A
The lighter block leaves the wall at the instant the spring reaches its natural length, when the heavier block moves at 1.15 m/s.
B
The largest speed ever reached by the lighter block is 1.73 m/s.
C
The maximum extension of the spring after the lighter block has left the wall is 0.071 m.
D
At the instant the lighter block is fastest, the heavier block is moving forward at 0.58 m/s.
13
A uniform ladder of mass MM leans against a vertical wall, making angle θ\theta with the horizontal floor, in a vertical plane perpendicular to the wall (figure). The coefficient of static friction is 0.350.35 between the ladder and the floor and 0.500.50 between the ladder and the wall. In each part the ladder is just at the verge of slipping where stated. Which of the following statements are correct?
Not attempted
topθμ_floor = 0.35wall: μ_wall = 0.50
A
The smallest angle at which the unloaded ladder can stand without slipping is 49.7∘49.7^\circ.
B
At that smallest angle the frictional force exerted by the wall on the ladder is 0.18 Mg0.18\,Mg.
C
A person of mass MM starts climbing the ladder when it stands at 60∘60^\circ; she can climb to about 0.830.83 of its length before it slips.
D
If the two coefficients of friction were such that their product is at least 1, an unloaded ladder could stay in equilibrium at every angle between the floor and the wall.
14
A box is made by joining five identical thin uniform square plates of side aa into an open cube: a square base and four vertical walls, with no lid (see the figure). Its total mass is MM and the zz-axis is vertical through the centre of the base. At the centre of mass the principal horizontal axes xx and yy are parallel to the base edges. Which of the following statements is correct? (More than one choice may be correct.)
Not attempted
aaopen topz
A
The centre of mass lies a/5a/5 below the geometric centre of the cube.
B
The moment of inertia about the vertical axis through the centre of the base is Ma2/4Ma^2/4.
C
The moment of inertia about the horizontal axis through the centre of mass, parallel to a base edge, is 77300Ma2\dfrac{77}{300}Ma^2.
D
For axes through the centre of mass, Ix+Iy=IzI_x+I_y=I_z, as the perpendicular axis theorem requires.
15
A uniform solid circular cylinder of radius R=0.20 mR=0.20\ \text{m} and length ℓ=0.60 m\ell=0.60\ \text{m} has a cylindrical hole of radius R/2R/2 drilled through its whole length, parallel to its axis, with the axis of the hole at distance R/2R/2 from the axis of the cylinder. The remaining body has mass MM. Let the xx-axis join the two axes (pointing from the cylinder axis towards the hole axis) in a cross-section, the zz-axis be along the cylinder axis, and the origin be at the geometric centre of the cylinder. Which of the following statements are correct?
Not attempted
A
The centre of mass lies on the xx-axis at 3.33 cm from the cylinder's axis, on the side away from the hole.
B
The moment of inertia about the cylinder's own axis is 0.0217 M0.0217\,M in SI units (that is, 1324MR2\tfrac{13}{24}MR^2).
C
The moment of inertia about the axis parallel to the cylinder's axis through the centre of mass is 0.0206 M0.0206\,M in SI units.
D
The moment of inertia about the line through the centre of mass parallel to the yy-axis is 0.0381 M0.0381\,M in SI units.
16
A uniform square plate of side 0.60 m0.60\ \text{m} has a concentric square hole of side 0.30 m0.30\ \text{m}, as in the figure. It is hung in a vertical plane from a horizontal pin passing through a small hole drilled in the material of the plate at a distance dd from the centre O, perpendicular to the plate, and oscillates through small angles (g=9.8 m s−2g=9.8\ \text{m s}^{-2}). The pin can be placed at any point of the material. Which of the following statements are correct?
Not attempted
O0.60 msquare hole of side 0.30 m, concentricplate
A
The radius of gyration of the plate about the axis through its centre O perpendicular to the plate is 0.245 m0.245\ \text{m}.
B
The shortest period obtainable by choosing the pin position is 1.49 s1.49\ \text{s}.
C
The period with the pin at a corner of the plate is shorter than the period with the pin at the midpoint of an outer edge.
D
Apart from the other three midpoints of the hole's edges, no point of the plate gives the same period as a pin at the midpoint of one edge of the hole.
17
A uniform rectangular block of mass MM has base length w=0.70 mw=0.70\ \text{m} (in the direction of pull) and height H=1.50 mH=1.50\ \text{m}. It rests on a rough horizontal floor with coefficient of static friction 0.350.35. A rope is tied to the midpoint of the block's top front edge and passes over the top of the block; it is pulled backwards and upwards in the vertical plane through the block's centre, at an angle φ\varphi above the horizontal. The person pulling can choose any angle with 0≤φ≤10∘0\le\varphi\le10^\circ but no other. The pull is increased slowly from zero. What is the smallest tension, in units of MgMg, at which the block begins to move (by sliding or by tipping about its back bottom edge) for the best choice of φ\varphi?
Not attempted
rope pulled back and up, angle φ above horizontalw = 0.70 mH = 1.50 mtied at top front edgeμ = 0.35
Your answer: —
Correct answer: 0.22
18
A thin planar lamina of unknown shape and mass lies in the plane of the page. Three small holes H1_1, H2_2, H3_3 in it lie on one straight line at distances 00, 0.25 m0.25\ \text{m} and 0.60 m0.60\ \text{m} from H1_1, measured along the line. The moments of inertia of the lamina, about axes perpendicular to its plane through the holes H1_1, H2_2, H3_3, are measured as 0.48930.4893, 0.29580.2958 and 0.4029 kg m20.4029\ \text{kg m}^2 respectively. The centre of mass of the lamina need not lie on the line. Find the least value of the moment of inertia about a perpendicular axis through any point of that line, in kg m2\text{kg m}^2.
Not attempted
Your answer: —
Correct answer: 0.28
19
A uniform thin square plate ABCD has side a=1.20 ma=1.20\ \text{m}. Divide it into nine equal squares by lines parallel to its sides and remove the central square (stage 1). Then treat each of the eight remaining squares in the same way (stage 2, drawn in the figure), and so on at every stage. Let MnM_n be the mass of the stage-nn plate and InI_n its moment of inertia about the axis through corner A perpendicular to the plane of the plate. Find lim⁡n→∞In/Mn\lim_{n\to\infty} I_n/M_n in m2\text{m}^2.
Not attempted
Astage 2 shown
Your answer: —
Correct answer: 0.99
20
A rigid body consists of a uniform thin rod AB of mass 2.4 kg2.4\ \text{kg} and length 1.2 m1.2\ \text{m} with a small bead of mass 1.2 kg1.2\ \text{kg} fixed to the end A. It lies at rest on smooth horizontal ice (top view in the figure; O is the midpoint of the rod). A small rubber ball moving on the ice perpendicular to the rod strikes it at a point 0.36 m0.36\ \text{m} from O on the side of B and rebounds. The ball's mass and speeds are not given. Immediately after the collision exactly one point of the rod is instantaneously at rest. Find its distance from the end A, in cm.
Not attempted
ABbeadOballtop view, smooth ice: AB = 1.2 m, impact point 0.36 m from O towards B
Your answer: —
Correct answer: 11.43
21
A U-tube of uniform bore (cross-section 2.0 cm22.0\ \text{cm}^2) has a horizontal limb of length b=0.70 mb=0.70\ \text{m} joining two vertical limbs. The tube is rigidly fixed to a small cart; the empty tube plus cart has mass 0.12 kg0.12\ \text{kg} and rolls without friction on a smooth horizontal floor. The tube holds water (density 1000 kg m−31000\ \text{kg m}^{-3}) of total length 1.00 m1.00\ \text{m} measured along the tube, so each vertical column is 0.15 m0.15\ \text{m} long at rest. The water levels are slightly displaced and released. Treat the water as ideal, neglect the corners and take g=9.8 m s−2g=9.8\ \text{m s}^{-2}. Find the period of the small oscillation of the levels, in seconds.
Not attempted
smooth floor, cart free to moveb = 0.70 mwaterwater
Your answer: —
Correct answer: 1.18
22
Paragraph A solid hemisphere of radius R=0.35 mR=0.35\ \text{m} and mass MM has a density that depends only on the distance rr from the centre O of its flat face: ρ=ρ0 (1+r/R)\rho=\rho_0\,(1+r/R). The hemisphere is placed with its curved surface on a rough surface (friction large enough that it never slips) and is free to roll. Take g=9.8 m s−2g=9.8\ \text{m s}^{-2}. The figure shows the hemisphere on an incline. Question 1 of 2: The surface is an incline of angle 12∘12^\circ with the horizontal. The hemisphere comes to rest in equilibrium on it, curved side down. Find the angle (in degrees) that its flat face makes with the horizontal.
Not attempted
OP12°orientation of the flat face is schematic, not to scale
Your answer: —
Correct answer: 32.62
23
Paragraph A solid hemisphere of radius R=0.35 mR=0.35\ \text{m} and mass MM has a density that depends only on the distance rr from the centre O of its flat face: ρ=ρ0 (1+r/R)\rho=\rho_0\,(1+r/R). The hemisphere is placed with its curved surface on a rough surface (friction large enough that it never slips) and is free to roll. Take g=9.8 m s−2g=9.8\ \text{m s}^{-2}. The figure shows the hemisphere on an incline. Question 2 of 2: Now the surface is horizontal. The hemisphere, resting with its curved side down in stable equilibrium, is rocked through a small angle and released. Find the period of the small oscillations, in seconds.
Not attempted
OP12°orientation of the flat face is schematic, not to scale
Your answer: —
Correct answer: 1.54
24
A uniform thin rod of mass MM and length LL lies at rest on a smooth horizontal table (the figure is a top view). A short horizontal impulse JJ is delivered at one end, perpendicular to the rod. Immediately after the impulse, exactly one point of the rod (or of its imaginary extension) has zero velocity. Each of the four figures below shows the rod with that point marked by a dot, together with the rod's centre CC (a plus sign). Which figure is correct? The figures are drawn with the struck end at the bottom, as in the stem figure, and the impulse pointing to the right.
Not attempted
+CJtop endstruck endsmooth table (top view)
A
Figure 1
+
B
Figure 2
+
C
Figure 3
+
D
Figure 4
+
25
A uniform thin rod of mass mm and length LL has a tiny smooth hole drilled at a distance xx from its centre, 0≤x≤L/20 \le x \le L/2. The rod is hung on a fixed smooth horizontal nail through the hole, held horizontal and released from rest. Let NN be the upward vertical force that the nail exerts on the rod immediately after release. Which graph correctly shows N/mgN/mg as a function of x/Lx/L? (All four graphs use the same axes; the horizontal axis is at N=0N = 0.)
Not attempted
centrehole (hinge)xuniform rod, length L
A
Graph 1
x/LN/mgO0.250.51−1−2
B
Graph 2
x/LN/mgO0.250.51−1−2
C
Graph 3
x/LN/mgO0.250.51−1−2
D
Graph 4
x/LN/mgO0.250.51−1−2
26
A closed box slides down a smooth fixed incline of angle 60∘60^\circ, with its floor always flat on the incline. A yo-yo (a uniform disc of mass mm and radius RR with a light axle of radius r=R/2r = R/2) is suspended inside the box by a light string wound on the axle; the other end of the string is fixed to the wall of the box that is farthest from the incline, and the string is taut and perpendicular to the incline. The yo-yo is released from rest relative to the box and unrolls without the string slipping on the axle, the string always remaining perpendicular to the incline. Find the tension in the string while the yo-yo unwinds.
Not attempted
60°smooth incline, 60°box slides freelyyo-yo hangs inside
A
2mg/32mg/3
B
mg/2mg/2
C
mg/6mg/6
D
mg/3mg/3
27
A uniform rod of mass mm and length LL is hinged at one end on a smooth fixed hinge, held horizontal and released from rest. Let θ\theta be the angle the rod has turned below the horizontal and PP the instantaneous rate at which the weight of the rod does work on it, for 0<θ<90∘0 < \theta < 90^\circ. Which graph shows PP as a function of θ\theta? (Each graph is scaled so that its highest point is at the same height; the markings on the horizontal axis are in degrees.)
Not attempted
θmguniform rod, mass m, length L, hinged at one end
A
Graph 1
θPO30°45°60°90°
B
Graph 2
θPO30°45°60°90°
C
Graph 3
θPO30°45°60°90°
D
Graph 4
θPO30°45°60°90°
28
A plank of mass M=2mM = 2m rests on a smooth horizontal floor. A uniform solid cylinder of mass mm and radius RR lies on it and rolls without slipping on the plank when a constant horizontal force FF is applied to the plank (the cylinder is always on the plank). The system starts from rest. At any later instant, what fraction of the work done by FF so far has been stored as the rotational kinetic energy of the cylinder about its own axis (the kinetic energy of the cylinder's centre of mass is not included)?
Not attempted
plank, mass M = 2mm, RFsmooth floorno slipping on the plank
A
2/212/21
B
1/91/9
C
1/51/5
D
1/31/3
29
A thin ring of mass M=2mM = 2m and radius R=0.4 mR = 0.4\ \text{m} carries a small bead of mass mm rigidly fixed on its rim. The ring rolls without slipping on a rough horizontal floor (the friction is large enough for rolling to continue). At the instant the bead is at the bottom, touching the floor, the centre of the ring has speed v0v_0. Find the largest v0v_0 (in m s−1^{-1}) for which the ring does not lose contact with the floor at any later time during the following revolution. Take g=9.8 m s−2g = 9.8\ \text{m s}^{-2}. You may assume the ring has enough speed to keep rolling over the top.
Not attempted
vring M = 2mbead m (at the contact point)rough floor
A
3.433.43
B
5.245.24
C
4.324.32
D
4.434.43
30
A uniform rod of mass M=3 kgM = 3\ \text{kg} and length L=1.2 mL = 1.2\ \text{m} lies at rest on a smooth horizontal table (top view shown). A particle of mass m=1 kgm = 1\ \text{kg} moving on the table at v=8 m s−1v = 8\ \text{m s}^{-1} perpendicular to the rod strikes it at a distance d=0.48 md = 0.48\ \text{m} from its centre CC and sticks to it. Immediately after the collision, what is the angular speed of the rod-plus-particle system about its own centre of mass? (Answer in rad s−1^{-1}.)
Not attempted
CvdmM, Lsmooth table, top view; rod initially at rest
A
6.506.50
B
7.217.21
C
3.313.31
D
5.415.41
31
A uniform rod of mass MM and length LL lies at rest on a smooth horizontal table (top view). A particle of mass m=Mm = M moving on the table perpendicular to the rod hits it elastically at a distance d=0.3Ld = 0.3L from the centre of the rod (and then moves on, with no second collision). What fraction of the particle's initial kinetic energy is carried away as rotational kinetic energy of the rod about its own centre?
Not attempted
Cvd = 0.3 Lrod: M, Lm = Msmooth table, top view; elastic collision
A
0.9990.999
B
0.4550.455
C
0.9010.901
D
0.8770.877
32
Four uniform bodies of the same mass mm are each hinged on a smooth fixed horizontal pin at the point shown by a dot, held in a vertical plane in the orientation drawn (the dashed line is horizontal) and released from rest. Let NN be the vertical component of the hinge force immediately after release, as a fraction of mgmg. List-IList-IIP) Square plate of side a, hinged at a corner, a diagonal horizontal1) 1/4Q) Same plate, hinged at a corner, a side horizontal2) 5/8R) Thin rod of length L, hinged at a point L/4 from its centre, horizontal3) 4/7S) Uniform disc, hinged at a point on its rim, a diameter horizontal4) 1/35) 1/2\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)}\ \text{Square plate of side } a \text{, hinged at a corner, a diagonal horizontal} & \text{1)}\ 1/4 \\ \text{Q)}\ \text{Same plate, hinged at a corner, a side horizontal} & \text{2)}\ 5/8 \\ \text{R)}\ \text{Thin rod of length } L \text{, hinged at a point } L/4 \text{ from its centre, horizontal} & \text{3)}\ 4/7 \\ \text{S)}\ \text{Uniform disc, hinged at a point on its rim, a diameter horizontal} & \text{4)}\ 1/3 \\ & \text{5)}\ 1/2 \end{array}
Not attempted
PQRS
A
P-2, Q-1, R-3, S-4
B
P-1, Q-2, R-3, S-4
C
P-1, Q-3, R-2, S-4
D
P-1, Q-2, R-5, S-3
33
A uniform rod of mass MM and length LL is rotating on a smooth horizontal table about a vertical axis at its end OO, with angular velocity ω\omega (taken positive in its initial sense). At one instant the axis at OO is removed and, at the same instant, the rod is suddenly pinned at one of the points marked in the figure (distances L/4L/4, L/2L/2, 2L/32L/3 and 3L/43L/4 from OO along the rod) by a vertical pin that exerts only an impulsive force during the pinning and is a smooth axis afterwards. List-I gives the pinning point; List-II gives the ratio ω′/ω\omega'/\omega of the angular velocity just after to just before. List-IList-IIP) Pinned at L/4 from O1) 0Q) Pinned at L/2 from O2) −2/7R) Pinned at 2L/3 from O3) 1S) Pinned at 3L/4 from O4) 10/75) −1/2\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)}\ \text{Pinned at } L/4 \text{ from } O & \text{1)}\ 0 \\ \text{Q)}\ \text{Pinned at } L/2 \text{ from } O & \text{2)}\ -2/7 \\ \text{R)}\ \text{Pinned at } 2L/3 \text{ from } O & \text{3)}\ 1 \\ \text{S)}\ \text{Pinned at } 3L/4 \text{ from } O & \text{4)}\ 10/7 \\ & \text{5)}\ -1/2 \end{array}
Not attempted
OL/4L/22L/33L/4rod rotating about O, anticlockwise
A
P-4, Q-3, R-1, S-2
B
P-4, Q-3, R-2, S-1
C
P-3, Q-4, R-1, S-2
D
P-4, Q-1, R-3, S-2
34
A uniform rod of mass mm and length LL is hinged at one end to a fixed smooth hinge. It is held horizontal and released from rest. Let θ\theta be the angle the rod has turned below the horizontal, so 0≤θ≤90∘0 \le \theta \le 90^\circ during the swing to the lowest position. The hinge force has a horizontal and a vertical (upward positive) component. Select ALL correct statements.
Not attempted
θrod: mass m, length L, released from restsmooth hinge
A
The magnitude of the total hinge force is smallest at the instant of release, where it equals mg/4mg/4.
B
The horizontal component of the hinge force has its largest magnitude, 9mg/89mg/8, when θ=45∘\theta = 45^\circ.
C
The upward vertical component of the hinge force equals mgmg at the instant when tan⁡θ=1/2\tan\theta = 1/\sqrt{2}.
D
In the lowest position the hinge force is purely vertical and equals 5mg/25mg/2.
35
A uniform thin rod of mass mm and length LL stands on a perfectly smooth horizontal floor, making an angle θ0\theta_0 with the vertical, where sin⁡θ0=0.6\sin\theta_0 = 0.6. It is released from rest and falls in a vertical plane (its lower end slides freely). Let NN be the normal force from the floor. Select ALL correct statements.
Not attempted
smooth floorθ₀rod: m, Lreleased from rest
A
Immediately after release, N=mg/4N = mg/4.
B
The upper end of the rod moves along a vertical straight line.
C
When the rod becomes horizontal (just before it lands), its angular speed satisfies ω2=9.6 g/L\omega^2 = 9.6\,g/L.
D
Just before the rod lands flat, N=mg/4N = mg/4 whatever the value of θ0\theta_0.
36
A small bob of mass mm hangs from a fixed point OO by a light inextensible string of length ℓ\ell and moves in a horizontal circle (a conical pendulum) with the string making a constant angle θ\theta with the vertical. Let CC be the centre of the circle, on the vertical through OO, and let L⃗O\vec L_O and L⃗C\vec L_C be the angular momenta of the bob about OO and CC. Select ALL correct statements.
Not attempted
OCθℓmbob moves in a horizontal circle with centre C
A
∣L⃗O∣=∣L⃗C∣ /sin⁡θ|\vec L_O| = |\vec L_C|\,/\sin\theta.
B
L⃗O\vec L_O is a constant vector, because the tension passes through OO and the bob's weight acts along the vertical through OO.
C
The magnitude of dL⃗O/dtd\vec L_O/dt is mgℓsin⁡θmg\ell\sin\theta.
D
The tension exerts no torque about CC, so L⃗C\vec L_C is conserved.
37
A long plank of mass M=2mM = 2m lies on a smooth horizontal floor. A uniform solid disc of mass mm and radius RR, spinning about its horizontal axis with angular speed ω0\omega_0 and with its centre at rest, is gently placed on the plank (spin sense such that its lowest point moves backward relative to its centre). The coefficient of kinetic friction between disc and plank is μ\mu. The plank is long enough that rolling without slipping is eventually established. Select ALL correct statements about the final state and the process.
Not attempted
plank, mass M = 2mω₀msmooth floordisc placed on the plank with its centre at rest
A
The plank finally moves with speed ω0R/7\omega_0R/7, opposite to the direction in which the disc's centre moves.
B
Exactly 4/74/7 of the initial kinetic energy is dissipated, whatever the value of μ\mu.
C
The final spin of the disc is 3ω0/73\omega_0/7.
D
The time taken to reach rolling is ω0R/(7μg)\omega_0R/(7\mu g).
38
Two coaxial uniform discs of radius R=0.20 mR = 0.20\ \text{m}, masses 2.0 kg2.0\ \text{kg} and 1.0 kg1.0\ \text{kg}, spin about a common smooth vertical axle with angular velocities +60 rad s−1+60\ \text{rad s}^{-1} and −30 rad s−1-30\ \text{rad s}^{-1} (opposite senses). The discs are then pressed together with a total normal force N=50 NN = 50\ \text{N} distributed uniformly over the whole face, so the kinetic friction coefficient between them is μ=0.30\mu = 0.30. No other torque acts. Select ALL correct statements.
Not attempted
disc 1: 2 kgdisc 2: 1 kgω₁ = 60ω₂ = −30N
A
The discs lock together at a common angular velocity of 30 rad s−130\ \text{rad s}^{-1}.
B
The time taken to lock is 0.4 s0.4\ \text{s}.
C
The heat generated is 54 J54\ \text{J}.
D
The two faces slip over each other through a relative angle of 18 rad18\ \text{rad}.
39
A uniform rod of mass mm and length LL hangs at rest from a smooth hinge at its upper end. A constant horizontal force F=mg/2F = mg/2 is applied permanently at the lower end (its direction does not change as the rod turns) and the rod swings out of the vertical in a vertical plane. Select ALL correct statements.
Not attempted
Frod: m, Lsmooth hingeF = mg/2, horizontal, always applied at the lower end
A
The rod comes momentarily to rest when it is horizontal.
B
At the instant the rod is momentarily at rest, the hinge exerts a purely vertical force on it.
C
The largest angular speed of the rod satisfies ωmax2=3(2−1) g/L\omega_{max}^2 = 3(\sqrt2 - 1)\,g/L.
D
Up to the position of momentary rest, the work done by FF equals mgL/2mgL/2.
40
A uniform flexible rope of mass M=3 kgM = 3\ \text{kg} and length L=2.4 mL = 2.4\ \text{m} hangs over a pulley that is a uniform disc of mass m=2 kgm = 2\ \text{kg} turning on a smooth fixed axle. The pulley radius is negligible compared with the lengths of the hanging parts, so the rope length lying on the pulley may be ignored. The rope does not slip on the pulley. Initially the rope is at rest with the two hanging parts differing in length by 0.3 m0.3\ \text{m}. It is then released. Take g=9.8 m s−2g = 9.8\ \text{m s}^{-2}. Find the time, in seconds, after release at which the shorter hanging end reaches the level of the pulley.
Not attempted
longer partshorter partmrope: mass M, length L, uniform; disc pulley mass m
Your answer: —
Correct answer: 1.12
41
A flywheel of moment of inertia I=1.2 kg m2I = 1.2\ \text{kg m}^2 on a fixed axle is driven by a motor that delivers a constant mechanical power P=450 WP = 450\ \text{W} to it at all times. The only other torque is a viscous bearing drag of magnitude bωb\omega opposing the motion, with b=0.02 N m sb = 0.02\ \text{N m s}. The flywheel starts from rest, and the motor can supply any torque needed to deliver the stated power. Its angular speed tends to a limiting value ω∞\omega_\infty. Find the time, in seconds, taken to reach ω=0.8 ω∞\omega = 0.8\,\omega_\infty.
Not attempted
Your answer: —
Correct answer: 30.65
42
A horizontal platform of radius RR has moment of inertia of moment of inertia I0=13mR2I_0 = \tfrac13 mR^2 about its vertical axle, which is smooth. A person of mass mm (treat as a point) stands at the rim and the platform rotates at ω0=1.5 rad s−1\omega_0 = 1.5\ \text{rad s}^{-1} with the person. The person now walks along a radius toward the axle, moving at constant speed relative to the platform and reaching the axle after τ=4 s\tau = 4\ \text{s}. Through what total angle, in radians, does the platform turn relative to the ground during these 44 s?
Not attempted
Your answer: —
Correct answer: 14.51
43
A particle of mass m=0.8 kgm = 0.8\ \text{kg} moves on a smooth horizontal table and is attached to a light inextensible string that passes through a small smooth hole in the table and carries a hanging block of mass M=0.6 kgM = 0.6\ \text{kg} below the table. The block moves only vertically. The particle is set into uniform circular motion at radius r0=0.4 mr_0 = 0.4\ \text{m} in a state where the block stays at rest. The particle is now given a very small radial nudge (its angular momentum about the hole is unchanged), so that rr oscillates slightly about r0r_0. Between two successive instants of greatest distance from the hole, through what angle (in radians) does the particle's line of position from the hole turn? Take g=9.8 m s−2g = 9.8\ \text{m s}^{-2} and give the answer to two decimal places.
Not attempted
Mmrsmooth table, light string through a small smooth holetable top
Your answer: —
Correct answer: 4.8
44
A rod of mass MM and uniform linear density, length L=1.4 mL = 1.4\ \text{m}, lies on a smooth horizontal table and can turn about a smooth vertical hinge at its end OO. A small ball of mass 0.30 M0.30\,M is fixed at its far end. A bullet of mass m=0.05 Mm = 0.05\,M moving horizontally at 40 m s−140\ \text{m s}^{-1} perpendicular to the rod strikes it at a distance xx from OO and embeds in it. For what value of xx (in metres) does the hinge exert no impulsive force during the collision?
Not attempted
O0.30Mvrod: M, Lbullet m (sticks)horizontal table, top view; hinge O
Your answer: —
Correct answer: 1.11
45
A uniform thin rod of mass mm and length LL stands vertically on a rough horizontal floor with coefficient of static friction μ=0.42\mu = 0.42 and is disturbed infinitesimally so that it begins to topple in a vertical plane. Its lower end does not slip until the friction needed exceeds the limiting value. Assume the lower end stays on the floor until it slips. Let θ\theta be the angle between the rod and the vertical. Find θ\theta, in degrees, at which the lower end first begins to slip.
Not attempted
θrod: m, Lrough floor, coefficient μ
Your answer: —
Correct answer: 51.54
46
A uniform ladder (a thin rod) of mass MM and length LL leans almost vertically against a smooth vertical wall, with its lower end on a smooth horizontal floor. A small bead of mass MM is fixed to the top end of the ladder (touching the wall). The ladder is disturbed infinitesimally and begins to slide, with its top end moving down the wall. At the instant the ladder loses contact with the wall, what is the height of the top end above the floor, as a fraction of LL? Give the answer to two decimal places.
Not attempted
bead m = Mrod: M, Lsmooth floor and wall
Your answer: —
Correct answer: 0.79
47
A uniform solid sphere of radius RR is dropped so that it first hits a rough horizontal floor with vertical speed 4.0 m/s4.0\ \text{m/s} and no horizontal velocity, spinning about a horizontal axis so that the surface speed of the spin at the contact point is Rω0=10 m/sR\omega_0=10\ \text{m/s}. At every impact the normal restitution coefficient is e=0.5e=0.5 and the coefficient of friction is μ=0.3\mu=0.3 (same value for sliding and sticking). Each impact is very short (gravity negligible during it), and the ball is a projectile between impacts. Take g=9.8 m/s2g=9.8\ \text{m/s}^2. What is the horizontal distance between the point of the first impact and the point of the third impact?
Not attempted
A
1.29 m1.29\ \text{m}
B
1.75 m1.75\ \text{m}
C
1.10 m1.10\ \text{m}
D
0.86 m0.86\ \text{m}
48
A uniform thin rod of mass MM and length LL lies on a smooth horizontal table and can turn about a smooth vertical hinge fixed at a point L/3L/3 from its centre. A particle of mass mm moving at speed uu perpendicular to the rod strikes it elastically (relative speed of the touching points reverses) at a point which can be chosen by the experimenter. For what value of m/Mm/M is it possible that, after the collision, the particle is at rest and the hinge has exerted no impulse?
Not attempted
Chingeustrike pointuniform rod, mass M, length Ltop view, smooth table; hinge at L/3 from centre
A
3/73/7
B
4/74/7
C
1/31/3
D
4/34/3
49
A spool of mass mm has outer radius RR, moment of inertia 12mR2\tfrac12mR^2 about its axis, and an axle of radius rr. It rests on a rough horizontal table and rolls without slipping. A light thread wound on the axle is pulled with a force FF in a vertical plane perpendicular to the axle. The thread leaves the axle tangentially, either from the lowest point or from the highest point of the axle, at an angle ϕ\phi above the horizontal in the direction of pulling. Match each arrangement in List-I with the acceleration of the axle in List-II, in units of F/mF/m (positive means in the direction of the horizontal component of the pull). List-IList-IIP)  r=R/2,  ϕ=0, thread from the lowest point1)  0Q)  r=R/2,  ϕ=60∘, thread from the lowest point2)  1/9R)  r=R/3,  ϕ=60∘, thread from the lowest point3)  1/3S)  r=R/2,  ϕ=0, thread from the highest point4)  15)  −1/6\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \; r=R/2, \; \phi=0, \text{ thread from the lowest point} & \text{1)} \; 0 \\ \text{Q)} \; r=R/2, \; \phi=60^\circ, \text{ thread from the lowest point} & \text{2)} \; 1/9 \\ \text{R)} \; r=R/3, \; \phi=60^\circ, \text{ thread from the lowest point} & \text{3)} \; 1/3 \\ \text{S)} \; r=R/2, \; \phi=0, \text{ thread from the highest point} & \text{4)} \; 1 \\ & \text{5)} \; -1/6 \end{array}
Not attempted
A
P-3, Q-1, R-2, S-4
B
P-3, Q-4, R-2, S-1
C
P-1, Q-3, R-2, S-4
D
P-3, Q-5, R-1, S-4
50
A long conveyor belt moves horizontally to the left at speed u=2vu=2v relative to the ground. A uniform wheel of radius RR rolls without slipping on the upper surface of the belt while its axle moves to the right at speed vv relative to the ground (both uu and vv constant). A small paint mark on the rim touches the belt at the lowest point at t=0t=0. Each figure below is the path of the mark seen from the ground, drawn to the same scale (the bar shows a length 2R2R; the dashed line is the level of the belt). Which path is correct?
Not attempted
A
Path 1
start2R
B
Path 2
start2R
C
Path 3
start2R
D
Path 4
start2R
51
A uniform solid sphere of radius RR rolls without slipping on a rough horizontal floor with centre speed vv and meets a fixed rough step of height h=0.30Rh=0.30R square-on. It strikes the sharp corner; the collision there is perfectly inelastic (no rebound, no slipping at the corner), the floor exerts no impulse because the sphere leaves it, and immediately afterwards the sphere pivots about the corner. Ignore rolling resistance. The sphere is said to climb successfully if it keeps pivoting on the corner, without the corner force ever becoming negative (the sphere never leaves the corner), until its centre is directly above the corner. For which range of vv does it climb successfully?
Not attempted
vhsolid sphere, radius Rstep corner
A
0.65gR≤v≤0.84gR0.65\sqrt{gR}\le v\le 0.84\sqrt{gR}
B
1.31gR≤v≤1.67gR1.31\sqrt{gR}\le v\le 1.67\sqrt{gR}
C
0.73gR≤v≤1.02gR0.73\sqrt{gR}\le v\le 1.02\sqrt{gR}
D
0.83gR≤v≤1.06gR0.83\sqrt{gR}\le v\le 1.06\sqrt{gR}
52
A thin-walled hollow cylinder (mass MM, radius RR, closed at both ends) is completely filled with an ideal (non-viscous, incompressible) liquid of total mass 3M3M. It rolls without slipping down a rough incline of angle θ\theta, starting from rest. What is the least coefficient of static friction for which it rolls without slipping? (Neglect the thickness of the wall in the moment of inertia of the shell.)
Not attempted
A
0.38tan⁡θ0.38\tan\theta
B
0.50tan⁡θ0.50\tan\theta
C
0.20tan⁡θ0.20\tan\theta
D
0.125tan⁡θ0.125\tan\theta
53
A uniform solid cylinder of mass m=2 kgm=2\ \text{kg} lies on a long plank of mass M=3 kgM=3\ \text{kg}. The plank rests on a rough horizontal floor with coefficient of friction 0.100.10 between plank and floor (static and kinetic equal). A constant horizontal force F=22 NF=22\ \text{N} is applied to the axle of the cylinder, which rolls on the plank without slipping; the friction between cylinder and plank is large enough for this. Both start from rest. Take g=10 m/s2g=10\ \text{m/s}^2. Select all correct statements.
Not attempted
Fcylinder, m = 2 kgplank, M = 3 kg, rough floor (mu = 0.10)
A
The plank stays at rest, because the friction F/3F/3 that the cylinder exerts on it is smaller than the largest friction the floor can supply.
B
The friction force exerted by the plank on the cylinder is 6.9 N6.9\ \text{N}.
C
The acceleration of the axle of the cylinder relative to the ground is 7.5 m/s27.5\ \text{m/s}^2.
D
Two seconds after the start, the rate at which heat is produced at the contact between plank and floor is 6.4 W6.4\ \text{W}.
54
Four uniform bodies of equal radius: a thin ring, a thin hollow sphere (I=23mR2I=\tfrac23mR^2), a solid cylinder and a solid sphere, are released from rest at the same time from the top of a long rough incline with tan⁡θ=34\tan\theta=\tfrac34. The coefficient of friction between each body and the incline is 0.280.28 (static and kinetic values equal). Any body that cannot roll without slipping slides while rotating. Select all correct statements.
Not attempted
A
The thin hollow sphere reaches the bottom before the solid cylinder does.
B
The ring and the thin hollow sphere reach the bottom at the same instant.
C
At every instant while the ring slides, its rim speed about its centre, ωR\omega R, exceeds 0.80.8 times its centre speed.
D
The ring takes less time to slide down than it would on an incline with enough friction to make it roll without slipping.
55
A uniform-length solid cylinder of mass mm and radius RR has two concentric parts: a core of radius R/2R/2 whose material has three times the density of the outer part. It rests on a rough horizontal floor. A horizontal force F=mg/2F=mg/2 is applied with its line of action at height hh above the axis (−R≤h≤R-R\le h\le R, negative below the axis), and the cylinder is to roll without slipping. Which graph shows the least coefficient of static friction needed, against h/Rh/R?
Not attempted
A
Graph 1
h/Rμ needed-1-0.50.510.250.5
B
Graph 2
h/Rμ needed-1-0.50.510.250.5
C
Graph 3
h/Rμ needed-1-0.50.510.250.5
D
Graph 4
h/Rμ needed-1-0.50.510.250.5
56
A long plank of mass M=1.5 kgM=1.5\ \text{kg} lies on a smooth horizontal floor and is attached to a fixed wall by a light spring of constant 40 N/m40\ \text{N/m}. A uniform solid sphere of mass 0.70 kg0.70\ \text{kg} rests on the plank; the coefficient of static friction between sphere and plank is 0.300.30. The plank is pulled aside by a distance AA and released from rest, with the sphere at rest on it. Take g=9.8 m/s2g=9.8\ \text{m/s}^2. What is the largest AA for which the sphere rolls on the plank without ever slipping?
Not attempted
ksolid sphere (mass m)plank (mass M) on a smooth floor
A
0.57 m0.57\ \text{m}
B
0.12 m0.12\ \text{m}
C
0.44 m0.44\ \text{m}
D
0.39 m0.39\ \text{m}
57
A thin-walled hollow cylinder (the shell) of mass 0.60 kg0.60\ \text{kg}, closed at both ends, is completely filled with an ideal (non-viscous, incompressible) liquid of total mass 1.20 kg1.20\ \text{kg}. It lies on a rough incline of angle 30∘30^\circ. A light spring of constant 60 N/m60\ \text{N/m} is attached to its axis, parallel to the incline, with its upper end fixed. It is released from rest with the spring at its natural length and rolls without slipping. Take g=10 m/s2g=10\ \text{m/s}^2. Select all correct statements.
Not attempted
30°km, R
A
The period of the oscillation is 1.26 s1.26\ \text{s}.
B
At the lowest point of its motion the friction force on the shell points down the incline and has magnitude 2.25 N2.25\ \text{N}.
C
The shell rolls without slipping throughout if the coefficient of static friction is at least 0.150.15.
D
The largest rotational kinetic energy of the shell is 0.34 J0.34\ \text{J}.
58
Each of four uniform bodies (a thin ring, a thin hollow sphere with I=23mR2I=\tfrac23mR^2, a solid cylinder and a solid sphere) rolls without slipping on a rough horizontal floor with centre speed v0v_0 towards a vertical smooth wall. The impact is very short, the wall is smooth (no friction on it) and the coefficient of restitution for the normal direction is e=0.6e=0.6. Afterwards the bodies are acted on by floor friction only and the wall is far enough that no further impact happens during the sliding stage. Select all correct statements.
Not attempted
A
The solid cylinder finally rolls away from the wall with speed v0/15v_0/15.
B
The thin ring finally rolls away from the wall with speed v0/5v_0/5.
C
The thin hollow sphere (moment of inertia 23mR2\tfrac23mR^2) keeps moving away from the wall for ever after the impact.
D
The solid sphere loses 64%64\% of its kinetic energy in the impact itself.
59
A uniform thin rod of length LL falls without rotating, tilted at 40∘40^\circ to the vertical, with speed uu. Its lower end strikes a rough horizontal floor. The collision is such that the vertical velocity of the contact point reverses (perfectly elastic normally) while the contact point does not slip at all (its horizontal velocity becomes zero). The floor exerts an impulse at the lower end only. Each figure shows the velocity components (horizontal to the right, vertical upward, in units of uu) of the lower end, the centre and the upper end immediately after the impact; the rod lower end is at the left. Which one is correct?
Not attempted
A
Velocity set 1
velocity components after impact (units of u)ACBA lower end: (-1.32, +1.00)C centre: (+0.00, -0.11)B upper end: (+1.32, -1.21)
B
Velocity set 2
velocity components after impact (units of u)ACBA lower end: (+0.00, +0.00)C centre: (+0.37, -0.31)B upper end: (+0.74, -0.62)
C
Velocity set 3
velocity components after impact (units of u)ACBA lower end: (+0.00, +1.00)C centre: (+0.00, +1.00)B upper end: (+0.00, +1.00)
D
Velocity set 4
velocity components after impact (units of u)ACBA lower end: (+0.00, +1.00)C centre: (+0.74, +0.38)B upper end: (+1.48, -0.24)
60
A uniform rod ABAB of length 1.2 m1.2\ \text{m} and mass 0.5 kg0.5\ \text{kg} lies on a smooth horizontal table and is free to turn about a frictionless vertical hinge fixed at a point 0.20 m0.20\ \text{m} from the end AA (see figure). A horizontal impulse JJ perpendicular to the rod is delivered at one point of the rod; the hinge may supply any impulsive reaction. Select all correct statements.
Not attempted
uniform rod AB, length 1.2 m, mass 0.5 kg; J is perpendicular to the rod0.20 mABhinge1.00 m from hinge to B
A
If J=0.70 N sJ=0.70\ \text{N s} is delivered at the end BB, the kinetic energy given to the rod is 1.75 J1.75\ \text{J}.
B
Because of the hinge, the energy given to the rod by J=0.70 N sJ=0.70\ \text{N s} at the end BB is 0.21 J0.21\ \text{J} less than the energy it would receive if the hinge were removed (rod free on the table).
C
Wherever along the rod JJ is delivered, the hinge impulse in the direction of JJ never exceeds 37J\dfrac37J.
D
If J=0.70 N sJ=0.70\ \text{N s} is delivered at the end AA, the rod turns at 1.0 rad/s1.0\ \text{rad/s}, in the sense opposite to that produced by a blow at BB.
61
A uniform rod of length LL lies on a smooth horizontal table. In each of four separate set-ups it is attached to the table by a frictionless vertical hinge at one of the points P, Q, R, S shown in the figure. The same horizontal impulse JJ, perpendicular to the rod, is given to the end of the rod that is farther from the hinge. The hinge then exerts an impulsive reaction. Match each hinge position in List-I with the impulse exerted by the hinge on the rod in List-II. List-IList-IIP) hinge at the centre1) zeroQ) hinge at L/6 from the centre2)  J/2 in the direction of JR) hinge at L/4 from the centre3)  J opposite to JS) hinge at the nearer end4)  2J/7 in the direction of J5)  2J/7 opposite to J\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \text{ hinge at the centre} & \text{1)} \text{ zero} \\ \text{Q)} \text{ hinge at } L/6 \text{ from the centre} & \text{2)} \; J/2 \text{ in the direction of } J \\ \text{R)} \text{ hinge at } L/4 \text{ from the centre} & \text{3)} \; J \text{ opposite to } J \\ \text{S)} \text{ hinge at the nearer end} & \text{4)} \; 2J/7 \text{ in the direction of } J \\ & \text{5)} \; 2J/7 \text{ opposite to } J \end{array}
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PQRSJuniform rod; possible hinge positions P, Q, R, SP: centre Q: L/6 from centre R: L/4 from centre S: end
A
P-1, Q-3, R-4, S-2
B
P-3, Q-1, R-4, S-2
C
P-3, Q-1, R-5, S-4
D
P-3, Q-4, R-1, S-2
62
A uniform disc of radius RR has a circular hole of radius 0.6R0.6R whose centre is at distance 0.4R0.4R from the centre OO of the disc (the hole touches the rim). The remaining disc has mass MM and its centre of mass is GG. The disc rolls without slipping on a rough horizontal floor in a vertical plane. Select all correct statements.
Not attempted
OholeGuniform disc, radius R, hole radius 0.6Rhole centre 0.4R from O; G is the centre of mass
A
For small oscillations about the stable rolling position (with GG vertically below OO) the period is 2π5.07 R/g2\pi\sqrt{5.07\,R/g}.
B
If the disc is released from rest with GG at the same height as OO, its initial angular acceleration is 0.38 g/R0.38\,g/R.
C
If released from rest with GG level with OO, the speed of OO at the instant GG is lowest is 0.87gR0.87\sqrt{gR}.
D
Immediately after release from rest with GG level with OO, the normal reaction of the floor is 0.97 Mg0.97\,Mg, where MM is the mass of the disc.
63
A yo-yo of mass m=1.0 kgm=1.0\ \text{kg} has a thread wound on its axle of radius rr; the flanges have radius R=3rR=3r and the yo-yo is a uniform disc for its moment of inertia, I=12mR2I=\tfrac12mR^2. The thread leaves the axle vertically, passes over a light smooth pulley fixed to the ceiling of a lift, and carries a block of mass 0.90 kg0.90\ \text{kg} on its other end (the yo-yo and the block hang from the pulley, and the thread unwinds from the yo-yo as it goes down). The lift moves upward with constant acceleration 2.8 m/s22.8\ \text{m/s}^2. The system is released from rest relative to the lift. Take g=9.8 m/s2g=9.8\ \text{m/s}^2. Find the acceleration of the block relative to the ground, in m/s2\text{m/s}^2, taken positive upward.
Not attempted
Your answer: —
Correct answer: 2.2
64
A thin uniform ring of mass M=2mM=2m and radius R=0.50 mR=0.50\ \text{m} carries two small beads fixed on its rim at opposite ends of a diameter, of masses 3m3m and mm. The ring rolls without slipping on a rough horizontal floor in a vertical plane. The equilibrium position is the one with the heavier bead at the lowest point. Take g=9.8 m/s2g=9.8\ \text{m/s}^2. Find the period (in seconds) of small oscillations about this equilibrium position.
Not attempted
3mmthin ring, mass M = 2m, radius RR
Your answer: —
Correct answer: 2.84
65
A cylindrical trough of inner radius R=0.45 mR=0.45\ \text{m} is fixed on a cart which accelerates horizontally with constant acceleration a=7.5 m/s2a=7.5\ \text{m/s}^2. A thin-walled hollow spherical shell (I=23mr2I=\tfrac23mr^2) of radius r=0.05 mr=0.05\ \text{m} rolls without slipping on the inner surface of the trough, in a vertical plane along the direction of motion. It is displaced slightly from its equilibrium position (as seen from the cart) and released. Take g=9.8 m/s2g=9.8\ \text{m/s}^2. Find the period (in seconds) of its small oscillations.
Not attempted
acart accelerates to the rightcylindrical trough, radius R
Your answer: —
Correct answer: 1.46
66
A uniform solid sphere of radius R=0.10 mR=0.10\ \text{m} is gently placed at rest on a rough plane inclined at angle θ\theta with tan⁡θ=12\tan\theta=\tfrac12. It is spinning at ω0=110 rad/s\omega_0=110\ \text{rad/s} about a horizontal axis in the sense in which the point of the sphere touching the plane moves down the slope. The coefficient of friction (static and kinetic) is 0.800.80. Take g=9.8 m/s2g=9.8\ \text{m/s}^2. Find the greatest distance (in metres) the centre of the sphere moves up the slope from its starting point.
Not attempted
contact point slides downspin ω0tan θ = 1/2
Your answer: —
Correct answer: 0.72
67
A uniform solid sphere of mass 1.0 kg1.0\ \text{kg} is launched along the top surface of a long, thin plank of mass 3.0 kg3.0\ \text{kg} that lies at rest on a smooth horizontal floor. At launch the centre of the sphere moves at 4.0 m/s4.0\ \text{m/s} along the plank and the sphere has backspin of surface speed Rω0=6.0 m/sR\omega_0=6.0\ \text{m/s} (so the contact point initially slides forward on the plank). The sphere-plank friction is constant, kinetic until rolling begins, and the plank is long enough for rolling to start before the sphere reaches its end. Find the final velocity of the plank (in m/s, along the initial direction of motion of the sphere).
Not attempted
Your answer: —
Correct answer: 0.87
68
A uniform rod of mass MM and length LL hangs at rest from a smooth horizontal hinge at its upper end. A lump of clay of mass 2M2M, moving horizontally with speed uu (small enough that the rod never swings above the horizontal), hits the rod at distance xx below the hinge and sticks to it. After the impact the rod with the clay swings up through a maximum angle θmax⁡\theta_{\max}. For which value of x/Lx/L is θmax⁡\theta_{\max} the largest? (The clay is treated as a point.)
Not attempted
uxrod: M, Lclay, mass 2Msmooth hinge at the top
Your answer: —
Correct answer: 0.56
69
A billiard cue gives a horizontal impulse J=0.30 N sJ=0.30\ \text{N s} to a solid ball of mass 0.20 kg0.20\ \text{kg} lying on a table, in a vertical plane through the ball's centre, at a height 1.8R1.8R above the table (RR is the ball's radius). The coefficient of kinetic friction between ball and table is 0.100.10. After its centre has moved 0.28 m0.28\ \text{m} (the ball is still sliding at this moment) it strikes head-on an identical stationary ball. The collision is elastic and the balls are smooth. Take g=10 m/s2g=10\ \text{m/s}^2. Find the speed (in m/s) of the first ball once it has settled into rolling without slipping afterwards. (Ignore friction during the very short collision.)
Not attempted
Your answer: —
Correct answer: 0.73
70
A thin uniform ring of mass MM and radius R=0.40 mR=0.40\ \text{m} carries a small bead of mass 2M2M fixed on its rim. The ring rolls without slipping on a rough horizontal floor (friction is always sufficient for no slipping), and the bead passes over the top on every turn. Take g=9.8 m/s2g=9.8\ \text{m/s}^2. Find the largest speed vv (in m/s) of the ring's centre at the instant the bead is at its lowest point, such that the ring never loses contact with the floor during any turn. Interpret this as: the floor reaction N≥0N\ge0 at all times (the limiting case N=0N=0 is allowed).
Not attempted
bead (2M)ring (mass M), radius R = 0.40 mv
Your answer: —
Correct answer: 6.45
The bead's vertical acceleration is the only vertical acceleration in the system. Minimise the floor reaction over the whole turn, not just at the top.
Let ψ\psi be the angle of the bead from the top and c=cos⁡ψc=\cos\psi. The ring's centre has no vertical motion, so the floor reaction is N=3Mg−2MR y¨relN=3Mg-2MR\,\ddot y_{\text{rel}} with the bead's vertical acceleration y¨=−R(c ψ˙2+sin⁡ψ ψ¨)\ddot y=-R\left(c\,\dot\psi^2+\sin\psi\,\ddot\psi\right). Kinetic energy about the instantaneous axis: 12MR2⋅2 [1+2(1+c)]ψ˙2\tfrac12MR^2\cdot2\,[1+2(1+c)]\dot\psi^2, i.e. MR2(3+2c)ψ˙2MR^2(3+2c)\dot\psi^2; potential 2MgRc2MgRc. With w=Rψ˙2/gw=R\dot\psi^2/g and ε=E/(MgR)\varepsilon=E/(MgR): w=ε−2c3+2cw=\dfrac{\varepsilon-2c}{3+2c}. The angular acceleration follows from the energy equation. Result: NMg=3−2cw−4(1−c2)(1+w)2(3+2c)\dfrac{N}{Mg}=3-2cw-\dfrac{4(1-c^2)(1+w)}{2(3+2c)}. Requiring N≥0N\ge0 for every cc: at the top (c=1c=1) the condition would be ε≤9.50\varepsilon\le9.50, i.e. vbottom=6.71 m/sv_{\text{bottom}}=6.71\ \text{m/s}. But the bead is not the only constraint: NN reaches zero first at c≈0.44c\approx0.44 (about 64∘64^\circ before the top), where N=0N=0 already holds for ε=8.62\varepsilon=8.62. At the bottom c=−1c=-1 the kinetic energy is MR2ψ˙2MR^2\dot\psi^2 and ε=wb−2\varepsilon=w_b-2 with wb=v2/(gR)w_b=v^2/(gR), so vmax⁡=gR (εmax⁡+2)=6.45 m/sv_{\max}=\sqrt{gR\,(\varepsilon_{\max}+2)}=6.45\ \text{m/s}. A numerical integration of the motion confirms that slightly below this speed NN stays positive and slightly above it NN becomes negative near 64∘64^\circ before the top. Checking only the top gives the larger, wrong value 6.71 m/s6.71\ \text{m/s}.