SYSTEM OF PARTICLES AND ROTATIONAL MOTION
ROTATIONAL MOTION AND MOMENT OF INERTIA
Centre of mass of a two-particle system. Centre of mass of a rigid body. general motion of a rigid body, nature of rotational motion, torque, angular momentum, its conservation and applications. Moment of inertia, parallel and perpendicular axes theorem, expression of moment of inertia for ring, disc and sphere.
INITIAL STEP EXERCISE
1. When an explosive shell, travelling in a parabolic path under the effect of gravity explodes, the centre of mass of the fragments will move (a) (b) (c) (d) 2. first vertically upwards and then vertically downwards vertically downwards along the original parabolic path first horizontally and then along a parabolic path.
Two blocks m1 and m2, having masses 10 kg and 5 kg respectively, are placed on a frictionless horizontal surface and are connected by a light spring of force constant 5 N/m. m1 is in contact with a rigid wall. m2 is pushed through a distance of 4 cm towards m1 and then released. The velocity of the centre of mass of the system when m1 breaks off the wall is (a) (c) 2/3 cm/s 2 cm/s (b) (d) 4/3 cm/s 4 cm/s
3.
Two spheres of masses M and 2M are initially at rest at a distance R apart. Due to mutual force of attraction they approach each other. When they are at separation R/2, the acceleration of their centre of mass would be (a) (c) 0 3g m/s2 (b) (d) g m/s2 12g m/s2
4.
An isolated particle of mass m is moving in a horizontal plane (x – y) along the x-axis, at a certain height above the ground. It suddenly explodes into two fragments of masses m/4 and 3m/4. An instant later, the smaller fragment is at y = + 15 cm. The larger fragment at this instant is at (a) (c) y = –5 cm y = +5 cm (b) (d) y = + 20 cm y = –20 cm
5.
A smooth sphere A is moving on a frictionless horizontal plane with angular speed ω and centre of mass velocity v. It collides elastically and head on with an identical sphere B at rest. Neglect friction everywhere. After the collision, their angular speeds are ωA and ωB respectively. Then (a) (c) ωA 1 (b) (d) µ 1/2
3.
At t = 0, the position and velocities of two particles are as shown in the figure. They are kept on a smooth surface and being mutually attracted by gravitational force. The position of centre of mass at t = 2s is
(a) (c) 4.
x=5m x=3m
(b) (d)
x=7m x=2m
A ball of mass m is allowed to roll down the wedge of mass M as shown in the figure. The displacement of wedge when the ball reaches from A to B
(a)
⎛ m ⎞ ⎜ ⎟(d + 2h cot θ) ⎝M+m⎠ towards left ⎛ m ⎞ ⎜ ⎟(d + 2h cot θ) ⎝ 2M − m ⎠ towards left ⎛m⎞ ⎜ ⎟(d + 2h cot θ) ⎝M⎠ towards right ⎛m+M⎞ ⎜ ⎟(d + 2h cot θ) ⎝ 2M ⎠ towards left
(b)
(c)
(d)
5.
What must be the relation between length and radius of a cylinder of given mass and density so that its moment of inertia about the axis through its centre of mass and perpendicular to its length may be minimum? (a)
3 2
3 2
(b)
2 3
2 3
(c) 6.
(d)
When a bicycle is in motion, the force of friction exerted by the ground on the two wheels is such that it acts (a) (b) (c) (d) in the backward direction on the front wheel and in the forward direction on the rear wheel. in the forward direction on the front wheel and in the backward direction on the rear wheel. in the backward direction on both the front and the rear wheels. both (a) and (c) are correct
7.
A cubical block of side a is moving with velocity v on a horizontal smooth plane as shown. It hits a ridge at point O. The angular speed of the block after it hits O is
(a) (c) 8.
3v/(4a) √3v/(√2a)
(b) (d)
3v/(2a) zero
A hollow sphere and a solid sphere, having the same mass, are released from rest simultaneously from the top of a smooth inclined plane. Which of the two will reach the bottom first? (a) (b) (c) (d) solid sphere hollow sphere the one which has the greater density both will reach the bottom simultaneously
9.
A particle of mass m is projected with a velocity v making an angle of 450 with the horizontal. The magnitude of the angular momentum of the projectile about the point of projection when the particle is at its maximum height h is
(a) 10.
zero
(b)
mv3 4 2g
(c)
mv3 2g
(d)
m gh 3
A uniform bar of length 6a and mass 8m lies on a smooth horizontal table. Two point masses m and 2m moving in the same horizontal plane with speeds 2v and v, respectively, strike the bar as shown in figure and stick to it after collision. Denoting angular velocity (about the centre of mass), total energy and centre of mass velocity by ω, E and Vc respectively, we have after collision. Choose the incorrect option?
ω=
(a) VC = 0 (b)
3v 5a
3mv 2 5
ω=
(c)
v 5a
E=
(d)
11.
Four spheres, each of mass M and diameter 2r, are placed with their centres on the four corners of a square of side a (> 2r). The moment of inertia of the system about one side of the square is (a) (b) (c) (d)
2 M (5r 2 + 4a 2 ) 5 2 M (5r 2 + 2a 2 ) 5 2 M (2r 2 + 5a 2 ) 5 2 M (4r 2 + 5a 2 ) 5
12.
A cord is wound round the circumference of a wheel of radius r. The axis of the wheel is horizontal and its moment of inertia about this axis is I. A weight mg is attached to the end of the cord and allowed to fall from rest. The angular velocity of the wheel, when the weight has fallen through a distance h, is
(a)
⎡ 2gh ⎤ ⎢ I + mr ⎥ ⎦ ⎣
1/ 2
(b)
1/ 2
⎡ 2mgh ⎤ ⎢ I + mr 2 ⎥ ⎦ ⎣
1/ 2
(c) 13.
⎡ 2mgh ⎤ ⎢ I + 2mr2 ⎥ ⎦ ⎣
(d)
(2gh)1/2
A body of mass M and radius r, rolling on a smooth horizontal floor with velocity v, rolls up an 3v 2 irregular inclined plane up to a vertical height 4g . The body may be (a) (c) sphere disc (b) (d) solid cylinder both (b) and (c)
14.
Two points masses m1 and m2 are joined by a mass less rod of length r. The moment of inertia of the system about an axis passing through the center of mass and perpendicular to the rod is (a)
(m1 + m 2 )
r2 4
(b)
(m1 − m 2 )
r2 4
(c) 15.
m1m 2 2 r m1 + m 2
(d)
m1m 2 r 2 m1 − m 2 4
A thin rod of length L and mass M is held vertically with one end on the floor and is allowed to fall. The velocity of the other end when it hits the floor, assuming that the end on the floor does not slip (a) (c)
3gL gL
(b) (d)
2gL 2 gL
16.
Three uniform rods each of mass m and length L, is used to form an equilateral triangle. The moment of inertia of this frame about an axis through the centroid and perpendicular to the plane of triangle is (a) (c)
mL2
mL 3
2
(b) (d)
mL2 2
mL2 4
17.
Three uniform rods each of mass m and length L, is used to form an equilateral triangle. The moment of inertia of this frame about one of the side (a) (c)
mL2
mL 3
2
(b) (d)
mL2 2 mL2 4
18.
A disc starts from rest with angular acceleration (9 – 12t) rad/s2 in anticlockwise direction where t is the time. The number of revolutions that the disc makes before it starts to move in clockwise direction is (a) (c)
3.375 2π 3.375 4π
(b) (d)
3.375 3π
none of these
19.
A diver makes 2.5 revolutions on the way from a 10 m high platform to the water. Assuming zero initial vertical velocity, the diver’s average angular velocity during a dive is (a) (c) 8 rad/s 10 rad/s (b) (d) 9 rad/s 11 rad/s
20.
A wheel has a constant angular acceleration of 3 rad/s2. During a certain 4s interval, it turns through an angle of 120 rad. Assuming that the wheel starts from rest, how long is it in motion at the start of this 4s interval ? (a) (c) 4s 12s (b) (d) 8s 16s
21.
A uniform disk, with mass M and radius R mounted on a fixed horizontal axle. A block with mass M hangs from a mass less cord that is wrapped around the rim of the disk. The cord does not slip, and there is no friction at the axle. The acceleration of the falling block is (a) (c) g 2/3 g (b) (d) g/2 g/3
22.
A uniform ladder of mass 10 kg rests against a smooth vertical wall making an angle of 530 with it. The other end rests on a rough horizontal floor. The friction force that the floor exerts on the ladder is (a) (c) 65 N 75 N (b) (d) 98 N 86 N
23.
Figure shows a mass m placed on a frictionless horizontal table and attached to a string passing through a small hole in the surface. Initially, the mass moves in a circle of radius r0 with a speed v0 and a person holds the free end of the string. The Person pulls on the string slowly to decrease the radius of
the circle to r. Let the tension in the string when the mass moves depends on radius r as rn . The value of n is (a) (c) –1 –3 (b) (d) –2 –4
24.
A cylinder of mass m and radius R is placed with pure linear speed v0 on the frictional ground having coefficient of friction µ. The time after which it starts pure rolling is (a) (c)
2v 0 3µg 2v0 5µg
(b) (d)
v0 3µg 5v 0 7µg
25.
A sphere of mass m and radius R is rolling without slipping with angular speed ω on a horizontal plane as shown.
The angular momentum of the sphere about any point lying on the surface is (a) (c) 26. 2/5 mR2ω 7/5 mR ω
2
(b) (d)
3/5 mR2ω 8/5 mR2ω
A cylinder is rolling with slipping on an inclined plane of inclination θ then the coefficient of static friction between plane and cylinder may be (a) (c)
1 tan θ 3
(b) (d)
1 tan θ 2 2 tan θ 3
1 tan θ 4
27.
A ring rolls without slipping on the ground. Its centre C moves with a constant speed u. P is any point on the ring. The speed of P with respect to the ground may not be (a) (c) √2u 2u (b) (d) 2√2u 0
28.
3R A billiard ball, initially at rest, is given a sharp impulse by a cue. The cue is held a distance 5 above the center line. The ball leaves the cue with a speed v0 and angular speed ω0. Then the relation between v0 and ω0 is
(a) (c) 3v0 = 2ω0R 2v0 = 3ω0R (b) (d) v0 = ω0R 5v0 = 3ω0R
29.
A wheel of radius r and mass m stands in contact with step of height h. The least horizontal force F which should be applied to the axle of the wheel to force it climb onto the step is
(a)
mg [h (2r − h )] r−h
(b) (c) (d)
mgh(2r − h ) r−h mgh r−h
None of these
30.
A circular hole of radius R/2 is cut from a homogenous circular disc of a radius R. The centre of mass of the remaining disc is (a) (b) (c) (d) R/6 towards left R/6 towards right R/3 towards left R/3 towards right
31.
A small sphere of radius R is held against the inner surface of a larger sphere of radius 6R. The masses of large and small sphere are 4M and M respectively. This arrangement is placed on a horizontal table. There is no friction between any surfaces of contact. The small sphere is now released. The coordinates of the center of the larger sphere when the smaller sphere reaches the other extreme position is (a) (b) (c) (d) {(L + 2R), 0} {(L – 2R), 0} {(L – R), 0} {(L + R), 0}
Solutions:
1. 7. 13. 19. 25. 31.
c a d d c a
2. 8. 14. 20. 26.
a d c b c
3. 9. 15. 21. 27.
b b a c b
4. 10. 16. 22. 28.
a b b a a
5. 11. 17. 23. 29.
c d b c a
6. 12. 18. 24. 30.
d b a b a
ANALYSIS
r r r Let F be the force acting on a particle having position vector r and T be the torque of this force about the origin. Then r r r r r . T ≠ 0 and F . T ≠ 0 (a) r r r r r . T = 0 and F . T = 0 (b) r r r r r . T = 0 and F . T ≠ 0 (c) r r r r r . T ≠ 0 and F . T = 0 (d)
1.
[Ans. : c] 2. A circular disc X of radius R is made from an iron plate of thickness t, and another disc Y of radius 4 R is made form an iron plate of thickness t/4. Then the relation between the moment of inertia IX and IY is (a) (c) [Ans. : b] 3. A particle performing uniform circular motion has angular momentum L. If its angular frequency is doubled and its kinetic energy halved, then the new angular momentum is (a) (c) [Ans. : c] 4. Two spherical bodies of mass M and 5 M and radii R and 2 R respectively are released in free space with initial separation between their centres equal to 12 R. If they attract each other due to gravitational force only, then the distance covered by the smaller body just before collision is (a) (c) 7.5 R 2.5 R (b) (d) 1.5 R 4.5 R 4L L/4 (b) (d) L/2 2L IY = IX IY = 32IX (b) (d) IY = 64IX IY = 16IX
[Ans. : a] 1. A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same which one of the following will not be affected? (a) (b) moment of inertia angular momentum
(c) (d) [Ans. : b] 2.
angular velocity rotational kinetic energy
One solid sphere A and another hollow sphere B are the same mass and same outer radii. Their moment of inertia about their diameters are respectively IA and IB such that (a) (c) IA = IB IA IB IA/IB = dA/dB
[Ans. : c] 3. An annular ring with inner and outer radii, R1 and R2 is rolling without slipping with a uniform angular speed. The ratio of the forces experienced by the two particles situated on the inner and outer F1 F parts of the ring, 2 is
(a)
1
(b)
R1 R2
(c) [Ans. : a] 4.
R2 R1
(d)
⎛ R1 ⎞ ⎜ ⎜R ⎟ ⎟ ⎝ 2⎠
2
A mass ‘m’ moves with velocity ‘v’ and collides inelastically with another identical mass. After v collision the 1st mass moves with velocity 3 in a direction perpendicular to the initial direction of motion. Find the speed of the 2nd mass after collision
(a) [Ans. : a] 5.
2 v 3
(b)
v 3
(c)
v
(d)
3v
A spherical ball of mass 20 kg is stationary at the top of a hill of height 100 m. It rolls down a smooth surface to the ground, then climbs up another hill of height 30 m and finally rolls down to a horizontal base at a height of 20 m above the ground. The velocity attained by the ball is (a) (c) 10 m/s 40 m/s (b) (d) 10√30 m/s 20 m/s
[Ans. : c] 6. The moment of inertia of a uniform semicircular disc of mass M and radius r about a line perpendicular to the plane of the disc through the centre is (a) (c) [Ans. : b] Mr2 (b) (d)
1 2 Mr2 2 5 Mr2
1 4 Mr2
7.
A body A of mass M while falling vertically downwards under gravity breaks into two parts; a body B 1 2 3 M and a body C of mass 3 M. The centre of mass of bodies B and C taken together shifts of mass compared to that of body A towards (a) (c) (d) body C does not shift (b) body B depends on height of breaking
[Ans. : d] 8. A ‘T’ shaped object with dimensions shown in the figure, is lying on a smooth floor. A force ' F' is applied at the point P parallel to AB, such that the object has only the translational motion without rotation. Find the location of P with respect to C (a) (b) (c) (d) [Ans. : c]
4 l 3 l 2 l 3 3 l 2