Problems 1. A solid body rotates with deceleration about a stationary axis with an angular deceleration α =k ω
0
; where k is a constant and is the angular velocity of the body. If the initial angular velocity is ω , then mean angular velocity of the body averaged over the whole time of rotation is
ω
Physics : Rotational Motion
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Problems 2. A uniform wheel of moment of inertia I is pivoted on a horizontal axis through its centre so that its plane is vertical as shown in the figure. A small mass m is stuck on the rim of the wheel as shown. The angular acceleration of the wheel when mass is at point A is and when mass is at point B is α .
Problems 3. A centrifuge consists of four solid cylindrical containers, each of mass m at radial distance r from the axis of rotation. A time t is required to bring the centrifuge to an angular velocity from rest under a constant torque applied to the shaft. The radius of each container is a and the mass of the shaft and the supporting arms is small compared to m. Then
τ ω
Problems 4. One quarter sector is cut from a uniform disc of radius R. This sector has mass M. It is made to rotate about a line perpendicular to its plane and passing through the center of the original disc. Its moment of inertia about the axis of rotation is
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Physics : Rotational Motion
Solution : If we assume complete disc to be present, then it would have a mass 4 times the mass of the sector. Then, moment of inertia of the complete disc is
( )
Problems 5. Let I be the moment of inertia of a uniform square plate about an axis AB that passes through its centre and is parallel to two of its sides. CD is a line in the plane of the plate that passes through the centre of the plate and makes an angle with AB. The moment of inertia of the plate about the axis CD is then
equal to θ passing through O and perpendicular to the plate, then by the perpendicular axis theorem
I0 =IAB +IA B' ' =2IAB .... (1) horizontal plane as shown in the figure. The magnitude of angular momentum of the disc about the origin O is
(a) 1 MR2
(b) the other from the motion of the body with respect to its centre of mass
( )
Lspinie., Lto tal =LC M. +rC M. × p
Physics : Rotational Motion
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⇒ Lto tal =LC M. .+M r
(
C M. ×vC M.)
For this problem
2 circular loop with centre at O as shown in the figure. The moment of inertia of the loop about the axis XX' is
(a)
Further if r is the radius of the loop, then 2π =r L
Problems 8. A uniform solid cylinder of mass M and radius R is resting on a horizontal platform (which is parallel to X-Y plane) with its axis along the Y-axis and free to roll on the platform. The platform is given a motion in X-direction given by x A . There is no slipping between the cylinder and the platform. The maximum torque acting on the cylinder as measured about its centre of mass
cos t
= ω
(a) 1 A 2
2MR ω (b) MRAω2
(c) 2mRAω2 (d) mRωA cos2 2ωt
Solution : 1 2 linear acceleration
I 2MR
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Physics : Rotational Motion
Problems 9. A metal ball of mass m is put at the point A of a loop track and the vertical distance of A from the lower most point of track is 8 times the radius R of the circular part. The linear velocity of ball when it rolls of the point B to a height R in the circular track will
be R
Solution : Applying the conservation of energy at points A and B, we have
( )
8 2 1 2Problems 10. A uniform solid cylinder of mass m can rotate freely about its axis which is kept horizontal. A particle of mass hangs from the end of a light string wound round the cylinder. When the system is allowed to move, the acceleration with which the particle descends is
m0 Solution : Suppose that the tension in the string is T.
Then, =m g T0 − =m a0
α = angular acceleration and T.r = moment of force acting on cylinder
or I
r T = α
Physics : Rotational Motion
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2
Solution : Due to law of conservation of angular momentum, = constant L i.e. L.L = constant surely change direction of .
τ L L
L Ans. (a), (b), (c)
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Physics : Rotational Motion
Problems 12. A particle of mass m is projected with 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) zero (b)
mv3
4 2g
(c) mv3
2g (d) m 2gh3
Solution : L =mv hx
(
cos 45)
2sin 4522 L m v v
g
° °
⇒ =
3
4 2 L mv
⇒ = g
Further L =mv hx
⇒ L =m v
(
cos 45°)
ho But 2sin 452 02g h =v
2
v gh
⇒ =
2 2
L mv gh h
⇒ =
2 3
L m gh
⇒ = .
Ans. (b) (d)
Problems 13. A constant force F is applied at the top of a ring as shown in figure. Mass of the ring is M and radius is R. Angular momentum of particle about point of contact at time t
F
(a) is constant (b) increases linearly with time (c) is 2F R t (d) decreases linearly with time Solution : From angular impulse = change in angular momentum
we haveL= τt or L=F
( )
2R ti.e., L varies linearly with time.
Physics : Rotational Motion
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Problems 14. A spherical body of radius R rolls on a horizontal surface with linear velocity v. Let L1 and L2 be the magnitudes of angular momenta of the body about centre of mass and point
of contact P. Then v
P (a) L2 = 2L1 if radius of gyration K = R (b) L2 = 2L1 for all cases
(c) L2 > 2L1 if radius of gyration K < R (d) L2 > 2L1 if radius of gyration K > R Solution : L1= ω =I MK2ω … (1)
L2= ω +I MRv
)
)
2
=MK2ω +MR R
(
ω (as v= ωR )=M Kω
(
2+R … (2) From equations (1) and (2), we can see that L2 = 2 L1when K = R and L2 > 2L1
when K > R
Problems 15. A hollow sphere of radius R rests on a horizontal surface of finite coefficient of friction. A point object of mass m moved horizontally and hits the sphere at a height of R/2 above its center. The collision is instantaneous and completely inelastic. Which of the following is/are correct ?
(a) total linear momentum of the system is not conserved
(b) total angular momentum about center of mass of the system remains conserved (c) the space gets finite angular velocity immediately after collision
(d) The sphere moves with finite speed immediately after collision
Solution : Impulse due to normal reaction is finite. So friction force gives finite impulse. Therefore frictional torque causes a finite angular impulse about center of mass of system so angular momentum about center of mass of system will change. All other options are correct.
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Physics : Rotational Motion