A boy is standing on a platform, which is free to rotate about its axis. The boy holds an open umbrella in his hand. The axis of the umbrella coincides with that of the platform. The moment of inertia of 'the platform plus the boy system' is 3 . 0 × 10 - 3 kg m 2 and that of the umbrella is 2 . 0 × 10 - 3 kg m 2 . The boy starts spinning the umbrella about the axis at an angular speed of 2 . 0 rev s - 1 with respect to himself. Find the angular velocity imparted to the platform?

Let the angular velocity of umbrella be ω 1 . The angular velocity of umbrella with reference to boy = ω 1 + ω 2 = 2 rev s - 1 Consider the platform, boy and umbrella to be a system, then the action of spinning the umbrella is done by the internal force/torque. Hence, the net external torque = 0 . Therefore, the angular momentum of the system remains conserved. Initial angular momentum, L net = 0 Final angular momentum = L platform + boy + L umbrella = I platfrom + boy ω 2 + I um ω 1 Now, as the net torque is zero, the angular momentum must be conserved. I platfrom + boy ω 2 = I um ω 1 ω 2 I p + b = I um ω 1 ⇒ ω 1 = 2 - ω 2 3 × 10 - 3 ω 2 = 2 × 10 - 3 × 2 - ω 2 ω 2 = 4 5 rev s - 1 .

A uniform rod of mass m and length l is struck at an end by a force F perpendicular to the rod for a short time interval t . Calculate (a) the speed of the centre of mass. (b) the angular speed of the rod about the centre of mass. (c) the kinetic energy of the rod. (d) the angular momentum of the rod about the centre of mass after the force has stopped to act. Assume that t is so small that the rod does not appreciably change its direction while the force acts.

Given, the mass of the rod m , the length of the rod l . The rod is uniform, hence, the moment of inertia about the C O M is m l 2 12 . As the force F acts for a short interval of time t , this will be acted as the impulse to end of the rod. So, the linear impulse of the magnitude, J = ∫ F d t = F t Also, the linear impulse equals the change in momentum. J = P f - P i P i = 0 ( ∵ the rod is at rest initially) Let the speed of the rod after the impact be v (which is equal to the velocity of the centre of the mass) Hence, the final linear momentum, P f = m v cm = m v Part (a): J = m v - 0 F t = m v v = F t m Part (b): Now, that linear impulse will give the angular impulse to the rod because it is acted at some distance from the centre of mass. So, applying the angular impulse about the centre of mass , J → ang = r → × J → J → ang = r → × J → = r J sin θ θ is the angle between the radius vector and the linear impulse. In the above case, θ = 90 ° and r = l 2 as taken from the centre of mass. J ang = l 2 × F t × sin 90 ° = F l t 2 . . . 3 Also, the angular impulse is given by the change in angular momentum. J ang = L f - L i L i = 0 (the rod is at rest initially) Let be the angular speed of the rod is ω . L f = I ω . . . 4 From the equation ( 3 ) and ( 4 ), J ang = I ω - 0 ⇒ F l t 2 = m l 2 12 ω ⇒ ω = 6 F t m l Part (c): The total kinetic energy is given to the rod is the sum of the translational kinetic energy and rotational kinetic energy. K E trans = 1 2 m v 2 cm = 1 2 m F t m 2 = 1 2 F 2 t 2 m K E rot = 1 2 I ω 2 = 1 2 m l 2 12 6 F t m l 2 = 1 2 3 F 2 t 2 m K E total = K E trans + K E rot K E total = 1 2 F 2 t 2 m + 1 2 3 F 2 t 2 m K E total = 4 F 2 t 2 2 m = 2 F 2 t 2 m Part (d): The final angular momentum is given by, L f = I ω = m l 2 12 6 F t m l = F t l 2 .

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A diver having a moment of inertia of $6.0 kg m^{2}$ about an axis through its centre of mass rotates at an angular speed of $2 rad s^{- 1}$ about this axis. If he folds his hands and feet to decrease the moment of inertia to $5.0 kg m^{2}$ what will be the new angular speed $(rad s^{- 1})$ ?

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Important Questions on Rotational Mechanics

MEDIUM

JEE Main

IMPORTANT

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 10 - Rotational Mechanics>EXERCISES>Q 53

A boy is seated in a revolving chair revolving at an angular speed of

120

revolutions per minutes. Two heavy balls form part of the revolving system and the boy can pull the balls closer to himself or may push them apart. If by pulling the balls closer, the boy decreases the moment of inertia of the system from

6 kg m^{2}

2 kg m^{2}

what will be the new angular speed?

HARD

JEE Main

IMPORTANT

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 10 - Rotational Mechanics>EXERCISES>Q 54

A boy is standing on a platform, which is free to rotate about its axis. The boy holds an open umbrella in his hand. The axis of the umbrella coincides with that of the platform. The moment of inertia of 'the platform plus the boy system' is

3.0 \times 10^{- 3} kg m^{2}

and that of the umbrella is

2.0 \times 10^{- 3} kg m^{2}

. The boy starts spinning the umbrella about the axis at an angular speed of

2.0 rev s^{- 1}

with respect to himself. Find the angular velocity imparted to the platform?

HARD

JEE Main

IMPORTANT

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 10 - Rotational Mechanics>EXERCISES>Q 55

A wheel of moment of inertia

0.10 kg m^{2}

is rotating about a shaft at an angular speed of

160 rev \min^{- 1}

. A second wheel is set into rotation at

300 rev \min^{- 1}

and is coupled to the same shaft, so that both the wheels finally rotate with a common angular speed of

200 rev \min^{- 1}

. Find the moment of inertia of the second wheel.

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JEE Main

IMPORTANT

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 10 - Rotational Mechanics>EXERCISES>Q 56

A kid of mass

M

stands at the edge of a platform of radius

R

, which can be freely rotated about its axis. The moment of inertia of the platform is

I

. The system is at rest when a friend throws a ball of mass

m

and the kid catches it. If the velocity of the ball is

v

horizontally along the tangent to the edge of the platform when it was caught by the kid, find the angular speed of the platform after the event.

HARD

JEE Main

IMPORTANT

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 10 - Rotational Mechanics>EXERCISES>Q 57

A kid of mass $M$ stands at the edge of a platform of radius $R$ , which can be freely rotated about its axis. The moment of inertia of the platform is $I$ . The system is at rest when a friend throws a ball of mass $m$ and the kid catches it. If the velocity of the ball is $v$ horizontally along the tangent to the edge of the platform when it was caught by the kid, find the angular speed of the platform after the event. Suppose the platform of the previous problem is brought to rest with the ball in the hand of the kid standing on the rim. The kid throws the ball horizontally to his friend in a direction tangential to the rim with a speed $v$ as seen by his friend. Find the angular velocity with which the platform will start rotating.

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CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 10 - Rotational Mechanics>EXERCISES>Q 58

Suppose the platform with the kid is rotating in anticlockwise direction at an angular speed $ω$ . The kid starts walking along the rim with a speed $v$ relative to the platform also in the anticlockwise direction.

Suppose the platform is brought to rest with the ball in the hand of the kid standing on the rim. The kid throws the ball horizontally to his friend in a direction tangential to the rim with a speed $v$ as seen by his friend. Find the angular velocity with which the platform will start rotating.

HARD

JEE Main

IMPORTANT

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 10 - Rotational Mechanics>EXERCISES>Q 59

A uniform rod of mass $m$ and length $l$ is struck at an end by a force $F$ perpendicular to the rod for a short time interval $t$ . Calculate

(a) the speed of the centre of mass.

(b) the angular speed of the rod about the centre of mass.

(d) the angular momentum of the rod about the centre of mass after the force has stopped to act. Assume that $t$ is so small that the rod does not appreciably change its direction while the force acts.

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JEE Main

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CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 10 - Rotational Mechanics>EXERCISES>Q 60

A uniform rod of length

L

lies on a smooth horizontal table. A particle is moving on the table strikes the rod perpendicularly at an end and stops. Find the distance travelled by the centre of the rod by the time it turns through a right angle. Show that if the mass of the rod is four times that of the particle, the collision is elastic.

A diver having a moment of inertia of 6.0 kg m2 about an axis through its centre of mass rotates at an angular speed of 2 rad s-1 about this axis. If he folds his hands and feet to decrease the moment of inertia to 5.0 kg m2 what will be the new angular speed(rad s-1)?

Important Questions on Rotational Mechanics

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A diver having a moment of inertia of $6.0 kg m^{2}$ about an axis through its centre of mass rotates at an angular speed of $2 rad s^{- 1}$ about this axis. If he folds his hands and feet to decrease the moment of inertia to $5.0 kg m^{2}$ what will be the new angular speed $(rad s^{- 1})$ ?