A man stands on a rotating platform, with his arms stretched...

A man stands on a rotating platform, with his arms stretched horizontally holding a $5 \mathrm{~kg}$ weight in each hand. The angular speed of the platform is 30 revolutions per minute. The man then brings his arms close to his body with the distance of each weight from the axis changing from $90 \mathrm{~cm}$ to $20 \mathrm{~cm} .$ The moment of inertia of the man together with the platform may be taken to be constant and equal to $7.6 \mathrm{~kg} \mathrm{~m}^{2}$. (a) What is his new angular speed? (Neglect friction.) (b) Is kinetic energy conserved in the process? If not, from where does the change come about?

A man stands on a rotating platform, with his arms stretched horizontally holding a $5 \mathrm{~kg}$ weight in each hand. The angular speed of the platform is 30 revolutions per minute. The man then brings his arms close to his body with the distance of each weight from the axis changing from $90 \mathrm{~cm}$ to $20 \mathrm{~cm} .$ The moment of inertia of the man together with the platform may be taken to be constant and equal to $7.6 \mathrm{~kg} \mathrm{~m}^{2}$.
(a) What is his new angular speed? (Neglect friction.)
(b) Is kinetic energy conserved in the process? If not, from where does the change come about?

Key Concepts

Rotational Kinetic Energy

Rotational kinetic energy is the energy due to the rotation of an object and is calculated as one-half times the moment of inertia times the square of the angular velocity. Altering the moment of inertia and angular speed in a rotating system can lead to changes in its kinetic energy, though angular momentum is conserved.

Work Done by Internal Forces

When an object reconfigures its mass distribution, such as pulling masses inward, internal forces (like muscular force) do work on the system. This work can result in an increase or decrease in the system's kinetic energy, even though external torques are absent and angular momentum remains conserved.

Conservation of Angular Momentum

This principle states that in the absence of external torques, the total angular momentum of a system remains constant. It is fundamental in rotational dynamics and explains why a decrease in moment of inertia leads to an increase in angular speed when nothing from the outside adds or removes angular momentum.

Moment of Inertia

The moment of inertia measures an object's resistance to changes in its rotational motion and depends on the distribution of mass relative to the axis of rotation. When the mass distribution changes, such as by moving masses closer to or farther from the rotation axis, the moment of inertia changes, directly affecting the rotational behavior of the system.

Transcript

00:04 In this problem, a person is standing on a rotating disk and he is carrying two masses of 5 kg each in each hand.

00:14 And at the time of stretched hand, the masses are at a distance of 90 centimetre from the axis of rotation.

00:23 So this distance is 90 centimeter.

00:25 Then what he does that he brings his arm closure in such a way that masses are now at a distance of 30.

00:34 At a distance of 20 cm from the axis of rotation.

00:37 So we have to find what will be the new angular speed.

00:41 If initial angular speed, omega was 30 revolution per minute.

00:48 All right? so we have to find new angular speed.

00:52 First of all, we will write what are the given quantities here.

00:57 So it is given that mass of each block is 5 kg.

01:03 Initially, it was, let us say this is situation one.

01:06 And this is situation 2 so at situation 1 its speed was 30 revolution per minute so omega 1 is 30 revolution per minute and initial distance was r1 is equals to 90 centimeter which is 0 .9 meter then when he close he when he closes his arm so at the final condition the distance is 20 centimeter so in meter we can write it as 0 .2 meter we have to find what will be the value of final angular speed now the person and the rotating disk has a combined moment of inertia i see so this c is indicating combined is 7 .6 kg meter square so first we will write what is the what is the total moment of inertia of this entire system considering the persian disk and both the masses so initial moment of inertia i i i will be equal to or i1 we can say here because this is the situation one so i1 will be equal to the combined mass moment of inertia plus mass moment of inertia because of the two blocks so both are at a at equal distance so we we can directly multiply by 2 with the moment of energy of one block and moment of inertia due to one block will be m into r1 square now ic is 7 .6 plus 2 into mass m is 5 kg and r1 is 0 .9 meter so i1 comes out to be i1 equals to 15 .7 kg meter square so we have got the value of i1 now we have to find what will be the total moment of inertia in case 2.

03:20 So i2 will be equal to combined mass of combined mass moment of energy plus 2 times m r2 square...

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