Question

What happens if a gymnast, sitting on a rotating stool with his arms outstretched, suddenly lowers his hands?
A. The angular velocity decreases
B. his moment of inertia decreases
C. the angular velocity stays constant
D. the angular momentum increases

Answer 1

Hint: Conservation of angular momentum is the basis for solving out this question. The law of conservation of angular momentum states that as long as no external torque acts on a body, there will be no variation of angular momentum taking place. Using this concept, we can solve this question.

Complete step by step solution:
First of all, let us take a look at the angular momentum of a particle or a body. Angular momentum is defined as the rotational equivalent of linear momentum. It is a prime quantity in physics as it is a conserved quantity.
Here the net torque on him is very close to zero as there is relatively smaller friction between his legs and the stool. And the friction is being exerted very close to the pivot point. As a result, he can spin for a long time. He can also increase his rate of spin by pulling his arms in.
As the angular momentum is given by the equation,
$L=I\omega $
Where $L$ the angular momentum,$\omega $ the angular velocity, and $I$ be the moment of inertia.
From this, we can see that as the hands are lowered the radius of gyration is getting reduced which will further reduce the moment of inertia. According to the equation mentioned above, as the moment of inertia is decreasing, the angular velocity will get increased.
  Therefore the correct answer is option B.

Note: The orbital angular momentum vector will always be parallel as well as directly proportional to the orbital angular velocity vector $\omega $ of the object, where the proportionality constant depends on the mass of the object and the distance from the origin. Angular momentum is a vector quantity. It is having both magnitude and direction along the axis.