Question

If all of a sudden the radius of the earth decreases, which one of the following statements will be true?
(A) The angular momentum of the earth will become greater than that of the sun
(B) The periodic time of the earth will increase
(C) The energy and angular momentum will remain constant
(D) The angular velocity of the earth will increase

Answer 1

Earth revolves in orbit, so it gains some amount of angular momentum. Here, we can use the concept of angular momentum. As we know that the angular momentum is conserved, so its value becomes constant, and with the help of expression of the angular momentum, we can determine the correct answer.

Complete step by step answer:
When any object rotates in the circular path with angular velocity, it will gain the angular momentum. This momentum does not change during its motion and is considered as a vector quantity. On the two quantities, angular momentum depends, one is angular speed, and another is the object's inertia moment.

Suppose the radius of the earth decreases suddenly. In that case, the momentum of inertia of the earth will decrease because the earth’s moment of inertia is directly related to the earth's radius. We know that the angular momentum's value does not change and is directly related to inertia's angular velocity and momentum. If anyone's quantity among angular velocity and momentum of inertia decreases, another one will increase. Hence, due to the decrease in the earth's radius, the momentum of inertia of the earth decreases, and decrement in inertia's momentum will increase angular velocity.

Therefore, if all of a sudden, the radius of the earth decreases, then the angular velocity of the earth will increase

So, the correct answer is “Option D”.

Note:
We can also determine the correct answer with the help of the expression of the angular momentum and momentum of inertia. Remember these expressions because they relate the angular momentum, moment of inertia with the other quantities, which may get affected due to variation in inertia momentum.