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I copied this question from my textbook.

An object is being rotated in the vertical plane with a rope of length 90cm by a circus man. And the object rotates 100 times per minute. Suddenly the object's mass becomes 2/3 of the previous mass. So, the man increases the length of the rope. Suppose the number of rotations remains the same. Find the final length of the rope?

The textbook solution says that the centripetal force will remain the same and found the final length. But my question is how can they conclude if angular velocity remains the same, centripetal force will not change?

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There is no requirement for the centripetal force to be the same in both cases.

I think the problem wants you to assume that the clown wants to exert the same force before and after for the simple reason that the clown just wants it so -- perhaps his arm is not strong enough or the string will break if the force gets any bigger.

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We need three relations for this:

$$v=\omega r\qquad\qquad a_c=\frac{v^2}{r} \qquad\qquad F_c=ma_c$$

The first is the geometric bond that ties angular velocity $\omega$ (which you want to keep constant) to speed $v$ and string length $r$. The second is the tie between this speed and both string length and centripetal acceleration $a_c$. The third is Newton's 2nd law where we in this case have only the centripetal force $F_c$ pulling inwards and causing the centripetal acceleration.

Merge together (input the first and third in the second):

$$\frac{F_c}m=\frac{(\omega r)^2}{r}\quad\Leftrightarrow\quad \frac{F_c}{r}=m\omega^2$$

Your task in the question is to keep $\omega$ unchanged when $m$ reduces. As you can see that is possible by simultaneously

  • increasing the string length $r$ in the same ratio or
  • decreasing $F_c$ in the same ratio.

(Or a mix of both.) You may even do an easy home-experiment to check this intuitively: Swing around something heavy and something light - the lighter object requires less force of you because you use the same string length.

So the answer sheet you have been given is not entirely correct. The clown should be able to adjust the force he pulls with in the string. If he doesn't want to do that (possibly the force becomes so small now that other issues become involved) then he must elongate the string. But there's no inherent requirement of centripetal force being constant when angular velocity is constant.

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