Angular Momentum: Mass Rotating About a Post

AI Thread Summary
In the discussion about angular momentum involving a mass rotating around a post, two cases are considered: one where the string passes through a hole at the center of the post and another where the string wraps around the post. In case (a), angular momentum is conserved due to the central tension force, leading to the conclusion that the final speed can be calculated using the relationship v0r = vfR. In case (b), the tension is not radial, introducing torque and resulting in the non-conservation of angular momentum, linear momentum, and energy. The participants explore the implications of work done on the mass in both scenarios, ultimately concluding that in case (b), the mass's velocity remains unchanged since no work is done. This analysis highlights the differences in energy conservation between the two cases.
Sam_Goldberg
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Homework Statement



This question is from Kleppner and Kolenkow's mechanics book, problem number 6.13:

Mass m is attached to a post of radius R by a string. Initially it is distance r from the center of the post and is moving tangentially with speed v0. In case (a) the string passes through a hole in the center of the post at the top. The string is gradually shortened by drawing it through the hole. In case (b) the string wraps around the outside of the post.

What quantities are conserved in each case? Find the final speed of the mass when it hits the post for each case.

Homework Equations





The Attempt at a Solution



Okay, in part (a), I believe I have the answer, but I would like to make sure. Since the rope is being pulled through a hole at the center of the post, the tension force is central (radially inward). Therefore the torque about the center of the post must be zero; from there, we may conclude that angular momentum (mvr) is conserved. Thus, v0r = vfR and the problem is solved.

Part (b), however, is different. Since the string is wrapping around the loop, the tension is not radial; thus, there is a torque on the mass about the center of the post, and angular momentum is not conserved. Clearly neither linear momentum nor energy are conserved. From here, I am not sure where to go. I could use the work-kinetic energy theorem, but this would involve a line integral about a path that I do not know...

Any help on this problem would be greatly appreciated. Thanks.
 
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Sam_Goldberg said:
Okay, in part (a), I believe I have the answer, but I would like to make sure. Since the rope is being pulled through a hole at the center of the post, the tension force is central (radially inward). Therefore the torque about the center of the post must be zero; from there, we may conclude that angular momentum (mvr) is conserved. Thus, v0r = vfR and the problem is solved.

Part (b), however, is different. Since the string is wrapping around the loop, the tension is not radial; thus, there is a torque on the mass about the center of the post, and angular momentum is not conserved. Clearly neither linear momentum nor energy are conserved. From here, I am not sure where to go. I could use the work-kinetic energy theorem, but this would involve a line integral about a path that I do not know...

Hi Sam_Goldberg! :smile:

(a) is ok (though you haven't mentioned why energy is not conserved)

(b): again, is energy conserved?

Hint: what is the work done ? :wink:
 
Well, in part (a), the force is always radially inward and the ball moves in such a way that it does have a radially inward component of motion, so F dot dr is nonzero and work is done on it. That's why energy is not conserved. In part (b), however, I get the feeling you're trying to tell me that the motion is always perpendicular to the rope. I see how this is initially true (when you stretch out the rope and give the ball its initial boosh to give it the speed v0). Yet, I do not see how this can continue...

Wait a minute: if the mass had a component of motion along the rope, then I bet the rope would collapse. In part (a), an external person was coming in and shortening the length of the rope, thus allowing the mass to have a component of motion along the rope. But in part (b), no one is doing this. Does this reasoning show that the mass in part (b) cannot have an F dot dr, and thus no work is done? Then the velocity would never change, of course. Is this correct?
 
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Sam_Goldberg said:
Wait a minute: if the mass had a component of motion along the rope, then I bet the rope would collapse. In part (a), an external person was coming in and shortening the length of the rope, thus allowing the mass to have a component of motion along the rope. But in part (b), no one is doing this. Does this reasoning show that the mass in part (b) cannot have an F dot dr, and thus no work is done? Then the velocity would never change, of course. Is this correct?

:biggrin: Woohoo! :biggrin:

Yes, instantaneously, the velocity is always perpendicular to the rope. :smile:
 
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