Self-studier w/ kleppner & kolenkow question

[pton265]

Hi,
This is my first post. I'm reviewing mechanics out of K&K and have a question about problem 4.5:

"Mass m whirls on a frictionless table, held to a circular motion by a string which passes through a hole in the table. The string is slowly pulled through the hole so that the radius of the circle changes from l₁ to l₂. Show that the work done in pulling the string equals the increase in kinetic energy of the mass."

I'm assuming the mass starts with uniform circular motion at radius l₁, and I analyze in polar coordinates with the center of this circle as the origin. My initial intuition about the motion:

(1) Angular acceleration is non-zero (positive), but there cannot be a $$\widehat{}\theta$$ component of acceleration (the force is always radial) - which is only true if
2$$\dot{}r$$$$\dot{}\theta$$ = -r$$\ddot{}\theta$$.

(2) The only way for the string to do work (increase the magnitude of m's velocity) is if m's trajectory (and, therefore, velocity) has some radial component. That is, the force must at some point have a non-orthogonal component with respect to the trajectory. Since the force is everywhere radial, the $$\widehat{}\theta$$ component of velocity is unchanged, while the radial component of velocity increases (in the negative $$\widehat{}r$$ direction). This statement does not contradict (1), where I state angular acceleration is non-zero (right?!). Physically, all of this corresponds to the mass breaking from uniform circular motion and spiraling inward toward the center of the table (i.e. where the hole is). When it reaches l₂, it will NOT be in uniform circular motion because it's velocity must have some radial component (inward).

Now, the only solution (http://hep.ucsb.edu/courses/ph21/problems/p7sol.pdf ) I've found takes (1) to be true, but (2) to be false. The final velocity in the solution has a magnitude such that the velocity can not possibly have a radial component. In other words, once the mass reaches l₂, it pops back into uniform circular motion (albeit, with higher velocity). How can the trajectory take this form (i.e. that of consecutively smaller concentric circles)?? Is the solution wrong? If not, where is my error?

Please bear in mind that the only assumed knowledge at this point is of translational motion, linear momentum, and the Work-energy theorem in one dimension (KK chs.1-4) - not angular momentum, rotational motion, etc.

My apologies if there is already a similar thread - I've searched PF pretty thoroughly.

Thanks very much for any and all help...

L

 

[anathema]

Hi L,

Thank you for your question and for providing such thorough analysis of the problem. Your intuition about the motion is correct - the mass will indeed break from uniform circular motion and spiral inward towards the center of the table as the string is pulled through the hole.

The solution you found may have overlooked the fact that the velocity of the mass must have a radial component in order for the string to do work and increase the kinetic energy. As you mentioned, this is due to the fact that the force is always radial and cannot have a non-orthogonal component with respect to the trajectory.

It is possible that the solution you found was for a different scenario where the string is pulled at a constant rate, resulting in the mass maintaining uniform circular motion. However, in the scenario described in the problem, the string is being pulled slowly, allowing for a change in velocity and a non-uniform circular motion.

I would suggest double-checking your calculations and making sure that you are considering the radial component of velocity in your analysis. If you are still having trouble, I would recommend seeking help from a professor or classmate who may have a better understanding of the problem.

I hope this helps and good luck with your review! Keep up the good work.