# Self-studier w/ kleppner & kolenkow question

1. May 18, 2010

### 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 [Broken]) 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

Last edited by a moderator: May 4, 2017