Pulling Strings

A mass m is attached to the end of a string. The mass moves on a frictionless table, and the string passes through a hole in the table (see Figure 1.1), under which someone is pulling on the string to make it taut at all times. Initially, the mass moves in a circle, with kinetic energy E₀. The string is then slowly pulled, until the radius of the circle is halved. How much work was done?

Figure 1.1

SOLUTION

In order to keep the mass traveling in a circular orbit of radius r, you must apply a force F equal to the mass times its centripetal acceleration v²/r

 Figure 1.2

(see Figure 1.2). Pulling on the rope exerts no torque on the rotating mass, so the angular momentum l = mvr is conserved. Therefore

Then the work W necessary to move the mass from its initial orbit of radius R to its final orbit of radius R/2 is

Solving in terms of E₀: