Answer to Question #158362 in Classical Mechanics for nasia

Question

A particle of mass m is attached to the end of a string and moves in a circle of radius r on a frictionless horizontal table. The string passes through a frictionless hole in the table and, initially, the other end is fixed. a) if the string is pulled so that the radius of the circular orbit decreases, how does the angular velocity change if it is ω0 when r = r0? b) what work is done when the particle is pulled slowly in from a radius r0 to a radius r0/2?

Answer 1

Answer

(a)The law of conservation of an angular momentum says

$I_0\omega_0=I\omega\\ mr_0^2\omega_0=mr^2\omega\\ r_0^2\omega_0=r^2\omega$

If the radius of the circular orbit decreases, then angular velocity increases.

(b)

The work done

$W=ΔK= \\ ​ ​ W=\frac{m(r_0/2)^2(4\omega_0)^2}{2}-\frac{m r_0 ^2 \omega_0 ^2}{2} ​ ​ ​ ​ ​ ​ = \frac{3mr_0^2\omega^2}{2} ​ ​$