A hockey puck of mass m slides on a horizontal frictionless plane. A massless st

Question

A hockey puck of mass m slides on a horizontal frictionless plane. A massless string connects it to a small hole at the origin, and its distance from the origin can be controlled by pulling the string down through the hole. The puck is set into circular motion about the origin with angular momentum L0. Write an expiration V(r) for its linear speed as a function of its distance r from the origin (and the constants L0 and m). Why is the angular momentum unchanged when the string is pulled down? Express the kinetic energy T = ½ m u2 as a function of r (and the constants L0 and m). Show that dt/dr = -L20/mr3 Then dT = F dr F = - L20/mr3 er. Verify that F is the centripetal force needed to maintain circular motion at each value of r.

Explanation / Answer

a) L = I

I = mr^2

= v/r

L = mr^2 v/r

L = m v r

v = L/mr

b) since the forces is perpindicular to the circle (i.e. radial) it won't change the angular momentum

c) T = 1/2 m v ^2 = 1/2 m (L/m r)^2 = 1/2 m (L^2/ m^2 r^2) = L^2/(2 mr^2)

d)

dT/dr = -2 L^2/(2m) r^(-3) = - L^2/(mr^3)

e) if centripetal force

F =- mv^2/r = -m (L/mr)^2 /r = - L^2/(mr^3) so it does match for all values of r.

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