Introduction to Classical Mechanics · Problem 3.35

Using F = ma — Morin

Physics — Mechanics · friction_on_sphere

Problem

Leaving the sphere ** A small mass rests on top of a fixed sphere of radius

R

. The coefficient of friction is

\mu

. The mass is given a sideways kick that produces an initial angular speed

\omega_0

. Let

\theta

be the angle down from the top of the sphere. In terms of

\theta

and its derivatives, what is the tangential

F = ma

equation? Depending on the value of

\omega_0

, the mass either comes to rest on the sphere or flies off it. If

g = 10

m/s

^2

R = 1

m, and

\mu = 1

, write a program to numerically determine the minimum

\omega_0

for which the mass leaves the sphere. For this cutoff case, give the angle at which the mass loses contact, and describe (roughly) what the plot of

\dot{\theta}

versus

\theta

looks like. See Prior and Mele (2007) for the exact solution for

\dot{\theta}

in terms of

\theta