Question

"a solid ball of inertia m rolls without slipping down a ramp that makes an angle θ with the horizontal."

Answer 1

A solid ball of inertia m rolls without slipping down a ramp that makes an angle θ with the horizontal. 

1. What frictional force is exerted on the ball? (Express your answer in terms of some or all of the variables m, θ, and the acceleration due to gravity g.) 

2. As a function of θ, what coefficient of friction is required to prevent slipping? (Express your answer in terms of θ.)

ANSWERS:-

1) 
m a = m g sinθ - f [translation] 
I a / r = f r [rotation] 
then 
f = m g sinθ / (1 + m r^2 / I) 
I = 2/5 m r^2 
then 
f = 5/7 m g sinθ 

2) 
must be 
f <= μ m g cosθ 
5/7 m g sinθ <= μ m g cosθ 
μ > = 5/7 tanθ