A hollow sphere (I = MR2) rolls without slipping along a flat surface then up a ramp. How far up the ramp, d, does it get before turning around? 

Challenge: In terms of the mass, m, and radius, R, of the sphere, what is the minimum coefficient of static friction between the slope and the sphere that would allow the sphere to roll the entire way up without slipping? 

(a) Using kinematics, what is the linear acceleration of the ball?

(b) What, then, is its angular acceleration? 

(c) What is the torque due to friction? 

(d) What, then, is the force due to friction? 

(e) Finally, what is the minimal coefficient of static friction? 

- 8 m/s 30