Question

A bowling ball rolls without slipping up a ramp that slopes upward at an angle β to the horizontal. Treat the ball as a uniform, solid sphere, ignoring the finger holes.

A) In order for the rotation of the ball to slow down as it rolls uphill, in which direction must the frictional force point?

B) Compared to a frictionless surface, does the ball roll farther uphill, the same distance, or not as far? Please explain.

C) What is the acceleration of the center of mass of the ball?

D) What minimum coefficient of static friction is needed to prevent slipping?

Answer 1

A)

For the ball to slow down, the rotations must slow down and for this the friction must be directed upwards so that it produces an opposite torque to slow it down.

B)

Compared to frictionless surface, the ball rolls farther uphill in this case with friction.

This can be explained using conservation of energy.

In the case with friction, all the total energy at the bottom is converted to gravitational PE at the top the hill. But in the case with no friction, all the total energy is not converted to gravitational PE as some of it remains as rotational KE (as not retarding torque is applied )

C)

-f + mg*sin() = ma

Now, f*R = (2/5)*mR^2*(a/R)

So, f = 2/5*ma

So, -(2/5)*ma + mg*sin() = ma

So, a = g*sin/(1 + 2/5) = 5*g*sin/7 <-------- answer

d)

For slipping, f = u*mg*cos = (2/5)*ma = (2/5)*m*(5*g*sin)/7

So, u*cos = (2/7)*sin

So, u = (2/7)*tan <-------- answer