A Ball Rolling Uphill. ?A bowling ball rolls without slipping up a ramp that slopes upward at an angle ? to the horizontal (see Example 10.7 in Section 10.3). Treat the ball as a uniform solid sphere, ignoring the finger holes. (a) Draw the free-body diagram for the ball. Explain why the friction force must be directed uphill. ?(b) What is the acceleration of the center of mass of the ball? (c) What minimum coefficient of static friction is needed to prevent slipping?
Solution 26E Step 1: a) Freebody diagram for the ball rolling down on a slope, Frictional force will be opposing the motion of the ball. So, the ball will move downward due to the influence of gravitational force. So, the frictional force will oppose this motion. Step 2: b) torque due to the frictional force, = f R f - frictional force, R - radius of the sphere We know that, = I Where, I - moment of inertia of the solid sphere, - angular acceleration So, we can write, f R = I Moment of inertia of a solid sphere through its center, I = MR 2 f R = MR So, we can write, f = MR So, we know that, R = linear acceleration of center of mass = a CM f = MaCM
