physics question-need answer fast thanks?

Question

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

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


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

Answer 1

If the ball or any other object were sliding, its "effective mass" (defined as force F/acceleration A) is simply mass M. In the case of a ball, the force is applied at radius R from the ball's CM. When rolling, an additional force is needed to accelerate the ball in rotation, thus increasing its effective mass. The moment of inertia I of a solid sphere is 2/5 * MR^2 (ref. 1). A torque T equal to I accelerates the ball's rotation 1 rad/s^2 or R/s^2. The force applied to the radius to generate this torque is T/R. Thus the ball requires a force at the radius of 2/5 * MR to accelerate at R/s^2. Thus the additional effective mass is 2/5 * M and the total effective mass is 1.4 * M. (See ref. 2 for indirect confirmation of this.)
However, the gravitational force remains equal to gM. So the rolling ball on the ramp accelerates at gM * sin(B) / (1.4 * M) or g * sin(B) / 1.4.
The sliding friction is independent of moment of inertia. The normal force is gM * cos(B). The ball will slide if gM * sin(B) > gM * sin(B). Thus the threshold coefficient of friction is sin(B) / cos(B).

Answer 2

1.
Fr=I(a/r)
I=2/5mr^2
F=2/5ma

mg*sinb-f=ma
mg*sinb-2/5ma=ma
g*sinb=7/5a
a=5/7g*sinb

2.
mg*sinb-(us)mg*cosb=m*5/7g*sinb
(us)*cosb=2/7*sinb
us=2/7*tanb