Question

A bowling ball of mass m and radius r has a coefficient of static friction µ when in contact with an oiled wood surface. Model the ball as a solid uniform sphere. The surface is inclined at an angle θ with respect to the horizontal. What is the maximum angle of inclination of the surface such that the ball rolls without slipping? How does the angle change if the ball is now a hollow shell with all its mass concentrated at the rim? Explain

Answer 1

For a ball rolling without slipping the torque provided by friction is the only factor which contributes in angular acceleration.

M = mass of the ball

R = radius of ball

I = moment of inertia of the ball

S = displacement of ball’s CM since it was at rest

y = the vertical distance the ball has descended since it was at rest

This can be formulated below

also

now conserving energy

k = I/(mR^2) because for no-slip

a=R*alpha and

V = R*omega

acceleration is given by a = v^2/(2S) Thus

Now using no-slip condition and torque equations at the top we get

for solid ball k = I/(mR^2) = 2/5

for shell ball k = I/(mR^2) = 2/3

Thus for hollow shell the maximum angle required for no-slip is lesser than that of solid ball.

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