A Classical Approach of Newtonian Mechanics

8 ROTATIONAL MOTION 8.11 Combined translational and rotational motion







b v^

cylinder

surface

f

since the perpendicular distance between the line of action of f and the axis

of rotation is the radius, b, of the cylinder. Note that the friction force acts to

decelerate the cylinder’s rotational motion. If the cylinder is slipping with respect

to the surface, then the friction force, f, is equal to the coefficient of friction, μ,

times the normal reaction, M g, at the surface:

f = μ M g.

Finally, the moment of inertia of the cylinder is

I =

1

M b^2.

2

The above equations can be solved to give

̇v = μ g,

b ω ̇ = − 2 μ g.

Given that v = 0 (i.e., the cylinder is initially at rest) and ω = ω 0 at time t = 0 ,

the above expressions can be integrated to give

v = μ g t,

b ω = b ω 0 − 2 μ g t,

which yields

v − b ω = −(b ω 0 − 3 μ g t).

Now, the cylinder stops slipping as soon as the “no slip” condition,

v = b ω,