A Classical Approach of Newtonian Mechanics

8 ROTATIONAL MOTION 8.7 Torque

The moment of inertia of a solid cylinder of mass M and radius b about the

cylindrical axis is

I =

1

M b^2.

2

The moment of inertia of a thin spherical shell of mass M and radius b about

a diameter is

I =

2

M b^2.

3

The moment of inertia of a solid sphere of mass M and radius b about a

diameter is

I =

2

M b^2.

5

8.7 Torque

We have now identified the rotational equivalent of velocity—namely, angular

velocity—and the rotational equivalent of mass—namely, moment of inertia. But,

what is the rotational equivalent of force?

Consider a bicycle wheel of radius b which is free to rotate around a perpen-
dicular axis passing through its centre. Suppose that we apply a force f, which

is coplanar with the wheel, to a point P lying on its circumference. See Fig. 79.
What is the wheel’s subsequent motion?

Let us choose the origin O of our coordinate system to coincide with the pivot
point of the wheel—i.e., the point of intersection between the wheel and the axis

of rotation. Let r be the position vector of point P, and let θ be the angle sub-
tended between the directions of r and f. We can resolve f into two components—

namely, a component f cos θ which acts radially, and a component f sin θ which
acts tangentially. The radial component of f is canceled out by a reaction at the
pivot, since the wheel is assumed to be mounted in such a manner that it can only
rotate, and is prevented from displacing sideways. The tangential component of

f causes the wheel to accelerate tangentially. Let v be the instantaneous rotation
velocity of the wheel’s circumference. Newton’s second law of motion, applied to