The torque that produces the angular acceleration of the particle is = rF, and is directed out of
the page. From the linear version of Newton’s Second Law, we know that F = ma or F = m r. If
we multiply both sides of this equation by r, we find:
If we compare this equation to the rotational version of Newton’s Second Law, we see that the
moment of inertia of our particle must be mr^2.
MOMENT OF INERTIA FOR RIGID BODIES
Consider a wheel, where every particle in the wheel moves around the axis of rotation. The net
torque on the wheel is the sum of the torques exerted on each particle in the wheel. In its most
general form, the rotational version of Newton’s Second Law takes into account the moment of
inertia of each individual particle in a rotating system:
Of course, adding up the radius and mass of every particle in a system is very tiresome unless the
system consists of only two or three particles. The moment of inertia for more complex systems
can only be determined using calculus. SAT II Physics doesn’t expect you to know calculus, so it
will give you the moment of inertia for a complex body whenever the need arises. For your own
reference, however, here is the moment of inertia for a few common shapes.
In these figures, M is the mass of the rigid body, R is the radius of round bodies, and L is the
distance on a rod between the axis of rotation and the end of the rod. Note that the moment of
inertia depends on the shape and mass of the rigid body, as well as on its axis of rotation, and that
for most objects, the moment of inertia is a multiple of MR^2.
EXAMPLE 1