Well, spinning objects also resist changes in their angular velocity. But that resistance, that rotational
inertia, depends less on the mass of an object than on how that mass is distributed. For example, a
baseball player often warms up by placing a weight on the outer end of the bat—this makes the bat more
difficult to swing. But he does not place the weight on the bat handle, because extra weight in the handle
hardly affects the swing at all.
The rotational inertia, I , is the rotational equivalent of mass. It tells how difficult it is for an object to
speed up or slow its rotation. For a single particle of mass m a distance r from the axis of rotation, the
rotational inertia is
To find the rotational inertia of several masses—for example, two weights connected by a thin, light rod
—just add the I due to each mass.
For a complicated, continuous body, like a sphere or a disk, I can be calculated through integration:
Exam tip from an AP Physics veteran:
On the AP exam, you will only very occasionally have to use calculus to derive a rotational inertia.
Usually you will either be able to sum the I due to several masses, or you will be given I for the object
in question.
—Joe, college physics student and Physics C alumnusNewton’s Second Law for Rotation
For linear motion, Newton says F (^) net = ma ; for rotational motion, the analog to Newton’s second law is
where t (^) net is the net torque on an object. Perhaps the most common application of this equation involves
pulleys with mass.
A 2.0-kg block on a smooth table is connected to a hanging 3.0-kg block with a light string. This string
passes over a pulley of mass 0.50 kg, as shown in the diagram below. Determine the acceleration of the