


Kinetic Energy
There is a certain amount of energy associated
with the rotational motion of a body, so that a ball rolling down
a hill does not accelerate in quite the same way as a block sliding
down a frictionless slope. Fortunately, the formula for rotational
kinetic energy, much like the formula for translational kinetic
energy, can be a valuable problemsolving tool.
The kinetic energy of a rotating rigid body is:
Considering that I is the
rotational equivalent for mass and is the rotational equivalent for velocity,
this equation should come as no surprise.
An object, such as a pool ball, that is spinning as it
travels through space, will have both rotational and translational
kinetic energy:
In this formula, M is the
total mass of the rigid body and is the velocity of its center of mass.
This equation comes up most frequently in problems involving
a rigid body that is rolling along a surface without sliding. Unlike
a body sliding along a surface, there is no kinetic friction to
slow the body’s motion. Rather, there is static friction as each
point of the rolling body makes contact with the surface, but this
static friction does no work on the rolling object and dissipates
no energy.
Example

Because the wheel loses no energy to friction,
we can apply the law of conservation of mechanical energy. The change
in the wheel’s potential energy is –mgh.
The change in the wheel’s kinetic energy is . Applying conservation of mechanical
energy:
It’s worth remembering that an object rolling down an
incline will pick up speed more slowly than an object sliding down
a frictionless incline. Rolling objects pick up speed more slowly
because only some of the kinetic energy they gain is converted into
translational motion, while the rest is converted into rotational
motion.
