The rotational energy of an object can be distinguished from the translational kinetic energy. If you lift the front wheel of your bicycle and apply a force to the tire in a way that it begins to rotate, you have done work. But the bicycle has no momentum because the sum of the forces is ZERO, that is your pushing on the wheel is balanced by your holding the bicycle back. The work does not appear as translational kinetic energy of the bicycle, rather the motion of the wheel can be viewed as motion of the many separate parts of the wheel.
It is quite convincing that a rotating barbell has energy, in fact Kinetic Energy, associated with it. We will convince ourselves of this in two ways: 1) by finding the of each ball, and 2) by finding the work necessary to arrest the rotation.
1) We'll calculate the total of the two balls assuming that the mass of the rod connecting them is negligible. For each ball of mass , moving with velocity of magnitude , . Therefore, quite simply, the total Kinetic Energy is . Note that the momentum of the barbell is ZERO.
2) One way to arrest the motion of the rotating barbell is to apply a force opposite the velocity of each ball. This force is not in a constant direction since the velocity is not in a constant direction. But it is possible to apply a steady force resulting in a constant rate of decrease of the magnitude of the velocity. The acceleration is
and the time it takes to reduce the magnitude of the velocity to ZERO is
During this time interval, the work done by the force is
The total work done to slow down the two barbells is twice this or
The kinetic energy of the balls of the barbell that rotates with total momentum ZERO is termed Rotational Kinetic Energy in order to contrast with Translational Kinetic Energy. The work done on a system can change rotational
, and it is necessary to take this into account in many situations.
Let's watch a race down an incline between two equal mass rolling cylinders. They start from the same height
above the bottom of the incline. We can consider either the work done by gravity as the cylinders descend, or we can use the gravitational
. Either approach produces the same result, of course. The total energy of each cylinder is
, a constant. If friction is not important, that total kinetic energy at the bottom of the incline is also
, and it is made up of translational
. If the two cylinders differ in the distribution of mass, the rotational
will generally be different and the one with more translational
at the bottom will win the race. In the rest of this lecture, we'll learn what is necessary to understand this outcome.
Fig 2 A rolling race down an incline of two equal mass cylinders. The total energy is the same for each, and the winner is the one with less rotational
(i.e. more translational
) at the bottom.