![]() |
Figure 1. A bicycle wheel rolls down an inclined board. |
A bicycle wheel rolls down an inclined board. We say that it is translating and rotating. As the bicycle wheel rolls, the velocity of its center of mass will increase. If it rolls without slipping, then its angular velocity will also increase. Since the object is rotating and translating, it will have rotational kinetic energy as well as translational kinetic energy. What effect does the rotational kinetic energy have on the total energy in this situation? Will the total energy be conserved if we ignore the rotational kinetic energy?
Figure 2 shows a diagram of this experiment. As the bicycle wheel travels down the board, its center of mass will have an acceleration, a. If it rolls without slipping, then the wheel will also have an angular acceleration, a. This means that both the translational and angular velocities will be changing. Since translational and rotational kinetic energies depend on these velocities, then they will both be changing too.
When an object is translating and rotating we can write the total mechanical energy as:
TE = KE + RE + PE
Here, KE is the translational kinetic energy of the center of mass, RE is the rotational kinetic energy relative to the center of mass and PE is the gravitational potential energy. These are written:
KE = 0.5mv2
RE = 0.5Icmw2
PE = mgh
The change in total energy is:
DTE = DKE + DRE + DPE
If the change in total energy is zero, then it is conserved and:
0 = DKE + DRE + DPE or -DPE = DKE + DRE
This means that a change in potential energy results in an opposite change in the sum of the translational and rotational kinetic energies.