When an object slides across a surface its center of mass is said to "translate." That is, its center of mass moves from position A to position B through a distance. When asked to calculate the magnitude of a moving object's translational kinetic energy, you use the formula KE = ½mv^{2} where **v** is the object's speed. Kinetic energy is a scalar quantity measured in joules where 1 J = 1 kg m^{2}/sec^{2}. In the following diagram, all four objects would have exactly the same kinetic energy if they all have the same mass and are moving at the same speed. The direction of the velocity vector does not affect your answer. Sometimes an object is in a state of pure rotation. That is, its center of mass does not translate, but the object rotates about a central axis. For example, a stationary exercise bike has a wheel which rotates as the rider pedals. The bike does not move, but the wheel spins on its axis. To calculate an object's rotational kinetic energy, you must know the following properties of the object:
- its mass
- its radius
- how its mass is distributed about its axis of rotation, and
- how fast it is rotating
The first three properties allow you to determine the object's moment of inertia, I. For a solid cylinder or disk, I = ½mr^{2}. For a hoop, where all of the mass is located along its rim, the moment of inertia is I = mr^{2}. The higher the wheel's moment of inertia, the harder it is to start the wheel rotation and, subsequently, the harder it is to stop the wheel's rotation. For this reason, many exercise bikes use flywheels, or very massive metal disks, like the one pictured above. An object's rotational inertia is a measure of its resistance to a change in its state of rotation. If instead of riding a stationary bike, someone rides a bicycle down a driveway, the wheels on the bike are rotating as well as translating. That is, the center of mass of each wheel is moving through a rectilinear distance while simultaneously each wheel is spinning on its axis. To calculate the rate at which a wheel rotates, called omega or w, we have to use the relationship the v = rw. Where **v** is the linear velocity of the center of mass in meters/sec,** r** is the radius of the wheel in meters, and **w** is the rate of rotation in radians/sec. To calculate the wheel's total kinetic energy, you would use the formula KE_{total} = KE_{linear} + KE_{rotational} = ½mv^{2} + ½Iw^{2} both of these energies are measured in Joules |