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 = ½mv² where v is the object's speed. Kinetic energy is a scalar quantity measured in joules where 1 J = 1 kg m²/sec². 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.

 

image courtesy of The New York times Health|Science, June 5th, 2008

 

To calculate an object's rotational kinetic energy, you must know the following properties of the object:

 

 

The first three properties allow you to determine the object's moment of inertia, I. For a solid cylinder or disk, I = ½mr². For a hoop, where all of the mass is located along its rim, the moment of inertia is I = mr². 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

           = ½mv² + ½Iw²  

both of these energies are measured in Joules

 

 

Before we leave the relationship v = r w, notice that different positions on a rotating surface have different linear velocities even though they all have the same angular velocity. This is because the circumference of the circle through which a point has to travel differs with its radial distance from the axis of rotation.

 

 

The pony located at the greatest radial distance travels a circumference of 2pR with each revolution; while the pony nearest the center might only travel through a circumference of 2p(1/6)R.

 

 

If you appreciate the sensation of speed while on a merry-go-round, sit at a position nearest to edge of the platform.

 

image courtesy of OCModShop.com

 

At this junction, we will always be referencing the tangential velocity and the angular velocity - that is, we will use the radial distance to always be the entire radius.

 

 

 

Conservation of energy states:

thus, for an incline that is 50 cm tall, the block would have a velocity of

 

 

when it reaches the bottom of the incline. Now suppose a metal ball bearing is released from rest and rolls down the same incline.

 

 

Conservation of energy now states:

 

 

Notice, that a new energy term has been added to the equation, KE_rot. This represents the energy that is found in the bearing's rotational motion.

 

The moment of inertia of a solid sphere is given by the formula I = 2/5(mr²). Substituting in this expression for I into our equation for rotational kinetic energy yields:

 

 

Our conservation of energy equation for a rolling ball bearing, or solid sphere, released from rest now becomes:

 

 

For our 50-cm tall incline, the ball bearing would arrive at the base of the incline traveling at

 

 

 

The ball bearing is not translating as quickly when it reaches the bottom of the incline since some of its potential energy (PE) is now in the form of rotational kinetic energy (KE_rot) as well as translational kinetic energy (KE). Consequently, rolling objects will not attain as high a final velocity as those objects which just slide.