Resource Lesson
Rotational Kinetic Energy
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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.
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:
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 = r
w
. 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}
+ ½
I
w
^{2}
both of these energies are measured in Joules
Refer to the following information for the next three questions.
Assume each wheel on the bike that Einstein is riding in the picture shown above has a mass of 1.75 kg, a radius of 0.28 meters, and a moment of inertia of 0.8mr
^{2}
.
If Einstein is traveling at 10 m/sec, what is the rotational kinetic energy of each of the wheels on his bike?
What is the translational kinetic energy of each wheel?
What is the total KE of each wheel?
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 2
p
R with each revolution; while the pony nearest the center might only travel through a circumference of 2
p
(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.
Now let's look at another example of translational and rotational kinetic energies. If a block slides (without friction) from rest down an incline, that block is said to only be translating.
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(mr
^{2}
). 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.
Refer to the following information for the next question.
Once again, assume each wheel on the bike that Einstein is riding in the picture shown above has a mass of 1.75 kg, a radius of 0.28 meters, and a moment of inertia of 0.8mr
^{2}
.
If Einstein's bike were to glide down a 15-meter tall hill, what would be the velocity of each wheel at the base of the hill?
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