Resource Lesson
Rotational Kinetic Energy
Printer Friendly Version
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?
Related Documents
Lab:
Labs -
A Battering Ram
Labs -
A Photoelectric Effect Analogy
Labs -
A Physical Pendulum, The Parallel Axis Theorem and A Bit of Calculus
Labs -
Air Track Collisions
Labs -
Ballistic Pendulum
Labs -
Ballistic Pendulum: Muzzle Velocity
Labs -
Bouncing Steel Spheres
Labs -
Collision Pendulum: Muzzle Velocity
Labs -
Conservation of Energy and Vertical Circles
Labs -
Conservation of Momentum in Two-Dimensions
Labs -
Density of an Unknown Fluid
Labs -
Inelastic Collision - Velocity of a Softball
Labs -
Loop-the-Loop
Labs -
Mass of a Paper Clip
Labs -
Moment of Inertia of a Bicycle Wheel
Labs -
Ramps: Sliding vs Rolling
Labs -
Roller Coaster, Projectile Motion, and Energy
Labs -
Rotational Inertia
Labs -
Rube Goldberg Challenge
Labs -
Spring Carts
Labs -
Target Lab: Ball Bearing Rolling Down an Inclined Plane
Labs -
Video Lab: Blow Dart Collides with a Stationary Cart
Labs -
Video Lab: M&M Colliding with Pop Can
Labs -
Video Lab: Marble Collides with Ballistic Pendulum
Resource Lesson:
RL -
A Chart of Common Moments of Inertia
RL -
A Further Look at Angular Momentum
RL -
APC: Work Notation
RL -
Center of Mass
RL -
Centripetal Acceleration and Angular Motion
RL -
Conservation of Energy and Springs
RL -
Discrete Masses: Center of Mass and Moment of Inertia
RL -
Energy Conservation in Simple Pendulums
RL -
Hinged Board
RL -
Introduction to Angular Momentum
RL -
Mechanical Energy
RL -
Momentum and Energy
RL -
Potential Energy Functions
RL -
Principal of Least Action
RL -
Rolling and Slipping
RL -
Rotary Motion
RL -
Rotational Dynamics: Pivoting Rods
RL -
Rotational Dynamics: Pulleys
RL -
Rotational Dynamics: Rolling Spheres/Cylinders
RL -
Rotational Equilibrium
RL -
Rotational Kinematics
RL -
Springs and Blocks
RL -
Symmetries in Physics
RL -
Tension Cases: Four Special Situations
RL -
Thin Rods: Center of Mass
RL -
Thin Rods: Moment of Inertia
RL -
Torque: An Introduction
RL -
Work
RL -
Work and Energy
Worksheet:
APP -
The Baton Twirler
APP -
The Jogger
APP -
The Pepsi Challenge
APP -
The Pet Rock
APP -
The Pool Game
APP -
The See-Saw Scene
CP -
Center of Gravity
CP -
Conservation of Energy
CP -
Momentum and Energy
CP -
Momentum and Kinetic Energy
CP -
Power Production
CP -
Satellites: Circular and Elliptical
CP -
Torque Beams
CP -
Torque: Cams and Spools
CP -
Work and Energy
NT -
Center of Gravity
NT -
Center of Gravity vs Torque
NT -
Cliffs
NT -
Elliptical Orbits
NT -
Escape Velocity
NT -
Falling Sticks
NT -
Gravitation #2
NT -
Ramps
NT -
Rolling Cans
NT -
Rolling Spool
NT -
Satellite Positions
WS -
Advanced Properties of Freely Falling Bodies #1
WS -
Advanced Properties of Freely Falling Bodies #2
WS -
Advanced Properties of Freely Falling Bodies #3
WS -
Energy Methods: More Practice with Projectiles
WS -
Energy Methods: Projectiles
WS -
Energy/Work Vocabulary
WS -
Force vs Displacement Graphs
WS -
Introduction to Springs
WS -
Kinematics Along With Work/Energy
WS -
Moment Arms
WS -
Moments of Inertia and Angular Momentum
WS -
Potential Energy Functions
WS -
Practice: Momentum and Energy #1
WS -
Practice: Momentum and Energy #2
WS -
Practice: Uniform Circular Motion
WS -
Practice: Vertical Circular Motion
WS -
Rotational Kinetic Energy
WS -
Static Springs: The Basics
WS -
Torque: Rotational Equilibrium Problems
WS -
Work and Energy Practice: An Assortment of Situations
WS -
Work and Energy Practice: Forces at Angles
TB -
Basic Torque Problems
TB -
Center of Mass (Discrete Collections)
TB -
Moment of Inertia (Discrete Collections)
TB -
Rotational Kinematics
TB -
Rotational Kinematics #2
TB -
Work, Power, Kinetic Energy
PhysicsLAB
Copyright © 1997-2015
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton