0.30 Phy1300: angular momentum -- rotational kinetic energy and  (Page 3/11)

When I was a youngster, I learned the hard way that a rotating rigid object has rotational kinetic energy. I had my bicycle turned upside down resting on the seat andthe handle bars with the rear wheel turning very fast. I was using a long thin triangular file to chip mud off of the bicycle. I accidentally allowed oneend of the file to come in contact with the tread on the spinning bicycle tire and ended up with a file sticking in the palm of my hand. Although I didn't knowthe technical term for rotational kinetic energy at the time, I did learn what rotational kinetic energy can do.

Calculating rotational kinetic energy

In principle, at least, we could calculate the rotational kinetic energy possessed by that spinning bicycle wheel by

• decomposing it into a very large number of small particles of mass,
• computing the kinetic energy of each particle of mass, and
• computing the sum of the kinetic energy values possessed by all of the particles of mass.

That would be a difficult computation. We need a simpler way to express the rotational kinetic energy of a rotating rigid body.

A simpler way

There is a simpler way that is based on the tangential speed of each particle of mass and theangular velocity of the rotating object.

You should recall that when the angular velocity is expressed in radians per second, the tangential speed of a point onthe circumference of a circle is given by

v = r * w

where

• v represents the tangential speed of the point in meters per second
• r represents the radius of the circle in meters
• w represents the angular velocity in radians per second

Thus, the tangential speed of our hypothetical particle of mass is equal to the product of the distance of that particle from the center of rotation (theaxle on my upturned bicycle) and the angular velocity of the wheel.

Terminology

The convention is to use the Greek letter omega to represent angular velocity, but I decided to use the "w" character because

• your Braille pad probably can't display the Greek letter omega
• the Greek letter omega looks a lot like a lower case "w"

Back to the bicycle wheel

Therefore, if we consider the bicycle wheel to be made up of an extremely largenumber of particles of mass, each located at a fixed distance from the axle, the kinetic energy of each particle would be given by

KEr = (1/2)*m*v^2, or

KEr = (1/2)*m*(r*w)^2, or

KEr = (1/2)*m*(r^2)*(w^2)

where all of the terms in this equation were defined earlier except for

• KEr, which represents rotational kinetic energy.

The total rotational kinetic energy of the bicycle wheel

Then the total rotational kinetic energy of the bicycle wheel would be

KErt = (1/2)*(sum from i=0 to i=N(mi*ri^2))*w^2

where

• KErt represents the total rotational kinetic energy of the wheel
• mi represents the ith mass particle in a set of N mass particles
• ri represents the distance of the ith mass particle from the axis of rotation
• w represents angular velocity in radians per second

Summation

It is conventional to use the Greek letter sigma to represent the sum with subscripts and superscripts providing the limits of the sum. However, since yourBraille display probably won't display the Greek letter sigma with subscripts and superscripts, we will have to settle for something like "sum from i=0 to i=N" to mean the same thing.