The Physics Philes, lesson 46: Rotational Motion Gets Kinetic!
Guys! I’m back and ready to science! I hope you missed me.
I’ll start today where I left it before the epic black hole detour. That’s right. We’re back in the domain of rotational motion. Specifically, we’ll discuss how to find the kinetic energy of a rigid, rotating body.
To understand the kinetic energy of rigid rotating body, I need to introduce a new concept: the moment of inertia.
The moment of inertia depends on what the mass of the body is and how it’s distributed. We need to think of a body as being made of up a bunch of different particles, each with a mass and a distance from the axis of rotation. Picture this rigid body is rotating around a fixed axis. We can find the speed v by multiplying the distance from the angular speed, or v = rω. You might remember from a billion years ago that the equation for kinetic energy is 1/2mv^2. You can replace v^2 with r^2ω^2. To get the total kinetic energy of the body, we need to find the sum of the kinetic energy of all the particles involved. Like this:
We use i to indicate all the particles. I don’t know why. It’s just one of those math things. If we factor out ω^2/2, we get:
All that stuff in the parenthesis – the mass of each particle times the square of its distance from the axis of rotation – is called the moment of inertia, denoted by a capital I.
The term moment means that the value of I depends on how the mass of the body is distributed through space. It has nothing to do with any moment in time. Which is confusing, but whatever.
Anyway, for a body with a given mass and a given rotational axis, the further away from the axis the particles are, the greater the moment of inertia. The greater moment of inertia, the greater the kinetic energy of a rigid body rotating with a given angular speed. The kinetic energy a body has is equal the amount of work done to accelerate that body from rest. So the greater I is, the harder it will be to start that body rotating and the harder it will be stop the rotation. (Which is why the moment of inertia is also called rotational inertia.) The units used for this measurement is the kilogram-meter^2. We have to measure the angular speed in radians. If you don’t then your answers will make no sense and you’ll want to Hulk smash all the things. Not that I know this from experience or anything…
That’s all I have time for today. Next week I’ll do some example problems and explain how it would differ if we had to compute gravitational potential energy. Let me know if I’ve gotten anything wrong in the meantime!
Featured image credit: RCabanilla