This week I’m teaching rotational motion to my students. Here’s an easy problem from their textbook, which comes from the idea of using a flywheel to store energy. I’m modifying it from problem 9.41 in Young and Geller:
Suppose we want to built a flywheel in the shape of a solid cylinder or radius 1.00m using concrete of density 2.2 x 103 kilograms per cubic meter. What must its length be to store 2.5 MJ of energy in its rotational motion, if for stability its speed is limited to 1.75 seconds for each revolution?
That seems a little slow – I bet some decent engineering could get it a lot faster, but that’s the setup. First let’s talk about what’s happening in this situation. A moving object has kinetic energy. Now this spinning cylinder is not going anywhere, but any given part is in motion. The parts of the wheel toward the outside will be moving faster and the parts near the center will be moving slower. There’s a convenient way of taking this into account when writing the total rotational kinetic energy of a spinning object:
Where I is the moment of inertia of the object, which is tabulated in many places for differently shaped objects. It describes the way the mass is distributed and how much total energy per angular velocity the object has. That lower case Greek letter omega is the angular velocity of the object. In needs to be in radians per second, but we’re given a value in seconds per revolution. We’ll invert that to get revolutions per second, and then multiply by 2π to get radians per second. I get 3.59 radians per second, though formally radians don’t actually count as a unit.
For a solid cylinder, the moment of inertia is:
M is the mass, R is the radius. Plugging in, we have:
So, we can find the total mass we need in order to store the requisite amount of energy. From there, we can find the required length. Solve for M, you scalawags!
Plug in our numbers for this problem and I get 776,000 kg. Heavy, but concrete is cheap. Given the density of the concrete, we have a length of 112 meters. Disaster, that’s preposterously large. That’s pretty hideously impractical, considering 2.5 MJ is not all that much energy – about what a good hairdryer uses in half an hour. This is why actual flywheel storage systems spin at a lot more than our pathetic 150ish rpm. Something like 20,000 rpm is more usual, and since energy is proportional to the rotational speed squared, all other things being equal a flywheel spinning at 20,000 rpm will hold something like 18,000 times more energy. That’s much more practical, and would allow the flywheel to be much more compact.
Indeed these kinds of systems have been in use for a long time. Today with the ever increasing interest in ways to generate and store energy, flywheels are not only still around, they’re finding even more new applications. Neat stuff, and a cool way to introduce rotation to a 201 class.