One of the last things we cover in Physics 201 is heat. You all know what heat is: the atoms in a substance jiggle around or fly around freely if the substance happens to be a gas. Like all moving massive objects, these atoms have a certain kinetic energy. Now the problem is that they’re all constantly moving and crashing into each other, exchanging energy back and forth. It’s hard enough to keep track of the energy exchange between two colliding objects (trust me!), much less a trillion trillion of them. So we treat them as a statistical ensemble and just look at the average energy. Modulo a few technical considerations, that average energy is just the temperature.

We know how to work with energy for macroscopic objects like baseballs in their trajectories and wheels rolling along the ground, but how can we convert those macroscopic and microscopic average energies into each other? We do it with a quantity called the heat capacity, which tells you how much energy it takes to change the temperature of a substance. It works like this:

Here Q is the amount of heat added – or equivalently the work done – m is the mass of the heated substance, c is its heat capacity, and delta T is the change in temperature. All units SI, of course. Here’s a quick quiz I gave my students last week:

A paddle wheel stirs a water tank at 50 RPM for one hour. The torque transmitted by the shaft is 20 N*m. The water in the tank has mass 10 kg and is initially at 20 degrees C. No heat is lost to the surroundings. What is the final temperature of the water, if its heat capacity is c = 4184 J/(kg*K) ?

Well, we have to find Q first, and then we can use the above equation to find the change in temperature. Q is the work done by the paddle, and work done by a torque is just torque times the angle it rotated through. That angle is 2 π times the number of revolutions, which is itself just 50 RPM * 60 minutes. In total, I get that the work is 376,992 J.

Now we’re looking for delta T, so divide that by the mass times the heat capacity. With the givens, I find a delta T of 9.01 degrees for a final temperature of 29.1 degrees.

Not a huge increase given the pretty hefty amount of energy transferred. But really this isn’t a shock. Catch a baseball thrown at you with high energy and it doesn’t turn red hot when you bring it to a stop. It takes a lot of energy to make temperature increase that’s noticeable at the human scale. But it can be done, from “burning rubber” when accelerating or braking a car to making fire with the friction of a stick.