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Heat is mechanical energy at a microscopic scale. The meaning of this is clearer if we consider an example:
When you heat a pot of water on the stove or in the microwave, you know the water gets hot (well, duh!) and eventually boils. Let's look in more detail at this process:
First the stove:
The stove heats the pot, which gets hot. The atoms that make up the pot are held together by chemical bonds, so they're not free to move about independently of one other (which is good, because what kind of pot would we have then?!) But they do move about. Each atom vibrates around its equilibrium position, like a tiny spring. When the stove heats the pot, the atoms in the pot vibrate more violently, and we say the pot get hotter. The liquid water in the pot consists of molecules (each of which has two hydrogen atoms and one oxygen atom) that, unlike the molecules in the solid pot, are free to move past one another. When a water molecule collides with a pot atom on the hot inner surface of the pot, the violently vibrating pot atom transfers some of its mechanical energy to water molecule, causing the water molecule to fly away from the pot surface faster than it collided with it. We say heat is transferred from the pot to the water. And so the water gets hotter.
Now the microwave:
An interesting thing about the microwave is that it heats the water, not the pot. The microwave causes the water molecules to rotate about their own axes more rapidly. This rotation is a form of _internal energy_, not heat energy. But as the water molecules collide with one another, some of their internal energy gets transferred to translational (kinetic, heat) energy, and the water heats up. The hot water then heats the pot that contains it, in the reverse of the process described for the water pot on the stove!
It's important and interesting to note that heat and mechanical energy are the same thing, but at different scales. Heat is mechanical energy on the scale of atoms and molecules. Most things we deal with are MUCH larger than atoms or molecules. (A grain of sand is about a million million million times bigger than an atom.) Since it's nearly impossible to figure out and not very useful to know the motion all of the atoms or molecules in an object, we just concentrate on the motion of the object itself, as a whole. The fact that at any given instant, the millions of millions of millions of molecules that make up the object are flying about in millions of millions of millions of different directions ordinarily concerns us in only one way: we want to know how fast, on average, the molecules are flying around. This average molecular speed defines the temperature of the object. As the molecules fly around faster and faster, we say heat has been transferred to the object, and the temperature goes up.