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Mechanical energy is energy of motion or of potential for motion on a macroscopic scale.
Take a Slinky, the good old metal kind. Let it rest in the palm of your hand so that you can hold on to one end with the base of your thumb and fingers. (If your hand is very small or your Slinky is very large, you can just set the slinky on a table or other horizontal surface and use your thumb and ring finger of one hand to pinch the end of the Slinky resting on the table to keep it from moving.) Now use your other hand to twist the free end of the Slinky around its axis, in the clockwise direction. Go two complete turns (you can tell how far you've twisted by watching the little band on the Slinky's last coil.) Feel the Slinky fighting you? (It gets harder to twist, the farther you go.) Now keep ahold of the end you've been holding while you let go of the end you've been twisting. What happened? Well, if you were careful and kept the Slinky balanced in your hand or on the table and didn't let it go sproinging down to the floor, you saw the free end of the Slinky rotate first counterclockwise, then clockwise, then counterclockwise again, then clockwise again, and so on, until it eventually came to rest.
Try it some more times to see if you get the same result. (Besides, it's fun!)
Now try one more thing: before you start twisting the Slinky, notice the position of the little band on the end of the last coil. Now go ahead and twist it and release it as before. (Keep a good grip on the base!) When the Slinky stops twisting, look at the position of the little band. Notice anything?
So how come the Slinky moved the way it did?
One way to think about it is in terms of energy: When the Slinky was just sitting about minding its own business, it was in its equilibrium position, which just means it was sitting still, not twisting on its own. We can say it had zero mechanical energy in this state. Then when you held one end and twisted the other, you added mechanical energy. The more you twisted the more mechanical energy you added. When you were finished twisting it but before you released it, it had the most mechanical energy. At this point, it was mechanical *potential* energy, because nothing was moving. (Compare this state to the one when the Slinky was just sitting around minding its own business, resting in its equilibrium position. The Slinky wasn't moving then, and it isn't moving now. But it had zero mechanical energy then, whereas now it has a positive amount of mechanical energy, which you so generously supplied. You can see the difference when you release the Slinky!)
When you release the Slinky, it begins untwisting itself. As it untwists, the amount of mechanical potential energy goes down. The mechanical potential energy is converted into mechanical *kinetic* energy (motion). If you'd twisted the Slinky two turns before letting go, it will untwist itself two turns (back to its original, untwisted position), at which point its mechanical potential energy is again zero. But this time it doesn't stay there! Most of the mechanical potential energy you gave the Slinky has now been converted into kinetic energy (motion). And so the Slinky keeps twisting, past its original equilibrium position. As it twists itself up (in the counterclockwise direction now), its kinetic energy is converted back into mechanical potential energy. The Slinky keeps twisting until *all* of its kinetic energy has been converted back into mechanical potential energy. When that happens, the Slinky has twisted itself almost 2 turns counterclockwise from its equilibrium position. Once again it's not moving (for a short period of time!) but it has positive mechanical energy. So it begins twisting in the clockwise direction.
This process repeats itself until eventually the Slinky stops, back in its original equilibrium position. Why does it stop? Because of friction. As the Slinky moves, its coils rub against one another. This rubbing action (friction) slows down the twisting motion, and converts the Slinky's mechanical energy into heat. When all of the mechanical energy has been converted to heat, the Slinky stops moving, and is back in its original state, with one difference: it's hotter! (But probably not enough hotter for you to feel the difference.)