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Conservative and Nonconservative Forces in Physics

In physics, it’s important to know the difference between conservative and nonconservative forces. The work a conservative force does on an object is path-independent; the actual path taken by the object makes no difference. Fifty meters up in the air has the same gravitational potential energy whether you get there by taking the steps or by hopping on a Ferris wheel. That’s different from the force of friction, which dissipates kinetic energy as heat. When friction is involved, the path you take matters — a longer path will dissipate more kinetic energy than a short one. For that reason, friction is a nonconservative force.

For example, suppose you and some buddies arrive at Mt. Newton, a majestic peak that rises h meters into the air. You can take two ways up — the quick way or the scenic route. Your friends drive up the quick route, and you drive up the scenic way, taking time out to have a picnic and to solve a few physics problems. They greet you at the top by saying, “Guess what — our potential energy compared to before is mgh greater.”

“Mine, too,” you say, looking out over the view. You pull out this equation:

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This equation basically states that the actual path you take when going vertically from hi to hf doesn’t matter. All that matters is your beginning height compared to your ending height. Because the path taken by the object against gravity doesn’t matter, gravity is a conservative force.

Here’s another way of looking at conservative and nonconservative forces. Say you’re vacationing in the Alps and your hotel is at the top of Mt. Newton. You spend the whole day driving around — down to a lake one minute, to the top of a higher peak the next. At the end of the day, you end up back at the same location: your hotel on top of Mt. Newton.

What’s the change in your gravitational potential energy? In other words, how much net work did gravity perform on you during the day? Gravity is a conservative force, so the change in your gravitational potential energy is 0. Because you’ve experienced no net change in your gravitational potential energy, gravity did no net work on you during the day.

The road exerted a normal force on your car as you drove around, but that force was always perpendicular to the road (meaning no force parallel to your motion), so it didn’t do any work, either.

Conservative forces are easier to work with in physics because they don’t “leak” energy as you move around a path — if you end up in the same place, you have the same amount of energy. If you have to deal with nonconservative forces such as friction, including air friction, the situation is different. If you’re dragging something over a field carpeted with sandpaper, for example, the force of friction does different amounts of work on you depending on your path. A path that’s twice as long will involve twice as much work to overcome friction.

What’s really not being conserved around a track with friction is the total potential and kinetic energy, which taken together is mechanical energy. When friction is involved, the loss in mechanical energy goes into heat energy. You can say that the total amount of energy doesn’t change if you include that heat energy. However, the heat energy dissipates into the environment quickly, so it isn’t recoverable or convertible. For that and other reasons, physicists often work in terms of mechanical energy.

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