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Gravitational Energy

[Article 62 by fmyhr, 2000-09-28 | 1 Review | Review this article]

Gravitational energy arises because of the gravitational force by which matter attracts other matter. Compared to other forces like electrical forces, gravitational forces are usually very weak. If you make a fist of each of your hands and hold them near each other in front of you, there is a gravitational force between them that tries to pull them together. But this force is very small! (Which explains why you don't feel your hands trying to move together.) What you do feel is the weight of each hand (and each forearm). The weight of each hand is the gravitational force between it and the earth. This gravitational force you can feel, because it is much larger than the gravitational force between your hands. If you relax your arms the gravitational force between them and the earth causes your hands to fall in your lap (if you're sitting down) or by your side (if you're standing up).

How come the gravitational force between your hand and the earth is so much bigger than that between one of your hands and the other? It's because the earth has much more mass than one of your hands does. (Let's say your hand has a mass of pound = 227 grams. The earth's mass is about 6 billion billion billion grams, or about 40 million billion billion times more than your hand's. That's a lot!)

So far we've been talking about forces, while this article is supposed to be about gravitational energy. The connection is this: when a gravitational force acts on a body and causes it to move through some distance, the gravitational energy is the gravitational force multiplied by the distance through which the body moved. For example, when your hand dropped to your lap or to your side, it lost gravitational energy. Where did this energy go? Well, before your hand hit your lap or jerked to a stop by your side, it was moving, right? Just before your hand stopped moving, almost all of the gravitational energy that it had while you held it out in front of you was converted to kinetic energy (energy of motion). And what about after your hand stopped moving? Where did its kinetic energy go then? To heat. (Your hand and your leg where your hand hit it are slightly hotter than they were before.)

Gravitational energy is also what makes roller coasters so much fun. As the coaster is pulled up the first big hill, its gravitational energy is increased. When the coaster reaches the back side of the hill, the gravitational force is what causes it to accelerate. Much of the coaster's gravitational energy is converted to kinetic energy on the back side of the first hill. (Some of the gravitational energy goes to heating up the coaster wheels and the track, and some more goes to pulling the air around the coaster along with it.) As the coaster climbs the second hill its kinetic energy is converted back into gravitational energy. Because the coaster's kinetic energy at the bottom of the first hill is less than its gravitational energy at the top of the first hill, the second hill is shorter than the first hill. (If the second hill were the same height as the first one, the coaster would stop before the top of the second hill, then start moving backwards.) Every hill in the track must be shorter than the previous one, unless the coaster is again towed.

Consider for a moment a roller coaster in outer space. (Weird idea, and a bad one to boot, as we're about to see.) There is essentially no gravitational force, and thus no gravitational energy, in outer space. (Assuming we're not near any black holes!) When the coaster got to the top of the first hill, it would not have any gravitational energy. On the back side of the first hill, it would just keep moving at about the same speed at which it had been towed up the first hill. In fact, it would slow down a bit because of friction between its wheels and the track. No matter how big the hills in the track were, the coaster would never speed up. It would keep going slower and slower until it finally stopped. Not a lot of fun! Roller coasters depend on gravity (gravitational force and gravitational energy).

We said before that the gravitational force between two objects depends on the mass of the objects. It also depends on the distance between the objects. Newton's law of gravitation allows you to calculate the size of the gravitational force Fg between an object of mass m1 and an object of mass m2 separated by a distance r:
      , (1)

where G = 6.6726 × 10-11 m3/kg-s2 is the universal gravitational constant.

In the very common case of an object of mass m1 = m being gravitationally attracted to the earth, we can set m2 = mEarth = 5.98 × 1024 kg and r = rEarth = 6.37 × 106 m in eqn. (1) to get
      , (2)

where g = 9.81 m2/s is the sea-level gravitational acceleration. (Note that g decreases slowly with altitude above sea-level.) Then the gravitational energy Eg of an object of mass m at some height h above sea level can be defined:
      . (3)

Back to our roller-coaster example, when the coaster of mass m is at the top of the first hill of height , it has gravitational energy . If all of this energy is converted to kinetic energy at the bottom of the first hill, we can find the coaster's speed at the bottom of the first hill as follows:
      , (4a)
      , (4b)
      . (4c)

As we discussed earlier, on its way down the backside of the first hill some of the coaster's energy is converted to heat and to motion of the surrounding air, instead of into kinetic energy. Therefore, the coaster's true speed at the bottom of the hill will be somewhat less than calculated using eqn. (4c).

Symbol List:

Symbol

Description

Eg

Gravitational energy

Gravitational force

G

Universal gravitational constant

g

Sea-level gravitational acceleration on Earth

h

Height above some reference level

m

Mass

r

Distance between objects

v

Velocity



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