Gravitational Potential Energy
Gravitational Potential Energy
In Chapter 4, we learned that the potential energy of a system is equal to the amount of work that must be done to arrange the system in that particular configuration. We also saw that gravitational potential energy depends on how high an object is off the ground: the higher an object is, the more work needs to be done to get it there.
Gravitational potential energy is not an absolute measure. It tells us the amount of work needed to move an object from some arbitrarily chosen reference point to the position it is presently in. For instance, when dealing with bodies near the surface of the Earth, we choose the ground as our reference point, because it makes our calculations easier. If the ground is h = 0, then for a height h above the ground an object has a potential energy of mgh.
Gravitational Potential in Outer Space
Off the surface of the Earth, there’s no obvious reference point from which to measure gravitational potential energy. Conventionally, we say that an object that is an infinite distance away from the Earth has zero gravitational potential energy with respect to the Earth. Because a negative amount of work is done to bring an object closer to the Earth, gravitational potential energy is always a negative number when using this reference point.
The gravitational potential energy of two masses, and , separated by a distance r is:
Example
A satellite of mass is launched from the surface of the Earth into an orbit of radius , where is the radius of the Earth. How much work is done to get it into orbit?
The work done getting the satellite from one place to another is equal to the change in the satellite’s potential energy. If its potential energy on the surface of the Earth is and its potential energy when it is in orbit is , then the amount of work done is:
Energy of an Orbiting Satellite
Suppose a satellite of mass is in orbit around the Earth at a radius R. We know the kinetic energy of the satellite is KE = 1/2 mv2. We also know that we can express centripetal force, , as = mv2/R. Accordingly, we can substitute this equation into the equation for kinetic energy and get:
Because is equal to the gravitational force, we can substitute Newton’s Law of Universal Gravitation in for :
We know that the potential energy of the satellite is , so the total energy of the satellite is the sum, E = KE + U:
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