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# Gravitation and Circular Motion: Of Special Interest to Physics C Students

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By McGraw-Hill Professional
Updated on Feb 10, 2011

Practice problems for these concepts can be found at:

Gravitation and Circular Motion Practice Problems for AP Physics B & C

### Kepler's Laws

In Physics C we have to deal with planetary and satellite orbits in a bit more detail. Johannes Kepler, the late 1500s theorist, developed three laws of planetary motion based on the detailed observations of Tycho Brahe. You need to understand each law and its consequences.

1. Planetary orbits are ellipses, with the sun as one focus. Of course, we can apply this law to a satellite orbiting Earth, in which case the orbit is an ellipse, with Earth at one focus. (We mean the center of the Earth—for the sake of Kepler's laws, we consider the orbiting bodies to be point particles.) In the simple case of a circular orbit, this law still applies because a circle is just an ellipse with both foci at the center.
2. An orbit sweeps out equal areas in equal times. If you draw a line from a planet to the sun, this line crosses an equal amount of area every minute (or hour, or month, or whatever)—see Figure 15.2. The consequence here is that when a planet is close to the sun, it must speed up, and when a planet is far from the sun, it must slow down. This applies to the Earth as well. In the northern hemisphere winter, when the Earth is slightly closer to the sun,1 the Earth moves faster in its orbit. (You may have noticed that the earliest sunset in wintertime occurs about two weeks before the solstice—this is a direct consequence of Earth's faster orbit.)
3. A planet's orbital period squared is proportional to its orbital radius cubed. In mathematics, we write this as T2 = cR3. Okay, how do we define the "radius" of a non-circular orbit? Well, that would be average distance from the sun. And what is this constant c? It's a different value for every system of satellites orbiting a single central body. Not worth worrying about, except that you can easily derive it for the solar system by solving the equation above for c and plugging in data from Earth's orbit … c = 1 year2/AU3, where an "AU" is the distance from earth to the sun. If you really need to, you can convert this into more standard units, but we wouldn't bother with this right now.

### Energy of Closed Orbits

When an object of mass m is in orbit around a sun, its potential energy is where M is the mass of the sun, and r is the distance between the centers of the two masses. Why negative? Objects tend get pushed toward the lowest available potential energy. A long way away from the sun, the r term gets big, so the potential energy gets close to zero. But, since a mass is attracted to a sun by gravity, the potential energy of the mass must get lower and lower as r gets smaller.

The kinetic energy of the orbiting mass, of course, is K = 1/2mv2. The total mechanical energy of the mass in orbit is defined as U + K. When the mass is in a stable orbit, the total mechanical energy must be less than zero. A mass with positive total mechanical energy can escape the "gravitational well" of the sun; a mass with negative total mechanical energy is "bound" to orbit the sun.2

All of the above applies to the planets orbiting in the solar system. It also applies to moons or satellites orbiting planets, when (obviously) we replace the "sun" by the central planet. A useful calculation using the fact that total mechanical energy of an object in orbit is the potential energy plus the kinetic energy is to find the "escape speed" from the surface of a planet … at r equal to the radius of the planet, set kinetic plus potential energy equal to zero, and solve for v. This is the speed that, if it is attained at the surface of the planet (neglecting air resistance), will cause an object to attain orbit.

Practice problems for these concepts can be found at:

Gravitation and Circular Motion Practice Problems for AP Physics B & C

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