Circular Motion, Centripetal Forces and Rotation

Newton's Law of Universal Gravitation describes the gravitational attraction between any two massive particles in the known universe. Since the circular motion of the planets in the solar system is caused by the force of gravity, the study of universal gravitation is pertinent to our study of circular motion.

As stated, the Law of Universal Gravitation is:

F = - G m₁m₂/r²

where G is the gravitational constant, 6.67 e -11 m³/s²kg, m₁ and m₂ are the masses of the two objects, and r is the distance between their centers of mass. The negative sign indicates that the force is always attractive.

This law is what is known as an inverse square law. The force of gravity is inversely proportional to the distance between the centers of mass of the two objects. As is easily seen the farther away the two are, the less they attract each other. The negative sign in the equation implies that the force is always attractive.

Galileo showed that all objects move toward the Earth at 9.81 m/s², which we call "g." Newton's law can determine this fact as well. To do so we assume that the force pulling an object toward the earth is the force of gravity, so we have:

F = ma = - GM_Em/R_E²

ma = - GM_Em/R_E²

a = - (GM_E/R_E²)

Where M_E is the mass of the Earth and R_E is the radius of the Earth, both constant values. So plugging in we find that...

a = - 9.81 m/s²

the same result that Galileo came up with.

We can use our centripital force equations together with the universal gravitation equations to find out things such as orbital velocities and geostationary orbits, etc. If we use them together with our analysis of the vertical circle, we can prove Kepler's Laws.

It is easily shown that a satellite traveling in a circular orbit or radius R around a body of mass M moves at a velocity given by:

v = sqrt (GM/R)

How does this whole orbiting thing work? Consider Newton's famous thought experiment. A cannon sits on a mountain. It is high enough so that the effects of the atmosphere can be neglected. It fires a cannonball at a certain horizontal velocity and it lands some distance away, tracing out a projectile path, a la projectile motion. The faster the ball is fired, the further the distance it travels and the greater the curve it traces. At some point, the curve of the ball's path is equal to the curvature of the Earth and instead of falling to the Earth, it continues to fall around it ( approx. 8 km/s). At a greater speed, the ball's path has a greater curvature than the Earth and will move in a more elliptical orbit. At some speed, ( >11.2 km/s )the stone will be able to come free of the Earth's pull altogether.

What about the "weightlessness" of things in orbit? If you do a quick calculation, you find that the astronauts aboard a space shuttle feel an acceleration due to gravity of about 5 m/s/s. That's hardly "weightless." The feeling of weightlessness is caused by the lack of a normal force pushing on the body the way it usually does. Consider a stationary elevator, with a person inside of it standing on a typical bathroom spring scale. The force of gravity acting on the person pulls down with a force mg against the scale. This makes the springs compress and the amount of compression indicates the weight of the person. The normal force from the elevator floor keeps the scale stationary.

If the cable is cut, and the elevator begins to fall, the elevator, the person within it, the scale, the tacky carpet, and everything else falls at the same rate of acceleration, 9.81 m/s/s. There is no normal force acting on the scale anymore. The scale reads zero while falling. The person can take out his wallet and watch it fall at the same rate as himself, seemingly floating in the air. (Being the good student he is, he tries to spend those last few seconds thinking about physics.)

Since the space shuttle is always "falling" around the Earth in its orbit, the astronauts aboard feel "weightless." Weightlessness can be simulated for relatively long periods of time (10 seconds or so) by a diving airplane. This is how they filmed the weightless scenes in Apollo 13.

Next stop, Rotation!