Circular Motion, Centripetal Forces and Rotation

Up until now, we have discussed primarily straight line motion. We did cover parabolic, projectile motion, but even in that case we split the motion into x and y straight line components. Well, no longer!

There are two main kinds of non straight line motion, circular and rotational. Circular motion, sometimes referred to as revolving motion, involves a body moving around an axis outside of itself. Rotational motion entails a body spinning around some self contained axis. Both of these examples can be seen in the rotation and revolution of the Earth. It's rotation causes the sun to rise and set, and its revolution causes the seasons.




The first case we will consider is that of the revolution, the circular motion. Let's consider the simple case of a ball attached to a string and being swung in a circle overhead. At every point along the circle, the ball's velocity has a different direction. Since an acceleration is any change in velocity, there must be some acceleration associated with this motion, even if the speed of the ball is the same at every point! In what direction does this acceleration point? We can see it using vectors:

The blue vector represents the difference between the two red velocity vectors, the change in velocity, AKA, the acceleration.

This center pulling acceleration is called centripital acceleration. It is given by the formula:

ac = v2/r

Sometimes it is useful to describe the acceleration in terms of a period (T), or how long it takes the moving object to make one complete revolution.

vT = 2ır

v = 2ır/T

ac = 4(pi)2r/T2

If a mass is being accelerated by such an acceleration, there must be some associated force, i.e.,

F = ma

Fc = mac

Fc = mv2/r


Fc = 4m(pi)2r/T2

This should make sense because from experience you probably know that if you release the ball, it will continue to move in a straight line. Thus the centripital force is what makes you stay moving in a circle. When you drive around a bank in a car, you experience such a centripital acceleration. It keeps the car moving in the circle, pulling the car toward the center. Since a passenger in the car has inertia, when a car turns, the person tends to stay in the same place, and thus feels a push outward toward the door of the car. This is called a centrifugal force, and it isn't really a force. It's just the misinterpretation of the sensation of centripital force. This concept can be useful when thinking about things such as artificial gravity for space stations.

As an example, we can figure out the maximum speed of a car moving along a circular track. See college prep page 143, or honors page 138.

It is obvious that it is wiser to have banked tracks. See college prep page 144, or honors page 139.

We can also discuss circular motion in the vertical plane. In this case the force of gravity has an effect on the motion. The object will speed up as it goes downward, since its kinetic energy is being converted to potential energy.

Similarly if the object is moving at constant speed, we can find the centripital force as a combination of the tension in the string and the part of gravity pushing toward the center, that is:

Fc = T + mg cos q

T = Fc - mg cos q

at the top of the circle, q = 0° and cos 0° = 1 so

Ttop = (mv2top/r) - mg

At the bottom of the circle, q = 180° and cos 180° = -1 so

Tbot = (mv2bot/r) + mg

So it is seen that for a constant velocity, the tension is least at the top of the circle and most at the bottom. With these equations we can find the g-forces acting on pilots during loops, etc. Does this help to explain your stomach on a roller coaster?

Next Stop, Universal Gravitation!