## Centrifugal Force

Suppose that you are riding in a train that starts to go around a bend rather quickly. You feel like you are being thrust outward toward the side of the car, and if you are really going quickly around a tight curve, you might have to hang onto something to stay put. The outward force that you are feeling is commonly known as centrifugal force. Strictly speaking, it is not a force at all (we'll explain that cryptic comment later), but it certainly feels like one. What is going on?

One of the most fundamental laws of physics, Newton's first law, says that, unless acted upon by a force, a body will remain at rest or continue moving in a straight line at a constant speed. That is, to change either direction or speed, a body must be acted upon by a force. Thus, in order for you to go around a bend, a force must act (and act on the train too), and this force, whatever it may be in a particular situation, is called the centripetal force. Without that force, you would continue to go in a straight line. The centrifugal force that you feel is caused by your inertia giving you a tendency to try to go in a straight line when your environment is undergoing a circular motion, so you feel that you are being pushed outward. You do end up going around the bend because your seat pushes against you, providing a real force (the aforementioned centripetal force) that accelerates you around the bend. The centripetal force that makes the train go around the bend comes from the rails pushing on the train wheels. With this discussion in mind, we see that there are two ways to think about the force balance as you go around the bend (literally).

1. From the point of view of someone standing by the side of the tracks, you are changing direction and a real force is causing you and the train to change direction, in accord with Newton's laws.

2. From your own point of view, you are stationary relative to the train. If you don't look out of the window, you don't know you are going around a bend. There appear to be two forces acting on you: the centrifugal force pushing you out and the seat pushing back (the centripetal force) in the other direction. In this frame of reference, Newton's law is satisfied because the two forces cancel each other out. That is, in the train's frame of reference you remain stationary and there is a balance of forces between the centripetal force from the seat pushing you in and the centrifugal force pushing you out.

The centrifugal force is, therefore, really a device that enables us to use Newton's laws in a rotating frame of reference: we can say that Newton's laws are satisfied in a rotating frame provided we introduce an additional force, the centrifugal force. There is a simple formula for this force that is derived in appendix A to this chapter. If an object of mass m is going around in a circle of radius r with a speed v, then the centrifugal force, Fcen, is given by c mv 2

The centrifugal force per unit mass is just v2/r.

A quite analogous situation occurs for us on Earth. Earth is actually rotating quite quickly; it goes around once a day, and the velocity of Earth's surface at the equator is a quite respectable 460 m s-1, or a little faster than 1,000 miles per hour. Sitting on the surface of Earth, we must therefore experience a centrifugal force that is trying to fling us off into space. The reason that we stay comfortably on the surface is that the force of gravity overwhelms the centrifugal force, as a quick calculation shows. The radius of Earth is about 6,300 km, so that using equation 3.1, the centrifugal force per unit mass is given by

This value should be compared to the force of gravity per unit mass (i.e., the gravitational acceleration) at Earth's surface, which is 9.8 m s-2. The centrifugal effect is therefore quite small, although not so small that we cannot measure it. In fact, because the centrifugal effect is largest at the equator and diminishes to zero at the poles, over time Earth has developed a slight bulge at the equator such that the line of apparent gravity (the real gravity plus the centrifugal force) is perpendicular to Earth's surface. (The distance from the center of Earth to the surface is in fact some 30 km larger at the equator than at the poles.) The centrifugal force is otherwise not terribly important, and its effect can be taken into account by slightly modifying the value of the gravitational force as necessary.