In this section, we derive an expression for the centrifugal force on a body moving in a circle of radius r with an angular velocity Q, as illustrated in figure 3.5.

Let us first consider what forces are involved in a body that is moving in a circle with a uniform speed. Newton's first law of motion says that if a body is left to its own devices it will either remain stationary or move in a straight line with a constant speed. If a body is undergoing uniform circular motion (for example, a child riding on a carousel or, more germanely for us, a person standing still on Earth and so rotating around Earth's axis once per day), then that body must be accelerating, with the acceleration directed toward the axis of rotation.

Let us suppose that in a small time, St, the body moves through a small angle SO, so that the angular velocity is X = SO/St. The speed of the body is a constant,

Figure 3.5. A body moving in a circle is constantly changing its direction and so accelerating. The acceleration is directed toward the center of the circle and has magnitude v2/r, where v is the speed of the body and r is the radius of the circle.

Figure 3.5. A body moving in a circle is constantly changing its direction and so accelerating. The acceleration is directed toward the center of the circle and has magnitude v2/r, where v is the speed of the body and r is the radius of the circle.

v = rX = r 50/5t, but its direction is changing such that at all times, the direction of motion is perpendicular to the radius; thus, if at the initial time the position vector is r1, then its velocity, v1, is in the perpendicular direction, and a short time later the velocity v2 is perpendicular to its new position vector, r2.

As can be seen in figure 3.5, the angle through which the radius vector has moved is related to the distance moved by

Because the velocity is always perpendicular to the radius, the direction of the velocity must move through the same angle as the radius and we have

The acceleration of the body is equal to the rate of change of the velocity; using equation 3.14, we obtain

where the last equality uses the fact that v = r86/8t. That is to say, the acceleration of a body undergoing uniform circular motion is directed along the radius of the circle and toward its center and has magnitude v2 , a r = r = X r. (3.16)

This kind of acceleration is called centripetal acceleration, and the associated force causing the acceleration (for there must be a force) is called the centripetal force and is directed inward, toward the axis of rotation.

Now let us consider the motion from the point of view of an observer undergoing uniform circular motion. An illuminating case to consider is that of a satellite or space station in orbit around Earth; when an astronaut goes for a space walk from the space station, she appears to be weightless, with no forces whatsoever acting upon her (figure 3.6). Now in fact, the astronaut is undergoing circular motion around Earth, and is therefore accelerating, and the force providing the acceleration is Earth's gravity.

a) Stationary or Inertial Frame

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b) Rotating Frame of Reference gravity ^-

centrifugal force

Figure 3.6. An astronaut orbiting Earth. Panel a views the motion in a stationary frame of reference, in which Earth's gravitational force provides the centripetal force that causes the astronaut to orbit Earth. Panel b views the situation from the astronaut's frame of reference, in which the gravitational force is exactly balanced by the centrifugal force and the astronaut feels weightless.

But from the point of view of an observer rotating with the astronaut, the astronaut is stationary and therefore no net forces are acting upon her. Now we know that gravity is acting, pulling her toward Earth, and we say that this force is balanced by another force, centrifugal force, which is pushing her out. The forces exactly balance, so the astronaut appears weightless; that is, in the rotating frame, the centripetal gravitational force pulling her toward Earth is exactly balanced by the centrifugal force pushing her out. Because the centripetal force has magnitude X2r, the centrifugal force must have this magnitude also, and we conclude that in a rotating frame of reference, there appears to be an additional force, the centrifugal force, which acts to accelerate a body along a radius, outward from the axis of rotation. The magnitude of the centrifugal force is X2r per unit mass, where r is the distance from the axis of rotation.

There is no centrifugal force in inertial frames of reference. However, given that we live on a rotating planet, it is extremely convenient to describe motions on Earth from a rotating frame of reference in which Earth's surface is stationary, and in this frame it appears that there is a centrifugal force that is trying to fling us into space. Rather fortunately for those of us living on the planet's surface, the centrifugal force is much weaker than Earth's gravity.

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