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Before discussing the dynamics of uniform circular motion, we must
explore its kinematics. Because the direction of a particle moving in
a circle changes at a constant rate, it must experience uniform
acceleration. But in what direction is the particle accelerated? To
find this direction, we need only look at the change in velocity over a
short period of time:

The diagram above shows the velocity vector of a particle in uniform
circular motion at two instants of time. By vector addition we can see
that the change in velocity, Δv, points toward the center of
the circle. Since acceleration is the change in velocity over a given
period of time, the consequent acceleration points in the same
direction. Thus we define centripetal acceleration as an
acceleration towards the center of a circular path. All objects in
uniform circular motion must experience some form of uniform
centripetal acceleration.

We find the magnitude of this acceleration by comparing ratios of
velocity and position around the circle. Since the particle is
traveling in a circular path, the ratio of the change in velocity to
velocity will be the same as the ratio of the change in position to
position. Thus:

= =

Rearranging the equation,

=

Thus

a =

We now have a definition for both the magnitude and direction of
centripetal acceleration: it always points towards the center of the
circle, and has a magnitude of v^{2}/r.

Let us examine the equation for the magnitude of centripetal
acceleration more practically. Consider a ball on the end of a string,
being rotated about an axis. The ball experiences uniform circular
motion, and is accelerated by the tension in the string, which always
points toward the axis of rotation. The magnitude of the tension of
the string (and therefore the acceleration of the ball) varies
according to velocity and radius. If the ball is moving at a high
velocity, the equation implies, a large amount of tension is required
and the ball will experience a large acceleration. If the radius is
very small, the equation shows, the ball will also be accelerated more
rapidly.

Centripetal Force

Centripetal force is the force that causes centripetal acceleration. By
using Newton's Second Law in conjunction with the equation for centripetal
acceleration, we can easily generate an expression for centripetal force.

F_{c} = ma =

Remember also that force and acceleration will always point in the same
direction. Centripetal force therefore points toward the center of the circle.

There are many physical examples of centripetal force, and we cannot completely
explore each one. In the case of a car moving around a curve, the centripetal
force is provided by the static frictional force of the tires of the car
on the road. Even though the car is moving, the force is actually perpendicular
to its motion, and is a static frictional force. In the case of an airplane
turning in the air, the centripetal force is given by the lift provided by its
banked wings. Finally, in the case of a planet rotating around the sun, the
centripetal force is given by the gravitational attraction between the two
bodies.

With a knowledge of physical forces such as tension, gravity and friction,
centripetal force becomes merely an extension of Newton's Laws. It is special,
however, because it is uniquely defined by the velocity and radius of the
uniform circular motion. All of Newton's Laws still apply, free body diagrams
are still a valid method for solving problems, and forces can still be resolved
into components. Thus the most important thing to remember regarding uniform
circular motion is that it is merely a subset of the larger topic of dynamics.