Circular motion occurs when a force acts tangentially to the direction of motion of a body at all times. For a body of mass m, travelling with a linear speed v, the force F and radius of curvature R (radius of the circle around which the body will travel) are related by the formula
F = m v^2 / R
We can introduce an angular speed w (usually the lowercase Greek letter omega), defined as the rate at which an object moves through some angle. This is usually defined with the units radians per second (in the SI unit system). There are 2 Pi radians in a full circle, so if a body moves around a complete circle once per second, the angular speed (or angular velocity) is 2 Pi radians per second.
Linear velocity v, in metres per second, is related to the angular velocity by the radius of curvature R:
w = v / R
This allows us to rewrite the force equation as:
F = m R w^2
Using the above formula, we can work out the gravitational force between the Earth and the Moon. The Moon rotates around the Earth once every 27.3 days. This means it travels 2 Pi radians; a complete circle around the Earth, in 27.3 * 24 * 60 * 60 = 2358720 seconds.
This gives us an angular velocity of 2 Pi / 2358720 = 6.66E-6 radians per second (rad/s). The Earth-Moon separation R = 384,000 km = 3.84E8 metres, and it has a mass m = 7.3E22 kg. Substituting these numbers into the force equation above,
F = 7.3E22 x 3.84E8 x (6.66E-6)^2
= 1.24E21 Newtons
We can compare this result with the gravitational force given by the equation for gravitation between two bodies:
F = G M m / R^2
The mass of the Earth is 5.97E24 kg, giving a force
F = 6.67E-11 x 7.3E22 x 5.97E24 / (3.84E8)^2
= 1.97E20 Newtons
The discrepancy arises from our assumption that the orbit is circular; the Moon is actually in a slightly elliptic orbit.
If the central force (the force directed in towards the centre of the circle about which an object is moving) is removed, the body will no longer move in a circle, but will move linearly in the direction tangent to the circle at the point when the force was removed. For example, if you swing a mass around on a string, and the string breaks, the mass will fly off in a straight line, its direction determined by the instantaneous direction of motion at the time the string broke. This fact is used by bowlers to let go of the ball at the top of their swing, sending the ball along a horizontal path!
In analogy to linear acceleration, which is the rate at which linear velocity changes, it is possible to define angular acceleration, usually denoted by the Greek letter alpha (here represented by a). Angular acceleration is the rate of change of angular velocity. Combined with the moment of inertia I, which plays the role of mass in circular motion, but depends on the shape, mass distribution and radius of the motion, it is possible to define a turning force, or torque T:
T = I a
Moments of Inertia are complicated and I will not go into the details here. Usually it is possible to look up the moment of inertia for a given shaped object (it differs for spheres, cylinders, rods, etc. and for the axis of rotation).
Another quantity which uses the moment of inertia is the kinetic energy T associated with a rotating body. This energy is given by
T = 0.5 I w^2
where w is the angular velocity.