Uniform circular motion
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The realm of physics consists of two types of circular motion: uniform circular motion and non-uniform circular motion.
Uniform circular motion describes motion in which an object moves with constant speed along a circular path.
[edit] Acceleration and velocity
Since the velocity is tangent to the circular path, no two velocities point in the same direction. Although the object has a constant speed, its direction is always changing. This change in velocity causes the object to accelerate. Similar to the velocity, the acceleration’s magnitude is held constant and the direction is forever changing. Such an object experiences a constantly changing acceleration pointing radially inwards (centripetally), which is perpendicular to its velocity. This acceleration is known as centripetal acceleration. 
The magnitude of the acceleration is given by a = v2 / r, where v is the speed of the object and r is the radius of its path, or a = (4π2r) / t2, where r is the radius of the object and t is the time it takes the object to travel a distance.
[edit] How to derive these equations of acceleration
Initial velocity = v1, Final velocity= v2, a = acceleration, r = radius of circular path, s = sector traveled
We follow the head-tail rule for adding vectors. Since the initial velocity points in the opposite direction as it is drawn on the circle diagram, we label it - v1. Thus getting the equation: 
v2-v1
a = 
(v2-v1) / t
a = (v2-v1) / t
at = |v2-v1| We use the absolute value of v2-v1 because we only focus on the magnitude of the velocity.
At this point we can set up a proportion: |v2-v1| / v = s / r
We substitute at for |v2-v1|, which gets us: (at) / v = s / r
(ar) / v = s / t
(ar) / v = v, s / t becomes v because we have a distance over time which gives us velocity. However, this is only when we have s as a very small value. This is because as s gets smaller, the closer it is to the curvature of the circular path.
a = v2 / r, this is an equation of centripetal acceleration with respect to velocity because the radius remains constant.
We can derive the other equation by using the equation v = d / t. The distance of a circle is 2πr. We can substitute 2πr for d, which gets us: v = (2πr) / t. With this new information we can substitute (2πr) / t for v in the equation a = v2 / r.
a = ((2πr) / t)2 / r = (4π2r) / t2
a = (4π2r) / t2, this is the equation of centripetal acceleration with respect to time because the radius remains constant.
[edit] Centripetal force
The acceleration is usually considered to be due to an inward acting force, which is known as the centripetal force. Centripetal force means “center seeking” force. It is the force that keeps an object in its uniform circular motion. We determine this force by using Newton's second law of motion, Fnet = ma, where Fnet is the net force acting on the object (this is the centripetal force, Fc, of an object in uniform circular motion), m is the mass of the object, and a is the acceleration of the object. Since the acceleration of the object in uniform circular motion is the centripetal acceleration, we can substitute v2 / r or (4π2r) / t2 for a. This gets us Fc = (mv2) / r or Fc = (4mπ2r) / t2
The centripetal force can be provided by many different things, such as tension (as in a string), and friction (as between a tire and the road).
An example of tension being the centripetal force is tying a mass onto a string and spinning it around in a horizontal circle above your head. The tension force is the centripetal force because it is the only force keeping the object in uniform circular motion. 
The m is the mass of the object, and the tension force is the centripetal force because it is keeping the object in uniform circular motion.
If a person were to cut the rope at one given point. The object would continue to move in the direction of the velocity. 
As one can see, the string holding mass m is cut about ¾ of the way. After the string is cut, the tension force/centripetal force is no longer acting upon the object so there is no force holding the object in uniform circular motion. Therefore it continues going in the direction when it was last in contact with the force.
Similar to the tension force, the friction force between the tires of a car and the road is the centripetal force because it keeps the car moving in a circular path. If this were a frictionless plane, the car would not be able to move in uniform circular motion, and will instead travel in a straight line. Without the friction force acting upon it, no force is keeping the car in uniform circular, so it moves in a straight line.
