Centripetal acceleration is how fast tangential velocity, or the speed at which an orbiting body moves, is changing. It includes both the magnitude and direction of the change in tangential velocity. When an object is moving with circular motion, acceleration always points directly at the circle's center. It has a magnitude related to the angular velocity and speed of the object.
If a body is moving in a straight line, then its acceleration describes how fast its velocity is changing. If an object is moving in a circular path, then centripetal acceleration explains how fast its tangential velocity is changing. The tangential velocity is a measure of how fast the object is changing direction, or going around the circle, as well as the actual speed that it’s moving.
Centripetal acceleration is a vector, which means it has both a magnitude and a direction. The direction always points inward to the circle's center, because this is the direction in which a rotating object is always accelerating. This is often a confusing concept, because an object undergoing circular motion doesn’t appear to be accelerating toward the circle's center. This is because, according to Newton’s laws, the acceleration of an object is always in the direction with which the force acts. For an object to move in a circle, there must be a force coming from the circle's center, so this is the direction of the acceleration.
In mathematics, the magnitude, or size, of the circular acceleration is proportional to the speed of the object and the square of its angular velocity. Angular velocity is the rate at which the angle of the object is changing. This means centripetal acceleration increases dramatically as the angular velocity increases.
The centripetal acceleration is closely related to the centripetal force. According to Newton’s laws, centripetal force is equal to centripetal acceleration multiplied by the mass of the object. In other words, centripetal force is the total force acting on an object that causes it to move in a circle.
An example of circular motion is the moon orbiting the Earth. As the moon orbits, it is under a force resulting from the gravity of the Earth. This means it’s constantly “falling” toward the Earth and, therefore, has a centripetal acceleration pointing toward the center of the Earth, even though it retains enough speed to stay in a circular orbit.