Uniform Circular Motion

Uniform circular motion is the state of an object moving in a circular path at a constant speed.

Of course, the object does not have constant velocity since its direction is changing at every point along the circle. If the object’s velocity is not constant, that must mean that it has an acceleration. This acceleration has constant magnitude but changing direction and is directed radially inwards. Because of its direction, the acceleration is known as centripetal acceleration.

It can be shown that the centripetal acceleration is given by where is the unit vector for the radius, directed from the position of the particle to the center of the circle. Since , the object then has a centripetal force . This centripetal force may manifest in any of several ways: in the case of an object swinging in a circular path on a string, it may be the tension in the string; in the case of gravitation, it is the gravitational force; in a car on a circular off-ramp from a highway, the frictional force between the tires and the road provides the centripetal force.

At any given time, the velocity vector’s direction is tangent to the circular path, as shown in the diagram to the right. Therefore, the velocity and acceleration vectors are at all times perpendicular.

If at any point the centripetal force is lost, the object will fly off tangentially. In the case of our example with a weight swinging circularly on a string, if the string is cut, the weight will leave its “orbit” in a straight line from where it was last held on by the string.

Since the path of an object in uniform circular motion is constant and predictable, we can find its period , the time interval the object takes to complete one revolution. We can derive an equation based on . We manipulate that equation to find and integrate for , or . A bit of arc length can be found by (we will discuss the cross product when we get to angular analogs of linear motion). Since we know the angular displacement all the way around the circle is , we find and plug that into our integrated equation to find . We’ve already defined , so that equation becomes .