Non-Uniform Circular Motion


Quick
Non-Uniform circular motion is when a particle moves along a circular path with non-constant, or varying, speed.


Details

An example is a roller coaster car that slows down and speeds up as it moves around a vertical loop. For non-uniform circular motion, Eq1 from the lesson Uniform Circular Motion still gives the radial component of acceleration, which is always perpendicular to the instantaneous velocity and directed toward the center of the circle. But since the speed v has different values at different points in the motion, the value of arad is not constant. The radial (centripetal) acceleration is greatest at the point in the circle where the speed is greatest.

With non-uniform circular motion there is also a component of acceleration that is parallel to the instantaneous velocity. This is the component apar, which is discussed in the lesson Parallel and Perpendicular Components of Acceleration. Here, this component is called atan to emphasize that it is tangent to the circle. From the Parallel and Perpendicular Components of Acceleration lesson, it is seen that the tangential component of acceleration atan is equal to the rate of change of speed. Thus:

arad = 
v2
R

and

atan = 
d|v|
dt

These equations are for non-uniform circular motion

The vector acceleration of a particle moving in a circle with varying speed is the vector sum of the radial and tangential components of accelerations. The tangential component is in the same direction as the velocity if the particle is speeding up, and is in the opposite direction if the particle is slowing down, as shown below.

For uniform circular motion there is no tangential component of acceleration, but the radial component is the magnitude of dv/dt. The two quantities |dv/dt| and d|v|/dt are in general not equal. With uniform circular motion the first quantity is constant and equal to v2/R; the second quantity is zero.