# Non-uniform vertical circular motion

A roller coaster in a vertical circle will have a velocity change. This change in velocity will change the radial acceleration. My text book states that

“the car is changing only direction, not speed, at the very top and very bottom locations so at these points the acceleration is purely centripetal so Newton’s Second Law applies”.

Why this is only relevant at the top and bottom of the circle?

• Odd that the quote implies Newton's Second Law only applies at those two points. – M. Enns Jan 8 '17 at 4:50

## 2 Answers

In the top of the path, there is no component of the weight along the direction of motion. As a result, the entire weight is providing the required centripetal force for circular motion and there is no tangential acceleration. Hence, the acceleration is purely centripetal.

But at other points on the path,there exist 2 components of the weight, one along the direction of the motion and another perpendicular to it. Due to this, the component of the weight along the direction of motion will increase its velocity, and the component perpendicular to it will change its direction

Please Note: Newtons laws can be applied at every point in the path by resolving the weight into its components and writing the equations of motion for each component

At the tope and bottom, the component of gravity is perpendicular to the motion. This means that gravity does no work on the car at this point - and that the velocity is therefore unchanged.

At every other point the car is either gaining (on the way down) or losing (on the way up) kinetic energy.