# Infinite acceleration of bob in pendulum with no friction or air resistance

So i'm a bit confused about something. If we take a mathematical pendulum and we apply a force to it. We ignore all friction and air resistance and only consider gravity and the force applied to bob. My question is and what i've figured out so far: If the force applied to the bob is less than the acceleration due to gravity, the bob will have a forever constant period of swings. However, if the force applied to the bob is greater than the acceleration due to gravity - does that mean that the bob will accelerate infinitely? Or does it only mean that the bob will have an infinite swing in one direction? I can't figure out if the acceleration will decay due to gravity when the initial force applied to the bob is greater than the acceleration due to gravity.

Anyone who care to help? :)

• It can move in a vertical circle. – Farcher Apr 14 '16 at 17:31
• It sounds as if you are thinking about forced harmonic oscillators aka driven harmonic oscillators, where a sinusoidally varying force is applied to the oscillator. If there is no damping then energy is continually being supplied and the amplitude of the oscillations increases without limit. In practice of course there is always damping. – John Rennie Apr 14 '16 at 17:31
• A force can be applied in many ways. To me at least it is unclear what your force is (which direction) and how you apply it (always the same? Periodically? – Martin Apr 14 '16 at 17:48
• @Martin as far as i understood it's when you apply the force at an initial angular displacement. So that you apply some force to the bob at the angle -Pi/2, e.g. give it a push with a force which is bigger than the gravitational pull. Will it then keep accelerating? – Peter Apr 14 '16 at 17:58
• Your description of the scenario is unclear in several ways. Maybe you could include a drawing and some formulas. Since you have come to the conclusion that the bob will accelerate infinitely, I assume that you have used a formula of some kind. – jkej Apr 14 '16 at 18:41

## 1 Answer

Whenever a pendulum moves, it accelerates since the velocity vector is constantly changing. So any pendulum that 'pendulates' forever will 'accelerate' forever. However, I think the acceleration you are talking about is better described as angular acceleration. Perhaps a better way to describe it is if it will complete a full rotation with increasing frequency. The answer to that question is no. The pendulum will have a constant oscillation frequency because the gravity that accelerates it on the way down is exactly equal to the gravity that decelerates it on the way up.