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? :)

  • $\begingroup$ It can move in a vertical circle. $\endgroup$
    – Farcher
    Apr 14, 2016 at 17:31
  • $\begingroup$ 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. $\endgroup$ Apr 14, 2016 at 17:31
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    $\begingroup$ 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? $\endgroup$
    – Martin
    Apr 14, 2016 at 17:48
  • $\begingroup$ @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? $\endgroup$
    – Peter
    Apr 14, 2016 at 17:58
  • $\begingroup$ 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. $\endgroup$
    – jkej
    Apr 14, 2016 at 18:41

2 Answers 2


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.


An pendulum released from rest will swing from side to side, going exactly as high as its initial position, repeating forever.

A pendulum given an initial "kick" in the form of some impulse has enough energy to rise higher than its initial position - if this height is higher than the pendulum string can reach, the pendulum will swing in a complete circle all the way around repeatedly, rather than oscillating side-to-side. If there is no force imparted after time 0, the overall pattern of the swing will not change from swing to swing.

A pendulum that has some force applied (like if the bob had a rocket motor attached) could remain stationary if the force balances gravity, or do an odd lopsided swing if it's weaker, or do vertical loops at ever-faster speeds. Different things may happen if the force has constant direction and magnitude, or if the force changes depending on the position of the bob. But in general, if there is a force applied over time, the period of the oscillations from swing to swig may change.


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