Pendulum why small angle

This model for the period of a pendulum only applies for the small angle approximation. As the amplitude becomes greater than 10 degrees, the period deviates from this equation. The motion is no longer sinusoidal as shown in the Physlet.

In this case the tension is always perpendicular to the motion of the pendulum. Thus although the tension changes the direction of the pendulum, it does not change the speed of the pendulum. The restoring force on a pendulum is caused by gravity. As gravity points down, we need to take the component of gravity which is parallel to its motion. Only the component of gravity which is parallel to the direction of motion will do work.

In this case, the force on a pendulum can be given as. In this case we can say that. Here we have the conditions for Simple Harmonic Motion where the angular acceleration is proportional to the angular displacement.

Two pendula with different masses but the same length will have the same period. Two pendula with different lengths will different periods; the pendulum with the longer string will have the longer period. How many complete oscillations do the blue and brown pendula complete in the time for one complete oscillation of the longer black pendulum?

From this information and the definition of the period for a simple pendulum, what is the ratio of lengths for the three pendula? Mathematica numerically solves this differential equation very easily with the built in function NDSolve[ ]. If the initial angle is smaller than this amount, then the simple harmonic approximation is sufficient.

We will start off with a diagram of the double pendulum. We will set the origin of the system to be the point where the double pendulum is connected to the ceiling.

The small angle approximation implies that the double pendulum will hang almost vertically, even during the oscillations. And there we have it!