A pendulum is a freely hanging weight that
is capable of swinging back and forth.
That weight is acted on by gravity, and moves according to
Newton's second law which states that an object will accelerate
in proportion to the net force acting on it, and inversely
proportional to the objects mass. (If you push harder, an object
will accelerate faster, if it is more massive, it is harder to push.)
This is often written as
F=ma, the angular version
When a pendulum is hanging, the tension in the rope or rod acts against the
force of gravity. As a result, only that component of gravity which
acts perpendicular to the pendulum rod will accelerate the pendulum.
Notice that as the pendulum moves, the forces always acts along a tangent
line to the pendulum's arc. We could then express this motion as
Note that we are using the angular form of Newton's second law, where the
rate of change of the angular momentum is proportional to the net torque.
With a little simplification, this can be written as
Small Angle Approximation
This is not an equation with a simple answer we can write on paper.
As a result, scientists often use an approximation known as the
small angle approximation, which states that is almost
equal to if the angle is very small.
Under the small angle approximation, the equation for a pendulum becomes
The only functions whose second derivatives are equal to -1 times
the original function are the sinusoidal functions sine and cosine.
This gives us for the equation of a simple pendulum under the small
where the amplitude of the oscillation is given by the initial
amount by which the pendulum was displaced, and the frequency
given by the square root of g/L.