Ml(d^2θ/dt^2) =-mgθ
1. from the linearized equation, justify galileo’s observation that the period of a pendulum depends only on its length and not on the mass or on the initial displacement.

The equation of motion of a pendulum is:

where it its length and is the gravitational acceleration. Notice that the mass is absent from the equation! This is quite hard to solve, but for small angles (), we can use:

Additionally, let us define:

We can now write:

The solution to this differential equation is:

where and are constants to be determined using the initial conditions. Notice that they will not have any influence on the period, since it is given simply by:

This justifies that the period depends only on the pendulum's length.

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