Question

How does the variation of the period change in a compound pendulum? How does the period change with different lengths and amount of pendulums? What equations can be used in a compound pendulum?

Answer 1

See below

Explanation:

The double pendulum is easiest worked out using the principle of Least Action and the Euler Lagrange Equation.

With the symbols (`theta_1, l_1, theta_2, l_2`), having their natural meaning, there are 2 independent variables (`theta_1, theta_2`), and 2 degrees of freeedom:

The Lagrangian for the double pendulum is:

`bbbL (theta_1, theta_2, dot theta_1, dot theta_2) = T - V`

`= 1/2 l_1^2 (m_1+m_2)dot theta_1^2 + m_2l_1l_2 dot theta_1 dot theta_2 cos (theta_1 - theta_2)+ 1/2 l_2^2m_2 dot theta_2^2 + (m_1 + m_2) g l_1 cos theta_1 + m_2 l_2 g cos theta_2`

The corresponding DE's from application of the Euler Lagrange Equation are:

• `l_1 ddot theta_1 cos (theta_1 - theta_2) - l_1 dot theta_1^2 sin (theta_1 - theta_2) + l_2 ddot theta_2 + g sin theta_2 = 0`

• `l_1 ddot theta_1 + (m_2l_2)/(m_1 + m_2) (ddot theta_2 cos (theta_1 - theta_2) + dot theta_2^2 sin (theta_1 - theta_2) ) + g sin theta_1 = 0`

These are second-order non-linear coupled DE's with no simple solutions. You could try model them but whilst deterministic the system is also chaotic.

To simplify further, you can look only at small oscillations (small angle approximations as per the simple pendulum) with `l_1 = l_2 = l`, and simplify the DE's to linear coupled DE's. The solutions in this highly idealised model are:

• `theta_1 = cos omega t`

• `theta _2 = +- a/sqrtmu cos omega t`

Where:

• `omega^2 = g/l 1/(1+- sqrtmu)`

• `mu = m_2/(m_1 + m_2)`

Remember though that the simple pendulum, outside of the small angle approximation, is already pretty horrible.