## Thursday, November 8, 2012

### Double pendulum

Question
In case of a double pendulum find the expression for the kinetic energy of the system
Solution
We take a simple case where lengths and masses are same.see below in the figure

Here on being displaced the co-ordinates of two pendulums are
$$\begin{array}{l}{x_1} = l\sin {\theta _1}\\{y_1} = l\cos {\theta _1}\end{array}$$
For the first pendulum where $${\theta _1}$$ is the angle through which the first pendulum have been displaced.
For second pendulum
$$\begin{array}{l}{x_2} = l\sin {\theta _1} + l\sin {\theta _2}\\{y_2} = l\cos {\theta _1} + l\cos {\theta _2}\end{array}$$
Where $${\theta _2}$$ is the angle through which second pendulum has been displaced.
The total kinetic energy of the system is given by the expression
$$T = \frac{1}{2}m(\dot x_1^2 + \dot y_1^2) + \frac{1}{2}m(\dot x_2^2 + \dot y_2^2)$$
Now

And

Which is the required result

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