Let be the length of the string. Then
which implies that
So, we need to find and as functions of . For , this can be done by equating the kinetic energy that the mass has when its angle is to the change in the mass's potential energy between its initial position and its position when its angle is :
In order to find , we use Lagrange's equation
The kinetic energy of the particle is
The potential energy of the particle is
Therefore the Lagrangian of the particle is
Using Langrange's equation, we find that
Plugging equations (3) and (4) into equation (1)-(2) gives
Therefore, answer (E) is correct.
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