Question

Consider the length of the simple pendulum is 1 and the mass of the pendulum bob is m.
Obtain the equation for the Lagrangian for the simple pendulum when its kinetic energy is
1
ml © and the potential energy is mgl(1-сoso)
2​

Answer 1

Kinetic energy of the bob is maximum at the mean position of oscillation i.e. at point A and conservation of energy states that this kinetic energy is converted into the potential energy as the bob reaches at the maximum height i.e. point B.

From figure, we get that AC=AO−OC

∴ AC=l−lcosθ=l(1−cosθ)

Potential energy of bob at point B P.E=mg(AC)

∴ P.E=mgl(1−cosθ)

Thus maximum kinetic energy of the bob K.E

max

=mgl(1−cosθ)

solution