Question

A pendulum of mass m and length b is attached to a massless support initially at rest and then moving with vertically upward acceleration a

( a) Determine the Lagrangian

( b) Find the equation of motion of the pendulum

Refer to the figure below

Answer 1

$x=b\sin\theta,~y=\frac 12at^2-b\cos \theta,$

$\dot x=b\dot {\theta}\cos \theta,~\dot y=at+b\dot{\theta}\sin \theta,$

$T=\frac 12m(\dot x^2+\dot y^2)=\frac 12 m(b\dot{\theta}^2+a^2t^2+2abt\dot{\theta}^2\sin \theta),$

$L=T-mgy,\implies$

$\frac d{dt}(mb^2\dot {\theta}^2+mabt\sin \theta)=mabt\dot \theta\cos \theta-mgb\sin\theta),$

$b^2\ddot \theta+ab\sin \theta+abt\dot \theta\cos \theta=abt\dot \theta\cos \theta-gb\sin \theta,$

$\ddot \theta+\frac{a+g}b\sin \theta=0,$ $\sin \theta=\theta,$

$\ddot \theta+\frac{a+g}b\theta=0,$

$\ddot \theta+\omega^2 \theta=0,\implies$

$T=\frac{2\pi}{\omega}=2\pi\sqrt{\frac b{a+g}}.$