Question

A particle of mass $m$ is attached to the ceiling of a cabin with an inextensible light string of length $l$. The cabin is moving upward with an acceleration $a$. The particle is taken to a position such that string makes an angle $\theta$ with vertical. When string becomes vertical, find the tension in the string.

Answer 1

In a frame associated with cabin
Work done on the particle when it comes in the vertical position
$=mgl(1-\cos \theta )+mal(1-\cos \theta )$
By work energy theorem
$\frac{1}{2}m{v}^2=(mgl+mal)(1-\cos \theta )$
$\frac{v^2}{2}=(g+a)l(1-\cos \theta )$
At vertical position
$T-(mg-ma)=\frac{{mv}^2}{l}$
$T=(mg+ma)+2m(g+a)(1-\cos \theta )$