Question

A table with smooth horizontal surface is placed in a cabin which moves in a circle of a large radius $R$. A smooth pulley of small radius is fastened to the table. Two masses of $m$ and $2\, m$ are placed on the table connected through a string going over the pulley. Initially the masses were at rest. The magnitude of the initial acceleration of the masses as seen from the cabin is arel and the tension in the string is $T$. Determine $\frac{T}{a_{\text{rel}}} .( m =1\, kg )$

Answer 1

Since $R$ is very large hence it can be assumed that masses $m _{1}$ and $m _{2}$ are moving in circles of radius $R$.

With respect to observer on table.
For motion of mass $m_{1}$,
$T - m _{1} \omega^{2} R = m _{1} a _{\text {rel }} \ldots $ (1)
For motion of mass $m _{2}$,
$m _{2} \omega^{2} R - T = m _{2} a _{\text {rel }} \ldots$ (2)
$(1)+(2) \Rightarrow\left( m _{2}- m _{1}\right) \omega^{2} R $
$= (m_1 + m_2) a_{rel}$
$a_{\text {rel }}=\frac{\left(m_{2}-m_{1}\right) \omega^{2} R}{\left(m_{2}+m_{1}\right)}=\frac{(2 m-m) \omega^{2} R}{(2 m+m)}$
$a_{\text{rel}} =\frac{\omega^{2} R}{3}$
and by (1), $T =m_{1} \omega^{2} R+m_{1} a_{\text{rel}}$
$=m_{1} \omega^{2} R+\frac{m_{1} \omega^{2} R}{3} $
$=\frac{4 m_{1} \omega^{2} R}{3}=\frac{4 m \omega^{2} R}{3}$