Question

A mass is kept on a horizontal frictionless surfaces. It is attached to a string and rotates about a fix centre at an angular velocity ${\omega }_0$. If the length of the string and angular velocity are doubled, find tension in the string which was initially $T_0$.

Answer 1

Initial Angular velocity $={\omega }_o$

Initial Length $=l$

Initial Tension, $T=ml{\omega }_o^2=T_o$

where, $m$ is mass of the particle. If length and angular velocity are doubled:

$l^{\prime }=2l$

${\omega }^{\prime }=2{\omega }_o$

New Tension, $T^{\prime }=m{l}^{\prime }{\omega }^{\prime 2}$

$⟹T^{\prime }=2ml\times 4{\omega }_o^2$

$⟹T^{\prime }=8ml{\omega }_o$

$⟹T^{\prime }=8T_o$