A bob is whirled in a horizontal plane by means of a string with an initial speed of w rpm. The tension in the string is T. If speed becomes 2ω while keeping the same radius, the tension in the string becomes :

We start with a bob of mass $m$ moving in a horizontal circle of radius $r$ with an initial angular speed $\omega$. The tension in the string supplies the required centripetal force.

The centripetal force (tension) is given by

$T=mr{\omega }^2$

Now the angular speed is doubled to $2\omega$ while the radius stays the same. The new tension $\mathrm{T\text{'}}$ is

$\mathrm{T\text{'}}=mr{2\omega }^2$

Because squaring the doubled speed gives

${2\omega }^2=4{\omega }^2$

we can rewrite the new tension as

$\mathrm{T\text{'}}=mr{\omega }^2·4$

Recognizing that the original tension $T$ equals $mr{\omega }^2$, we obtain

$\mathrm{T\text{'}}=4T$

Hence, when the speed becomes 2ω, the tension in the string becomes 4T.