A bob is whirled in a horizontal plane by means of a string with an initial speed of $\omega \text{rpm}$. The tension in the string is $T$. If speed becomes $2\omega$ while keeping the same radius, the tension in the string becomes:

The tension $T$ in the string providing the centripetal force for horizontal circular motion is given by:
$$T=m{\omega }^{2}R$$
Where:$m$ = mass of the bob,$\omega$ = angular speed,$R$ = radius of the circle.Given that the radius $R$ and mass $m$ remain constant, we have:
$$T\propto {\omega }^2$$
Initially, $T_1=T$ for ${\omega }_1=\omega$.When the speed becomes ${\omega }_2=2\omega$:
$$T_2=m(2\omega {)}^{2}R=4m{\omega }^{2}R=4T_1$$
Therefore, the new tension is $4T$.