Answer
(a) The tension in string A is greater than the tension in string B.
(b) The tension in string A is less than the tension in string B.
Work Step by Step
The centripetal force $F_c$ is the net force which keeps objects moving around in a circle. In general: $F_c = \frac{mv^2}{r} = m\omega^2~r$
(a) We can write an expression for $T_A$:
$\sum F = \frac{mv^2}{r}$
$T_A+mg = \frac{mv^2}{r}$
$T_A = \frac{mv^2}{r} - mg$
We can write an expression for $T_B$:
$\sum F = \frac{mv^2}{2r}$
$T_B+mg = \frac{mv^2}{2r}$
$T_B = \frac{mv^2}{2r} - mg$
Since $\frac{mv^2}{r}$ is greater than $\frac{mv^2}{2r}$, then $T_A$ is greater than $T_B$.
The tension in string A is greater than the tension in string B.
(b) We can write an expression for $T_A$:
$\sum F = m\omega^2~r$
$T_A+mg = m\omega^2~r$
$T_A = m\omega^2~r - mg$
We can write an expression for $T_B$:
$\sum F = m\omega^2~(2r)$
$T_B+mg = 2m\omega^2~r$
$T_B = 2m\omega^2~r - mg$
Since $m\omega^2~r$ is less than $2m\omega^2~r$, then $T_A$ is less than $T_B$.
The tension in string A is less than the tension in string B.
