A mass m is revolving in a vertical circle at the end of a string of length 20 cms. By how much does the tension of the string at the lowest point exceed the tension at the topmost point.

The correct option is 6 mg.

Let us analyze the motion of a mass $m$ revolving in a vertical circle of radius $r$ (which is the length of the string, $l=20\text{cm}$) under the influence of gravity $g$.

Let $v_1$ be the velocity of the mass and $T_1$ be the tension in the string at the lowest point. At this lowest position, the centripetal force is directed upward towards the center of the circle, while gravity acts downward. Therefore, the net force towards the center is:
$T_1-mg=\frac{m{v}_1^2}{r}$
Rearranging for $T_1$, we get:
$T_1=mg+\frac{m{v}_1^2}{r}$ ---- (Equation 1)

Let $v_2$ be the velocity of the mass and $T_2$ be the tension in the string at the topmost point. At the highest point, both the tension $T_2$ and the gravitational force $mg$ act downwards towards the center of the circle. Thus, the equation of motion is:
$T_2+mg=\frac{m{v}_2^2}{r}$
Rearranging for $T_2$, we get:
$T_2=\frac{m{v}_2^2}{r}-mg$ ---- (Equation 2)

To find by how much the tension at the lowest point exceeds the tension at the topmost point, we calculate the difference $T_1-T_2$:
$T_1-T_2=\left(mg+\frac{m{v}_1^2}{r}\right)-\left(\frac{m{v}_2^2}{r}-mg\right)$
Simplifying this expression:
$T_1-T_2=2mg+\frac{m}{r}\left(v_1^2-v_2^2\right)$ ---- (Equation 3)

According to the law of conservation of mechanical energy, the total energy of the mass remains constant throughout its motion. As the mass moves from the lowest point to the highest point, it rises by a vertical height equal to the diameter of the circle, $h=2r$.
Therefore, the gain in potential energy is equal to the loss in kinetic energy:
$mg\left(2r\right)=\frac{1}{2}m{v}_1^2-\frac{1}{2}m{v}_2^2$
Dividing both sides by $m$ and multiplying by $2$, we get:
$v_1^2-v_2^2=4gr$

Substitute this relation into Equation 3:
$T_1-T_2=2mg+\frac{m}{r}\left(4gr\right)$
$T_1-T_2=2mg+4mg=6mg$

Thus, the tension at the lowest point exceeds the tension at the topmost point by exactly 6 mg, which is independent of the length of the string.