Question

A mass 'm' placed on a frictionless horizontal table and attached to a string passing through a small hole in the surface. Initially, the mass moves in a circle of radius $r_0$ with a speed $v_0$ and the free end of the string, is held by a person. The person pulls on the string slowly to decrease the radius of the circle to 'r'. Find the tension in the string when the mass moves along the circle?

• A. $\frac{m{r}_0^2{v}_0^2}{r^2}$
• B. $\frac{m{r}_0^2{v}_0^2}{r^3}$
• C. $\frac{2m{r}_0^2{v}_0^2}{r^3}$
• D. $\frac{m{r}_0^2{v}_0^2}{2r^3}$

Answer 1

The correct option is B

$\frac{m{r}_0^2{v}_0^2}{r^3}$

The torque acting on the mass m about the vertical axis through the hole is zero. The angular momentum about the axis, therefore, remains constant. If the speed of the mass is v when it moves in the circle of radius r, we have

$m{v}_0{r}_0=mvr$

or, $v=\frac{r_0}{r}{v}_0$ ...(i)

The tension $T=\frac{m{v}^2}{r}=\frac{m{r}_0^2{v}_0^2}{r^3}$.