Question

A stone of mass 1 kg is tied up with an inextensible string of length l = 10/3 m. It is rotated in a vertical circle of l radius. If a ratio of maximum and minimum tension in the string is 4 and the value of g is 10$m/s^2$ then calculate a speed of stone at the highest point of a circle.

Answer 1

Given that,

Mass $m=1kg$

Length $L=\frac{10}{3}m$

Radius $r=1m$

Ratio of tension $\frac{T_2}{T_1}=4$

Now,

The ratio of maximum $T_2$ to the minimum tension is

$\frac{T_2}{T_1}=4$

$T_2=4T_1$

Now, the difference between the two tensions should be 6 mg

Now,

$T_2-T_1=6mg$

$4T_1-T_1=6mg$

$3T_1=6mg$

$T_1=2mg$

Now, the tension at the top of the circle is

$T_1+mg=\frac{m{v}_1^2}{r}$

$T_1+mg=\frac{m{v}_1^2}{L}$

$2mg+mg=\frac{m{v}_1^2}{\frac{10}{3}}$

$3mg=\frac{3m{v}_1^2}{10}$

$v_1^2=\frac{30mg}{3m}$

$v_1=\sqrt{10\times 10}$

$v_1=10m/s$

Hence, the speed of stone at highest point of circle is 10 m/s