Question

A small sphere of mass $m$ is connected by a string to a nail at $O$ and moves in a circle of radius $r$ in the smooth plane include at an angle $\theta$ with the horizontal. If the sphere has a velocity $u$ at the top position $A$. Mark the correct options.

• A. Minimum velocity at $A$ so that the string does not get slack instantaneously is $\sqrt{\frac{3}{5}gr}.$
• B. Tension at $B$ is $9mg/5$ if the sphere has the required velocity as in option (a).
• C. Tension at $C$ is $18mg/5$ in situation of option (b)
• D. None of these

Answer 1

Answer: (A), (B), (C)

The correct options are
A Minimum velocity at $A$ so that the string does not get slack instantaneously is $\sqrt{\frac{3}{5}gr}.$
B Tension at $B$ is $9mg/5$ if the sphere has the required velocity as in option (a).
C Tension at $C$ is $18mg/5$ in situation of option (b)

Minimum velocity at $A$ should be
$v_A=\sqrt{g\sin {37}^{o}r}=\sqrt{\frac{3}{5}gr}$

Velocity at $B:$

$v_B^2=v_A^2+2gr\sin {37}^o$
$\Rightarrow v_B^2=\frac{3}{5}gr+\frac{6}{5}gr$

$\Rightarrow v_B=\sqrt{\frac{9gr}{5}}$
So, the tension at B is:

$T_B=\frac{m{v}_B^2}{r}=\frac{9mg}{5}$

Velocity at $C:$

$v_c^2=v_A^2+2g2r\sin {37}^o$
$\Rightarrow v_C^2=\frac{3}{5}gr+\frac{12}{5}gr$

$\Rightarrow v_c=\sqrt{3gr}$

So, tension at C is:
$T_C-mg\sin {37}^o=\frac{m{v}_c^2}{r}$
$\Rightarrow T_C-\frac{3}{5}mg=3mg\Rightarrow T_C=\frac{18}{5}mg$