Questions

A long steel wire of length 'L' is suspended from the ceiling of a room. A sphere of mass 'm' and radius 'r' is attached to the lower end of the wire. The height of the ceiling is (L+e). When the sphere is made to oscillate as a pendulum, its lowest point just touches the floor. The velocity of the sphere at the lowest point will be (L >> r,l and r is the radius of the wire)

a

$\sqrt{\frac{\pi {r}^{2}eY}{m}-gL}$

b

$\sqrt{\frac{\pi {r}^{2}Y}{me}-gL}$

c

$\sqrt{\frac{\pi {r}^{2}m}{Ye}-gL}$

d

None

detailed solution

Correct option is A

During oscillation, when the string is vertical, pseudo force mV^{2}/L acting on the sphere is vertically downward so tension in the string becomes maximum.

$\therefore {T}_{max}=mg+\frac{m{V}^{2}}{L}$

Elongation in the string = Height of ceiling - Length of string = (L + e) - L = e

$\therefore e=\frac{{T}_{max\cdot L}}{AY}=\frac{\left(mg+\frac{m{V}^{2}}{L}\right)L}{\pi {r}^{2}\cdot Y}$

$\Rightarrow V=\sqrt{\frac{\pi {r}^{2}eY}{m}-gL}$

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