Question

A sphere of mass 20 kg is suspended by a metal wire of unstretched length 4 m and diameter 1 mm. When in equilibrium, there is a clear gap of 2 mm between the sphere and the floor. The sphere is gently pushed aside so that the wire makes an angle θ with the vertical and is released. Find the maximum value of θ so that the sphere does not rub the floor. Young modulus of the metal of the wire is 2.0 × 1011 N m−2. Make appropriate approximations.

Answer 1

Maximum Value of \theta is 36°

Explanation:

The additional force $f = l \times (1 - cos\theta)$ can get additional through $\frac {m \times v^2}{l}$

which says - $mg l(1 - cos\theta) = \frac {m\timesv^2}{2}$

$mg\times l(1 - cos\theta) = \frac {m\times v^2}{2}$

As per the figure, (Attached for Reference)

$F = \frac {m\times v^2}{l} = 2mg (1 - cos\theta)$

Hence, $Y = \frac {FL}{A \Delta l}$,

where Y is Young's Modulus

$2\times 10^1^1 = \frac {2\times20\times10\times4(1 - cos\theta)}{\pi\times(5\times10^-^4)^2 \times 2\times10^-^3}$

$1 - cos\theta = \frac {\pi}{16}, cos\theta = \frac {16 - \pi}{16}$

$\theta = cos^-(\frac {12.84}{16})$

= 36°