Question

A sphere of mass M kg is suspended by a metal wire of length L and diameter d. When in equilibrium, there is a gap of ∆l between the sphere and the floor. The sphere is gently pushed aside so that it makes an angle θ with the vertical. Find θ_max so that sphere fails to rub the Floor. Youngs modulus of

the wire is Y.

• A. sin^-1$\left( 1 - \frac{Y\pi d^{2}\Delta l}{8MgL} \right)$
• B. tan^-1$\left( 1 - \frac{Y\pi d^{2}\Delta l}{8MgL} \right)$
• C. cos^-1$\left( 1 - \frac{Y\pi d^{2}\Delta l}{8MgL} \right)$
• D. None of these

Answer 1

Answer: (C)

cos^-1$\left( 1 - \frac{Y\pi d^{2}\Delta l}{8MgL} \right)$

Answer 2

Y = $\frac{Fl}{A\Delta l} = \frac{2Mg\left( 1 - \cos\theta \right)\Delta l}{\pi\frac{d^{2}}{4}\Delta l}$

or 1 – cos θ = $\frac{Y\pi d^{2}\Delta l}{8Mgl}$ or cos θ = 1 - $\frac{Y\pi d^{2}\Delta l}{8Mgl}$

$\frac{mv^{2}}{2}$ = mgl(1 – cos θ)

or $\frac{mv^{2}}{l}$ = 2 mg (1 – cos θ)

θ = $\cos^{- 1}\left( 1 - \frac{Y\pi d^{2}\Delta l}{8Mgl} \right)$