Question

A small sphere is given vertical velocity of magnitude $v_0=5{\text{ms}}^{-1}$ and it swings in a vertical plane about the end of a massless string. The angle $\theta$ with the vertical at which the string will break, given that it can withstand a maximum tension equal to twice the weight of the sphere, is

• A. ${\cos }^{-1}\left(\frac{2}{3}\right)$
• B. ${\cos }^{-1}\left(\frac{1}{4}\right)$
• C. ${60}^{\circ }$
• D. ${30}^{\circ }$

Answer 1

The correct option is B ${\cos }^{-1}\left(\frac{1}{4}\right)$

FBD of sphere:


Applying law of conservation of energy:
Loss in Potential Energy of sphere = Gain in Kinetic Energy of Sphere
$\frac{m{v}^2}{2}-\frac{m{v}_o^2}{2}=mgl\cos \theta$
$\Rightarrow v^2=v_o^2+2gl\cos \theta \dots \left(i\right)$

From the FBD,
$T-mg\cos \theta =\frac{m{v}^2}{l}$
Given, $T_{max}=2mg$
$2mg=mg\cos \theta +\frac{m}{l}(v_0^2+2gl\cos \theta )$ (substituting (i))
$3mg\cos \theta +\frac{m{v}_0^2}{l}=2mg$
Putting $l=2\text{m},v_0=5\text{m/s}$
$3g\cos \theta =\frac{15}{2}$$\Rightarrow \cos \theta =\frac{1}{4}$
$\Rightarrow \theta ={\cos }^{-1}\left(\frac{1}{4}\right)$