question_answer

A uniform rope of mass 'm' is placed on a smooth fixed sphere of radius 'R' as shown in the figure. Find the tension T in the rope when it is in a horizontal plane at a vertical distance \[h=R/2\]below the top of sphere.

A) \[\frac{mg}{2\pi }\]                   

B)        \[\frac{mg}{2}\]

C) \[\frac{\sqrt{3}mg}{\pi }\]          

D)        \[\frac{\sqrt{3}mg}{2\pi }\]

Correct Answer: D

Solution :

[d] For equilibrium of an elemental part 'dm? \[2T\left( \frac{d\theta }{2} \right)\sin \alpha =dmg\cos \alpha \] \[Td\theta =\left( \frac{m}{2\pi }d\theta g \right)\cot \alpha \] \[T=\frac{mg}{2\pi }\left( \frac{\sqrt{3}R}{2(R/2)} \right)\] \[T=\frac{\sqrt{3}mg}{2\pi }\]