question_answer

A simple pendulam of bob mass m & length (. Is oscillating in vertical plane, when string of pendulam is making an angle of \[37{}^\circ \] with the vertical its tangential and radial acceleration have same magnitude. The velocity of the bob when its crosses the mean position is

A) \[\sqrt{g\ell }\]                                

B) \[\sqrt{3g\ell /5}\]

C) \[\sqrt{2g\ell /5}\]

D)  No such condition is possible

Correct Answer: A

Solution :

Let u is the speed of particle (bob) when it is at mean position and v when it is making an angle of \[37{}^\circ \] with the vertical. For given condition, \[g\sin 37=\frac{{{v}^{2}}}{\ell }\] \[\Rightarrow \]\[{{v}^{2}}=g\ell \times \frac{3}{5}=\frac{3g\ell }{5}\] From work energy theorem \[\frac{m{{u}^{2}}}{2}=\frac{m{{v}^{2}}}{2}=mg\times \ell (1-cos37)=\frac{mg\ell }{5}\] \[\Rightarrow \]\[{{u}^{2}}={{v}^{2}}+\frac{2g\ell }{5}=\frac{3g\ell }{5}+\frac{2g\ell }{5}=a\ell \]