• # 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

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$

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