A frictionless wire AB is fixed on a sphere of radius R. A very small spherical ball slips on this wire. The time taken by this ball to slip from A to B is

(1) $\frac{2\sqrt{gR}}{g\mathrm{cos}\theta }$

(2) $2\sqrt{gR}.\frac{\mathrm{cos}\theta }{g}$

(3) $2\sqrt{\frac{R}{g}}$

(4) $\frac{gR}{\sqrt{g\mathrm{cos}\theta }}$

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Concept Questions :-

Uniformly accelerated motion

(3) Acceleration of body along AB is $g\mathrm{cos}\theta$

Distance travelled in time t sec = $AB=\frac{1}{2}\left(g\mathrm{cos}\theta \right){t}^{2}$

From $\Delta ABC,\text{\hspace{0.17em}}AB=2R\mathrm{cos}\theta ;\text{\hspace{0.17em}}2R\mathrm{cos}\theta =\frac{1}{2}g\mathrm{cos}\theta {t}^{2}$

${t}^{2}=\frac{4R}{g}$ or $t=2\sqrt{\frac{R}{g}}$

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