First slide

Rolling motion

Question

Figure shows a rough track, a portion of which is in the form of a cylinder of radius R. With what minimum velocity a sphere of radius r be set rolling on the horizontal part so that it completely goes around the circle on the cylindrical part?

Difficult

A

$\sqrt{\frac{27}{7} g (R - r)}$
B

$\sqrt{\frac{20}{7} g (R - r)}$
C

$\sqrt{\frac{20}{7} gR}$
D

$\sqrt{\frac{17}{7} g (R - r)}$

Solution

KE of sphere = $\frac{7}{10} {mv}^{2}$
$\frac{7}{10} {mv}^{' 2} + mg 2 (R - r) = \frac{7}{10} {mv}^{2}$
but ${v_{\min}}^{'} = \sqrt{g (R - r)}$
put the value of $v^{'}$ and solve to
get $v = \sqrt{\frac{27}{7} g (R - r)}$

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