Two bodies of mass m and 4m are attached with string as shown in the figure. The body of mass m hanging from a string of length l is executing oscillations of angular amplitude $θ_{0}$ while the other body is at rest. The minimum coefficient of friction between the mass 4m and the horizontal surface should be:

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$(\frac{2 - \cos θ_{0}}{3})$

$(\frac{3 - \cos θ_{0}}{4})$

$2 \cos^{2} (\frac{θ_{0}}{2})$

$(\frac{1 - \cos θ_{0}}{2})$

answer is D.

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Detailed Solution

Maximum tension in the string is in its lowest position.
$∴$ Speed of mass m in its lowest position is
$v^{2} = 2 gh = 2 g (1 - \cos θ)$

$T_{\max} - mg = \frac{{mv}^{2}}{l}$

$T_{\max} = mg + 2 mg (1 - \cos θ)$

= $mg (3 - \cos θ)$

Block of mass 4m does not move.

So ${μ(4mg) ≥ T}_{\max}$

or $4 μmg \geq mg (3 - \cos θ) or μ \geq (\frac{3 - \cos θ_{0}}{4})$

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