Two bodies of mass m and 4 m are attached by a string as shown in the figure. The body of mass m hanging from a string of length l is executing simple harmonic motion with amplitude A while the other

Question

Two bodies of mass m and 4 m are attached by a string as shown in the figure. The body of mass m hanging from a string of length l is executing simple harmonic motion with amplitude A while the other body is at rest on the surface. The minimum coefficient of friction between the mass 4 m and the horizontal surface to keep it at rest is:

TestHub · Accepted Answer

The swinging mass acts as a pendulum; for amplitude $A$ the speed at the lowest point is
$v^{2} = \frac{g A ^{2}}{l}$
so the maximum string tension is
$T_{ma x} = m g + \frac{m v ^{2}}{l} = m g (1 + \frac{A ^{2}}{l ^{2}}) .$
To keep $4 m$ at rest: $μ \times 4 m g \geq T_{ma x}$ , giving
$μ_{min} = \frac{1}{4} (1 + \frac{A ^{2}}{l ^{2}}) .$
Option B.

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