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A uniform cyl...

Question

A uniform cylinder of mass $M$ lies in a fixed plane inclined at angle $θ$ with horizontal. A light string is tied to the cylinder at the right most point, and a mass $m$ hangs from the string, as shown in. Assume that the coefficient of friction between the cylinder and the inclined plane is sufficiently large to prevent slipping. For the cylinder to remain static, the value of m is

A

\frac{M sin θ}{1 - sin θ}

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B

\frac{M cos θ}{1 + sin θ}

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C

\frac{M sin θ}{1 + sin θ}

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D

\frac{M cos θ}{1 - sin θ}

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Solution

The correct option is A $\frac{M sin θ}{1 - sin θ}$
According to diag.
$T = m g$
$f = M g sin θ + T sin θ$
$= M g sin θ + m g sin θ$
Balancing torque about $C$ ;
$T r = f r$
$\Rightarrow m g r = (M g sin θ + m g sin θ) r$

$\Rightarrow m = \frac{M sin θ}{1 - sin θ}$

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