A metallic chain of length $^{'} l_{0}^{'}$ is placed on a rough horizontal table with some part hanging over edge of the table. If coefficient of friction between chain and table is $0.5$ , then maximum length of chain which can hang over edge without slipping is

\frac{I_{0}}{3}

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\frac{I_{0}}{4}

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\frac{3 I_{0}}{4}

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\frac{2 I_{0}}{3}

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Solution

The correct option is A $\frac{I_{0}}{3}$
Answer (1)
Let 'x' length is hanging over edge of table.
$\Rightarrow T = \frac{M . x . g}{I_{0}}$ also $f = T$ [For part of chain on table]
$\Rightarrow T \leq μ_{s} \frac{M}{I_{0}} (I_{0} - x) \times g$
$\Rightarrow \frac{M}{I_{0}} . x . g \leq μ_{s} \frac{M}{I_{0}} (I_{0} - x) \times g$
$x \leq 0.5 (I_{0} - x) \Rightarrow x \leq \frac{I_{0}}{2} - \frac{x}{2}$
$\Rightarrow \frac{3 x}{2} \leq \frac{I_{0}}{2}$
$x \leq \frac{I_{0}}{3} \Rightarrow x_{max} = \frac{I_{0}}{3}$

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