A uniform chain of $6\, m$ length is placed on a table such that a part of its length is hanging over the edge of the table. The system is at rest. The co-efficient of static friction between the chain and the surface of the table is $0.5$, the maximum length of the chain hanging from the table is.......$m.$
JEE MAIN 2022, Medium
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Mass per unit length $=\lambda$
$N = mg =\lambda( L - x ) g$
$fs _{\max }=\mu_{ s } N$
$fs _{\max }=(0.5)(\lambda)( L - x ) g$
And also $fs _{\max }= m _{ x } g$
$0.5 \lambda( L - x ) g =\lambda xg$
$\frac{ L - x }{2}= x$
$\frac{ L }{2}=\frac{3 x }{2} \Rightarrow x =\frac{ L }{3}=\frac{6}{3}=2 \,m$
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