Part of a uniform chain of length L is hanging out of a table. If the coefficient of friction between the chain and the table is ,`mu` estimate the maximum length of the chain that can hang without slipping.

Explanation

Let the maximum length of the chain that can hang out without slipping be `l`.
Length of the chain on the table = L- `l`. If m is mass per unit length of the chain weight of the chain resting on the table = (L-`l`) mg
`:.` Normal force of the table R = (L-`l`) mg
Downward force on the hanging part = weight of the hanging part = `l` mg
To avoid slipping this downward force has to be balanced by the limiting friction f i.e, f = `l` mg Hence coefficient of friction ,
`mu = (f)/(R ) = (l mg)/((L-l)mg) = (l)/(L-l)`
or, `" " l= mu L-mul " ""or", l (1+mu)= muL`
`:. l = (muL)/(1+mu)`.