First slide

Friction on horizontal surface

Question

A uniform rope of length L lies on a table. If the coefficient of friction is µ, then the maximum length l₁ of the part of this rope which can overhang from the edge of the table without sliding down is:

Moderate

A

$\frac{1}{μ}$
B

$\frac{1}{μ + 1}$
C

$\frac{μL}{μ + 1}$
D

$\frac{μL}{μ - 1}$

Solution

weight of the hanging part = frictional force on the table
if l₁ part is hanging its weight $= \frac{M}{L} I_{1} g$
$L - I_{1}$ is on the table and frictional force due to this $= μ \frac{M}{L} (L - I_{1}) g$
$\begin{array}{l} \frac{M}{L} I_{1} g = μ \frac{M}{L} (L - I_{1}) g \\ I_{1} = μ (L - I_{1}) \\ I_{1} (1 + μ) = μL \\ I_{1} = \frac{μL}{1 + μ} \end{array}$

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