Question

What is the minimum coefficient of friction necessary so that the ladder does not slip on the incline?

• A. $\frac{1}{3}$
• B. $\frac{1}{4}$
• C. $\frac{1}{6}$
• D. None of these

Answer 1

The correct option is A $\frac{1}{3}$

The force of friction will also start acting when incline plane is rough
From FBD of ladder
The forces in horizontal direction
$N_{2}cos{45}^{\circ }+fcos{45}^{\circ }=N_1$...(i)
The forces in vertical direction
$N_{2}cos{45}^{\circ }-fcos{45}^{\circ }=mg$...(ii)
writing torque balance equation abount point B
$mg\frac{l}{2}cos{45}^{\circ }-N_{1}cos{45}^{\circ }=0$...(iii)
$\Rightarrow N_1=2mg$
for minimum frictional coefficient the frictional force should be maximum
$f_{max}\le \mu N_2$
substituting values of f and N_1 in equation (i) and (ii)

$\frac{N_2(1+\mu )}{\sqrt{2}}=2mg$...(iv)
$\frac{N_2(1-\mu )}{\sqrt{2}}=mg$...(v)
dividing the equation (iv) by (v)
$\frac{1+\mu }{1-\mu }=2$
$\mu =\frac{1}{3}$