If a ladder weighing 250N is placed against a smooth vertical wall having coefficient of friction between it and floor is 0.3, then what is the maximum force of friction available at the point of contact between the ladder and the floor

The correct answer is 75N.

This problem involves a ladder in static equilibrium, leaning against a smooth vertical wall. We are asked to find the maximum force of friction available at the contact point between the ladder and the floor.

Given Information:

Weight of ladder, W = 250 N
Coefficient of friction between ladder and floor, μ = 0.3
The wall is smooth (frictionless), meaning the wall exerts only a horizontal normal reaction on the ladder.

Step 1: Identify the forces acting on the ladder

Since the wall is smooth, it can only push the ladder horizontally — there is no friction from the wall.
At the floor, two forces act:
- Normal reaction from the floor (N) — acting vertically upward
- Friction force (f) — acting horizontally, preventing the base of the ladder from sliding outward

Step 2: Apply vertical equilibrium to find the Normal Reaction (N)

Since the system is in static equilibrium, the sum of all vertical forces must be zero:

N=W

N=250\text{N}

The floor must push up with a normal force equal to the entire weight of the ladder, because the smooth wall provides no vertical support.

Step 3: Calculate the Maximum Force of Friction

The maximum (limiting) force of friction available at a contact surface is given by:

f_{max}=\mu \times N

Substituting the known values:

f_{max}=0.3\times 250

f_{max}=75\text{N}

Key Insight: The phrase "maximum force of friction available" refers to the limiting friction — the maximum friction that the floor can provide before the ladder starts to slide. This is always calculated using the formula f_max = μN, regardless of whether the ladder is actually on the verge of slipping or not. Since the wall is smooth (frictionless), the entire weight of the ladder is borne by the floor, making N = W = 250 N.

Therefore, the maximum force of friction available at the point of contact between the ladder and the floor is 75 N.