A uniform ladder 6m long, weighing 300N, is resting against a vertical wall with which
it makes 30. A man weighing 750N climbs up the ladder. At what position along the
ladder from the bottom end does the ladder slips? The coefficient of friction for both
the wall and the ground with the ladder is 0.2.

To determine the position along the ladder where it begins to slip, we need to analyze the forces acting on the ladder and apply the principles of equilibrium and friction. The ladder is in static equilibrium, meaning the sum of forces and the sum of moments (torques) acting on it must equal zero. Let's break down the problem step by step.

Understanding the Forces Involved

First, we identify the forces acting on the ladder:

• The weight of the ladder (300 N) acting downwards at its midpoint (3 m from either end).
• The weight of the man (750 N) acting downwards at a distance 'x' from the bottom of the ladder.
• The normal force from the ground (N_g) acting upwards at the base of the ladder.
• The normal force from the wall (N_w) acting horizontally at the top of the ladder.
• The frictional force at the ground (F_f) acting horizontally towards the wall.

Setting Up the Equations

For the ladder to be in equilibrium, the following conditions must hold:

1. Sum of Vertical Forces

The sum of the vertical forces must equal zero:

N_g - (Weight of ladder + Weight of man) = 0

N_g = 300 N + 750 N = 1050 N

2. Sum of Horizontal Forces

The sum of the horizontal forces must also equal zero:

N_w - F_f = 0

F_f = N_w

3. Moment about the Base of the Ladder

Taking moments about the base of the ladder, we have:

Moment due to the weight of the ladder + Moment due to the weight of the man = Moment due to the normal force from the wall

300 N * 3 m * cos(30°) + 750 N * x * cos(30°) = N_w * 6 m * sin(30°)

Calculating the Normal Force from the Wall

We know that the maximum frictional force (F_f) can be expressed as:

F_f = μ * N_g = 0.2 * 1050 N = 210 N

Since F_f = N_w, we have N_w = 210 N.

Substituting Values into the Moment Equation

Now we can substitute N_w into the moment equation:

300 N * 3 m * cos(30°) + 750 N * x * cos(30°) = 210 N * 6 m * sin(30°

Calculating the trigonometric values:

• cos(30°) = √3/2 ≈ 0.866
• sin(30°) = 1/2 = 0.5

Substituting these values gives:

300 N * 3 m * 0.866 + 750 N * x * 0.866 = 210 N * 6 m * 0.5

Now simplifying:

259.8 + 649.5x = 630

649.5x = 630 - 259.8

649.5x = 370.2

x = 370.2 / 649.5 ≈ 0.57 m

Final Position of the Man on the Ladder

The man will begin to slip when he is approximately 0.57 meters from the bottom of the ladder. This analysis shows how the forces and moments interact to determine the stability of the ladder as the man climbs it. Understanding these concepts is crucial in physics and engineering, especially in scenarios involving static equilibrium.