# A uniform ladder of mass 46 kg and 3.0 m in length is leaning against a frictionless vertical wall at an inclination of 60 degrees to the horizontal. A Window cleaner of mass 67.3 kg climbs up the ladder. What coefficient of friction between the ladder and ground is required so that he can climb to the top of the ladder safely?

## To calculate the coefficient of friction required for the window cleaner to climb safely, we need to consider the forces acting on the ladder.

Let's break it down step by step:

Step 1: Analyzing the forces

The forces acting on the ladder are:

1. Weight of the ladder (Wladder): This force acts downward and has a magnitude of mg, where m is the mass of the ladder (46 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).

2. Weight of the window cleaner (Wcleaner): This force also acts downward and has a magnitude of mg, where m is the mass of the window cleaner (67.3 kg) and g is the acceleration due to gravity.

3. Normal force (N): This force acts perpendicular to the contact surface between the ladder and the ground. It balances the weight of the ladder and the window cleaner and prevents the ladder from sinking into the ground.

4. Frictional force (f): This force acts parallel to the contact surface between the ladder and the ground. It opposes the motion of the ladder.

Step 2: Resolving the forces

Since the ladder is at an angle of 60 degrees to the horizontal, we need to resolve the weight of the ladder and the window cleaner into components parallel and perpendicular to the contact surface.

Drawing a free-body diagram, we can see that the component of the ladder's weight acting parallel to the contact surface is Wladder sin(60°), and the component of the window cleaner's weight acting parallel to the contact surface is Wcleaner sin(60°).

The component of the ladder's weight acting perpendicular to the contact surface is Wladder cos(60°), and the component of the window cleaner's weight acting perpendicular to the contact surface is Wcleaner cos(60°).

Step 3: Equilibrium condition

The ladder will be in equilibrium if the sum of the forces acting parallel to the contact surface (in this case, f) is zero.

Since the frictional force is equal to the coefficient of friction (μ) multiplied by the normal force (f = μN), we can write:

μN = Wladder sin(60°) + Wcleaner sin(60°)

Now, we need to calculate the normal force (N).

Step 4: Calculating the normal force

The normal force is equal to the sum of the perpendicular components of the ladder's weight and the window cleaner's weight:

N = Wladder cos(60°) + Wcleaner cos(60°)

Step 5: Substituting values and solving for the coefficient of friction

Substituting the given values into the equations, we can solve for the coefficient of friction (μ):

μ(Wladder cos(60°) + Wcleaner cos(60°)) = Wladder sin(60°) + Wcleaner sin(60°)

μ = (Wladder sin(60°) + Wcleaner sin(60°)) / (Wladder cos(60°) + Wcleaner cos(60°))

By substituting the values (m = 46 kg and m = 67.3 kg) and solving the above equation, you can determine the required coefficient of friction for the window cleaner to climb safely.