1 - Determine the distance to which the 90 kg painter can climb without causing the 4 m ladder to slip at its lower end A as shown in Figure 1. The top of the 15 kg ladder has a smaller roller, and at the ground the coefficient of static friction is 0.25. The mass center of the painter is directly above his feet.

2.55m

To determine the distance to which the painter can climb without causing the ladder to slip, we need to consider the forces acting on the ladder and calculate the maximum force of static friction at its lower end.

Let's start by analyzing the forces acting on the ladder:

1. Weight of the ladder (Wl): This is equal to the mass of the ladder (15 kg) multiplied by the acceleration due to gravity (9.8 m/s^2).

Wl = 15 kg * 9.8 m/s^2

2. Weight of the painter (Wp): This is equal to the mass of the painter (90 kg) multiplied by the acceleration due to gravity (9.8 m/s^2).

Wp = 90 kg * 9.8 m/s^2

3. Normal force (N): The normal force is the force exerted by a surface to support an object resting on it. In this case, it is equal to the sum of the weight of the ladder and the weight of the painter.

N = Wl + Wp

Next, we need to calculate the maximum force of static friction at the lower end of the ladder. The maximum force of static friction (Fmax) can be found by multiplying the coefficient of static friction (μs) by the normal force (N). In this case, the coefficient of static friction is given as 0.25.

Fmax = μs * N

Now, we can calculate the distance to which the painter can climb without causing the ladder to slip. This distance, denoted as d, can be determined using the torque equation:

d = Fmax * l / Wp

Where:
- d is the distance to which the painter can climb without slipping the ladder.
- l is the length of the ladder (4 m in this case).
- Wp is the weight of the painter.

Substituting the known values, we can calculate the distance (d).