A ladder is leaning against a vertical wall. There is negligible friction between the ladder and the wall, and the coefficient of static friction between the ladder and the ground is 𝜇s=0.245. The ladder is uniform, with its center of gravity located at its geometric center.

The figure shows a cut away side view of the ladder described in the problem. The ladder rests diagonally with the top of the ladder against a vertical wall at the left of the figure and the bottom of the ladder against the floor at the bottom of the figure. The acute angle between the vertical wall and the angled ladder is labeled alpha. Separately, an X Y coordinate system is drawn in the upper right corner of the figure. The Y axis points up and the X axis points to the right.
At what maximum angle 𝛼, relative to the wall, can the ladder lean without slipping?

To find the maximum angle α relative to the wall at which the ladder can lean without slipping, we need to consider the forces acting on the ladder.

There are two main forces to consider: the gravitational force acting vertically downward (mg) and the static friction force acting horizontally between the ladder and the ground.

Let's break down the forces acting on the ladder:

1. Gravitational Force (mg): The ladder's weight acts vertically downward, perpendicular to the wall. The magnitude of this force is given by mg, where m is the mass of the ladder and g is the acceleration due to gravity.

2. Normal Force (N): The normal force acts perpendicular to the surface of contact between the ladder and the ground. Since there is no vertical acceleration, the normal force exactly balances the gravitational force: N = mg.

3. Static Friction Force (f): The static friction force acts horizontally between the ladder and the ground. It prevents the ladder from sliding or slipping. The maximum static friction force is given by the equation f_max = μ_s * N, where μ_s is the coefficient of static friction and N is the normal force.

To prevent the ladder from slipping, the static friction force must balance the component of the gravitational force parallel to the ground. This component is given by m * g * sin(α), where α is the angle of the ladder relative to the wall.

Therefore, the maximum angle α can be found by setting the maximum static friction force equal to the component of the gravitational force parallel to the ground:

μ_s * N = m * g * sin(α)

Substituting N = mg, we have:

μ_s * mg = m * g * sin(α)

Simplifying and canceling out the mass:

μ_s = sin(α)

To find the maximum angle α, we can take the inverse sine (arcsine) of the coefficient of static friction:

α = arcsin(μ_s)

Plugging in the given coefficient of static friction (μ_s = 0.245), we can calculate the maximum angle α at which the ladder can lean without slipping.