) A uniform ladder of mass m and length leans at an angle against a frictionless wall. If the coefficient of static friction between the ladder and the ground is determine a formula for the minimum angle at which the ladder will not slip.
To determine the minimum angle at which the ladder will not slip, we need to consider the forces acting on the ladder.
1. Weight force (mg): This is the force due to gravity acting on the ladder. It acts vertically downward at the center of mass of the ladder.
2. Normal force (N): This is the force exerted by the ground on the ladder perpendicular to the surface of contact. It acts vertically upward through the point of contact between the ladder and the ground.
3. Friction force (f): This is the force of static friction between the ladder and the ground. It acts parallel to the surface of contact, opposing the tendency of the ladder to slip.
To prevent the ladder from slipping, the friction force must be equal to or greater than the maximum possible friction force, which is given by the coefficient of static friction (μ) multiplied by the normal force (N).
The normal force can be determined by considering the equilibrium of forces in the vertical direction:
N - mg cos(θ) = 0
N = mg cos(θ)
The friction force can be determined by considering the equilibrium of forces in the horizontal direction:
mg sin(θ) - f = 0
f = mg sin(θ)
Setting the friction force equal to the maximum possible friction force gives:
μN = mg sin(θ)
μmg cos(θ) = mg sin(θ)
μ cos(θ) = sin(θ)
μ = tan(θ)
Now, to find the minimum angle at which the ladder will not slip, we need to find the angle (θ) for which the coefficient of static friction (μ) is equal to the tangent of that angle.
The minimum angle (θ_min) can be found using the inverse tangent function:
θ_min = arctan(μ)
So, the formula for the minimum angle at which the ladder will not slip is:
θ_min = arctan(μ)