Final answer:
The critical angle at which a ladder starts to slip is found using static equilibrium principles, where the angle θ is the arctan of the coefficient of static friction (μ). For μ=0.4, the critical angle is tan⁻¹(0.4). This result is length and mass independent.
Step-by-step explanation:
To determine the critical angle at which a ladder begins to slip, we can use the principles of static equilibrium. The condition for no slip is that the frictional force at the base must be equal to or greater than the component of the ladder's weight parallel to the floor. This frictional force is also the product of the normal reaction force and the coefficient of static friction (μ).
From the condition of no slip, we can derive that tan θ = μ, where θ is the angle between the ladder and the ground. To find the angle where the ladder starts to slip, we take the inverse tangent of the coefficient of static friction: θ = tan⁻¹(0.4). Therefore, the critical angle, θ, at which the ladder will begin to slip is given by the inverse tangent of the coefficient of static friction.
Note that this result is independent of the ladder's length or mass, as these factors cancel out in the equilibrium equations. This principle is also applied in similar physics problems involving static friction, such as objects resting on inclines, as shown in various textbook problems.