The Static Ladder Problem with Two Sources of Friction

The problem of a ladder leaning against a wall in static equilibrium is a classic example encountered in introductory mechanics texts.¹ Most discussions of this problem assume that the static frictional force between the ladder and wall can be ignored. A few authors^2,3 consider the case where the static friction coefficients between ladder/wall (μ_s,wall) and ladder/floor (μ_s,floor) are equal. This situation is indeterminate (i.e. without modeling the elasticity of the ladder, it is not possible to solve for all the external forces), but the critical angle beyond which the ladder will not remain in equilibrium can nevertheless be determined if the ladder is assumed to be perfectly rigid. The case where μ_s,wall ≠ μ_s,floor has also been considered by several authors.^4–6 Assuming a perfectly rigid ladder, we will show that the critical angle in this case can be determined through an interesting application of the inequality describing the magnitude of the static friction force. We will also show how it is possible to experimentally investigate the critical angle in this situation using commonly available lab equipment.