The 180-lb man climbs up the ladder and stops at the...

The 180-lb man climbs up the ladder and stops at the position shown after he senses that the ladder is on the verge of slipping. Determine the inclination $\theta$ of the ladder if the coefficient of static friction between the friction pad $A$ and the ground is $\mu_{s}=0.4 .$ Assume the wall at $B$ is smooth. The center of gravity for the man is at $G$. Neglect the weight of the ladder.

Key Concepts

Static Equilibrium

This concept involves ensuring that the sum of all forces and the sum of all moments (torques) acting on the ladder are zero. It is fundamental in problems where structures or bodies remain at rest, meaning they are not undergoing linear or rotational acceleration. Understanding static equilibrium allows us to set up the conditions under which the forces due to weights, friction, and normal reactions balance each other.

Free Body Diagram

A free body diagram is a graphical representation that isolates a body and shows all external forces acting on it. In this context, the diagram would include the weight of the man, the normal forces at the base and the wall, the friction force at the ground, and any applied forces resulting from the ladder’s geometry. This visual breakdown is critical in setting up the equations based on the equilibrium conditions.

Friction and Coefficient of Friction

Friction is the force that opposes motion between two contacting surfaces. In static friction problems like this, the coefficient of static friction (?s) defines the maximum friction force available before sliding occurs. It relates directly to the normal force at the contact surface. This concept is essential in determining the limits of the ladder’s stability against slipping.

Torque/Moment Equilibrium

Torque or moment equilibrium involves ensuring that the sum of all moments about any pivot point is zero. This is particularly useful in ladder problems where forces act at different distances from the pivot. By choosing a strategic pivot point, often where unknown forces are applied, one can set up an equation that simplifies the computation of the ladder’s inclination angle.