A uniform ladder 5.0 m long rests against a frictionless, vertical wall with its lower end 3.0 m from the wall. The ladder weighs 160 N. The coefficient of static friction between the foot of the ladder and the ground is 0.40. A man weighing 740 N climbs slowly up the ladder. Start by drawing a free-body diagram of the ladder. What is the maximum friction force that the ground can exert on the ladder at its lower end?

Verified step by step guidance

1. Begin by drawing a free-body diagram of the ladder. Identify all the forces acting on the ladder: the weight of the ladder acting downward at its center of gravity, the weight of the man acting downward at a variable position along the ladder, the normal force exerted by the ground acting upward at the base of the ladder, the frictional force acting horizontally at the base of the ladder, and the normal force exerted by the wall acting horizontally at the top of the ladder.
2. Calculate the maximum static friction force that the ground can exert on the ladder using the formula: \( f_{\text{max}} = \mu_s \cdot N \), where \( \mu_s \) is the coefficient of static friction and \( N \) is the normal force. Since the ladder is in equilibrium, the normal force \( N \) is equal to the total weight of the ladder and the man when he is on the ladder.
3. Determine the total weight acting on the ladder. This is the sum of the weight of the ladder and the weight of the man: \( W_{\text{total}} = W_{\text{ladder}} + W_{\text{man}} = 160 \text{ N} + 740 \text{ N} \).
4. Substitute the total weight into the equation for the normal force: \( N = W_{\text{total}} = 900 \text{ N} \).
5. Calculate the maximum static friction force: \( f_{\text{max}} = 0.40 \times 900 \text{ N} \). This gives the maximum frictional force that the ground can exert on the ladder at its lower end.

Key Concepts

Free-Body Diagram

A free-body diagram is a graphical representation used to visualize the forces acting on an object. In this scenario, it helps identify forces such as the weight of the ladder, the weight of the man, the normal force from the ground, and the frictional force. Drawing this diagram is crucial for understanding the balance of forces and solving for unknowns like the maximum friction force.

Static Friction

Static friction is the force that prevents surfaces from sliding past each other. It acts at the contact point between the ladder and the ground, opposing any motion. The maximum static friction force is calculated using the coefficient of static friction and the normal force. Understanding static friction is essential to determine the ladder's stability and the maximum force before slipping occurs.

Torque and Equilibrium

Torque is the rotational equivalent of force, calculated as the product of force and the perpendicular distance from the pivot point. For the ladder in equilibrium, the sum of torques around any point must be zero. This concept helps analyze how the ladder remains stable as the man climbs, ensuring that the forces and torques are balanced to prevent rotation or slipping.