Friction directions on a leaning ladder A ladder rests on a smooth horizontal floor and leans against a rough vertical wall. Considering impending slip conditions, what is the correct direction of the frictional force at the upper contact with the wall?

Introduction / Context:
Equilibrium of a ladder against a wall is a classic statics problem that develops understanding of friction directions at multiple contacts. Here, the ground is smooth (no friction at floor) and the wall is rough (friction available at wall). Determining the correct direction of the frictional force is essential for writing equilibrium equations correctly.

Given Data / Assumptions:

• Smooth floor → zero friction at the floor; only a vertical normal from the floor.
• Rough vertical wall → normal is horizontal; friction acts vertically along the wall.
• Ladder tends to slide down at the top under its weight.
• Impending motion is considered for friction direction.

Concept / Approach:
Friction acts to oppose the relative motion at the contact. The top end of the ladder tends to move downward along the wall due to gravity. Therefore, at the wall contact, friction must act upward to oppose this impending downward motion.

Step-by-Step Solution:

Verification / Alternative check:
If you incorrectly assume friction is downward at the wall, the vertical forces would both point downward (weight and wall friction), making vertical equilibrium impossible because the floor provides no vertical friction (smooth). Thus the only self-consistent direction is upward at the wall.

Why Other Options Are Wrong:

• Horizontal directions (towards/away from wall) describe the wall normal, not friction.
• Downward at wall would assist the slide, not resist it.
• None of these is unnecessary because one option matches the correct direction.

Common Pitfalls:
Forgetting the floor is smooth; mixing up normal and friction directions; assuming directions first and fixing later—always start by considering the likely slip direction.

Final Answer:
Upwards at its upper end (vertical)