In one line
Treat the ladder as a rigid body with normal and friction forces at floor and wall, put friction at its limiting value where slipping is impending, and use three equilibrium equations.
Explained simply
A ladder leaning against a wall tends to slide its foot outward and its top down. Floor friction stops the foot sliding; wall friction (if rough) stops the top sliding down. A steeper ladder is safer because it needs less friction.
Technically, the ladder is a general coplanar system with \( N_A, F_A \) at the floor and \( N_B, F_B \) at the wall; at limiting state \( F_A = \mu_1 N_A \) and \( F_B = \mu_2 N_B \).
How it works
- Draw the ladder AB at angle \( \theta \) to the floor, weight W at the middle, plus any person's weight at its position.
- At the foot A: \( N_A \) up, \( F_A \) towards the wall.
- At the top B: \( N_B \) away from the wall, \( F_B \) upward along the wall (zero if the wall is smooth).
- \( \Sigma F_x = 0 \): \( F_A = N_B \). \( \Sigma F_y = 0 \): \( N_A + F_B = \Sigma W \).
- \( \Sigma M_A = 0 \): moments of \( N_B \), \( F_B \) and the weights about the foot.
- Substitute limiting friction and solve for the unknown (angle, \( \mu \), or how far a person can climb).
Key formula
For a uniform ladder, smooth wall, rough floor: \( \tan\theta_{min} = \dfrac{1}{2\mu} \), where \( \theta \) is the angle with the floor and \( \mu \) the floor coefficient.
Quick example
Uniform ladder of weight 200 N, smooth wall, floor \( \mu = 0.4 \). \( N_A = 200 \) N, \( F_A = 0.4 \times 200 = 80 \) N \( = N_B \). Moments about A (length L): \( N_B L\sin\theta = 200 \cdot \tfrac{L}{2}\cos\theta \Rightarrow \tan\theta = 100/80 = 1.25 \), so \( \theta_{min} = 51.3^\circ \).
Common mistakes
- Drawing wall friction downward; when the top tends to slide down, friction on it acts upward.
- Using the ladder length instead of the perpendicular distances for moments.
- Assuming the wall is rough when the question says smooth (then \( F_B = 0 \)).
How MU asks it
A classic 6 to 8-mark numerical: 'A ladder of length ... rests against a rough wall ... find how far up a man can climb before slipping' or 'find the minimum angle'. Full marks: FBD, the three equations, limiting conditions stated, answer with units.