12 2-2 (e) A ladder against a smooth wall
Setting
- A uniform ladder of weight and length rests with its foot on rough horizontal ground and its top against a smooth vertical wall.
- Forces: weight at the middle; vertical ground reaction and friction at ; the wall reaction (horizontal, because the wall is smooth) at .
- Conditions: and ; moments about give .
- Limiting equilibrium: .
✍️ Worked examples
Example 17:
- A uniform ladder of weight makes an angle with the ground and is about to slip.
- The coefficient of friction at the ground is .
- Find .
Solution:
- and .
- Moments about the foot: .
- , so .
- With the ladder is on the point of slipping at ; it is safe for larger angles.
Example 18:
- A uniform ladder of weight 20 kg·wt rests at to the horizontal against a smooth wall on rough ground with .
- Show that a girl of weight 60 kg·wt can climb half the length of the ladder before it slips.
Solution:
- Let the ladder have length and the girl be along it from the foot.
- Vertical: kg·wt. Limiting friction: , and .
- Moments about the foot: .
- Left side: . Right side: .
- gives .
Example 19:
- A uniform ladder of weight 30 kg·wt and length 4 m rests on a smooth floor against a smooth wall at , held by a horizontal rope at the foot.
- Show the tension grows as a man of weight 80 kg·wt climbs.
Solution:
- Floor reaction ; the rope tension balances the wall reaction : .
- Moments about the foot with the man at distance along the ladder: , so .
- increases with .
- If the rope can bear at most 67 kg·wt, then gives m.