Short Answer:
The equilibrium of ladder problem deals with the conditions under which a ladder remains at rest while leaning against a smooth wall and resting on a rough horizontal surface. The ladder is said to be in equilibrium when all the forces acting on it—such as the weight, normal reactions, and friction—are balanced.
In simple terms, the equilibrium of a ladder means the ladder neither slips nor rotates under the influence of forces. To maintain equilibrium, the sum of all horizontal and vertical forces, as well as the moment about any point, must be equal to zero.
Detailed Explanation :
Equilibrium of Ladder Problem
The equilibrium of ladder is a classic problem in engineering mechanics and statics that helps to understand how different forces act on a rigid body. A ladder leaning against a wall represents a non-concurrent force system, where multiple forces act at different points and directions.
In this condition, for the ladder to remain stationary (i.e., in equilibrium), all the acting forces and moments must balance each other. The problem involves analyzing these forces and applying the conditions of equilibrium to determine unknown quantities such as frictional force, reaction forces, and angles at which the ladder will slip or remain stable.
Definition
The equilibrium of ladder can be defined as:
“The state in which a ladder remains stationary and does not slip or rotate under the action of various forces acting on it, due to the balance of all horizontal, vertical, and moment forces.”
This means that the ladder neither moves upward or downward, nor slides along the wall or ground.
Forces Acting on the Ladder
Consider a uniform ladder of length and weight , leaning against a smooth vertical wall and resting on a rough horizontal floor.
Let:
- = Point of contact of the ladder with the ground.
- = Point of contact of the ladder with the wall.
- = Angle of inclination of the ladder with the horizontal.
The following forces act on the ladder:
- Weight of the Ladder (W):
Acts vertically downward through the center of gravity (midpoint of the ladder). - Normal Reaction at the Wall (R₂):
Acts horizontally at the top end of the ladder (point B) and perpendicular to the wall. - Normal Reaction at the Floor (R₁):
Acts vertically upward at the bottom end of the ladder (point A). - Frictional Force at the Floor (F₁):
Acts horizontally at the bottom end (A), opposing the tendency of slipping.
Since the wall is smooth, no frictional force acts at the wall, only a normal reaction exists there.
Conditions of Equilibrium
For the ladder to remain in equilibrium, the following three conditions must be satisfied:
- Sum of all horizontal forces = 0
Therefore,
(The frictional force at the floor balances the reaction from the wall.)
- Sum of all vertical forces = 0
Therefore,
(The vertical reaction at the floor balances the weight of the ladder.)
- Sum of all moments about any point = 0
Taking moments about point A (the bottom of the ladder):
Simplifying,
Substituting :
This expression shows the frictional force required to keep the ladder in equilibrium at a given inclination.
Condition for No Slipping
The maximum frictional force at the floor is given by:
where is the coefficient of friction between the ladder and the floor.
For equilibrium (no slipping),
Substituting the expressions,
Simplifying,
or,
Hence, for equilibrium, the ladder must be inclined at an angle greater than or equal to .
If the angle of inclination is smaller, the ladder will slip.
Explanation of the Problem
When a person climbs the ladder, the total weight on the ladder increases. This causes a higher frictional force to be required at the base to maintain equilibrium. If the friction available at the floor is not enough to balance the increased load, the ladder will start slipping.
Thus, while solving ladder equilibrium problems, both the weight of the ladder and the weight of the person (if any) are considered together, acting at their respective centers of gravity.
Important Observations
- Wall Smoothness:
Since the wall is smooth, only a horizontal reaction acts at the top. - Frictional Force Location:
Friction acts only at the base (floor), opposing the sliding tendency. - Weight Distribution:
The ladder’s weight acts at its center, and additional load (like a person) shifts the center of gravity, affecting equilibrium. - Angle of Inclination:
A smaller angle of inclination increases the tendency to slip, while a larger angle increases stability. - Coefficient of Friction:
A higher coefficient of friction between the floor and ladder increases the ladder’s stability.
Example
A ladder of weight and length rests on a rough horizontal surface with . Determine the minimum angle for equilibrium.
Using the equilibrium condition:
Hence, the ladder must be inclined at an angle of at least 63.43° with the horizontal to remain in equilibrium.
Applications
- Design of fire escape ladders and construction ladders.
- Stability analysis in scaffolding and supports.
- Determining safety angles for ladders used in industrial and maintenance work.
- Used in statics problems to study non-concurrent force systems.
Precautions for Safety
- The angle of inclination should be sufficient to prevent slipping.
- The floor should not be oily or slippery.
- Rubber pads can be used at the bottom to increase friction.
- The weight on the ladder should be centered and not exceed the limit.
Conclusion
The equilibrium of ladder problem involves analyzing the forces and moments acting on a ladder to ensure it remains stable without slipping. For equilibrium, the sum of horizontal and vertical forces and the moment about any point must be zero. The friction at the base of the ladder plays a vital role in maintaining this balance. By understanding these conditions, engineers can design safe and stable ladder systems for real-life applications, preventing accidents and structural failures.