Short Answer:

Limiting friction is the maximum value of the frictional force that can act between two surfaces before motion begins. It is the frictional force just before a body starts to move when an external force is applied. When the applied force increases gradually, the frictional force also increases until it reaches its maximum value — this point is known as the limiting condition. Once the applied force exceeds the limiting friction, the body starts to slide, and kinetic friction takes over.

Detailed Explanation :

Limiting Friction

In Engineering Mechanics, friction plays an important role in determining whether a body remains at rest or starts moving when an external force is applied. When a body is placed on a surface, and a small external force is applied, the body initially does not move because friction opposes this force. This opposing force increases with the applied force up to a certain limit. The maximum frictional force that develops just before the body begins to slide is called limiting friction.

It represents the transition point between static and kinetic friction. Once the applied force slightly exceeds this limit, the body overcomes resistance and begins to move, and the frictional force acting afterward is kinetic friction, which is smaller in magnitude.

Definition

The limiting friction is defined as:
“The maximum value of the static frictional force which comes into play when a body is just about to move over the surface of another body.”

In mathematical form,

where,

= Limiting friction
= Coefficient of static friction
= Normal reaction between the two surfaces

The limiting friction is directly proportional to the normal reaction between the surfaces.

Explanation of Limiting Friction

When an external force is applied to a body at rest, the frictional force acts in the opposite direction of the applied force to resist motion. Initially, the frictional force is equal in magnitude to the applied force, keeping the body in equilibrium.

As the applied force is gradually increased, the frictional force also increases up to its maximum value. At this stage, the body is on the verge of motion, but not yet moving. This highest value of static frictional force is called limiting friction.

Once the applied force exceeds the limiting friction, the body starts moving, and the frictional force drops slightly to a lower value known as kinetic friction.

Conditions at Limiting Friction

At the point of limiting friction:

The body is just about to move, but motion has not yet started.
The frictional force reaches its maximum possible value.
The applied force is equal to the limiting friction.
The resultant force of the normal reaction and frictional force makes a constant angle with the normal, known as the angle of friction (φ).

Mathematically,

Thus, the angle of friction gives a geometric interpretation of the limiting condition.

Factors Affecting Limiting Friction

Several factors influence the value of limiting friction between two surfaces:

Nature of the Surfaces:
Rough surfaces produce greater interlocking between surface irregularities, resulting in higher limiting friction. Smooth surfaces have lower limiting friction.
Normal Reaction (N):
The limiting friction is directly proportional to the normal reaction. Heavier objects have higher normal forces and thus greater limiting friction.
Material Properties:
The type of material in contact affects the coefficient of friction, such as wood on wood, steel on steel, or rubber on concrete.
Surface Condition:
Presence of dust, oil, or lubrication changes the surface interaction, reducing limiting friction.
Temperature and Wear:
High temperatures and worn surfaces can alter the frictional behavior, either increasing or decreasing limiting friction.

Experimental Verification

The limiting friction can be verified through simple experiments:

Block on a Horizontal Surface:
- Place a block on a flat surface.
- Gradually increase the applied force using a spring balance.
- Note the force just before the block starts to move — this value gives the limiting friction.
Inclined Plane Method:
- Gradually tilt the plane until the body just begins to slide.
- The angle at which the body starts to move is the angle of repose (θ).
- The coefficient of friction can be found using , and thus .

Graphical Representation of Limiting Friction

If we plot a graph between the applied force and the frictional force:

Initially, the frictional force increases linearly with the applied force.
It reaches a maximum value at the point of limiting friction.
Beyond this point, when the body starts to move, the frictional force slightly decreases and remains constant as kinetic friction.

This graph visually demonstrates the transition from static friction → limiting friction → kinetic friction.

Importance of Limiting Friction

The concept of limiting friction is essential in many areas of mechanical and civil engineering:

Design of Brakes and Clutches:
Limiting friction determines the maximum force that can be transmitted without slipping.
Structural Stability:
Prevents sliding of objects or materials on inclined surfaces like slopes and embankments.
Machine Components:
Helps in calculating power loss due to friction and designing efficient contact surfaces.
Transportation:
Tire-road grip is based on limiting friction, ensuring vehicle safety during braking or turning.
Inclined Plane Analysis:
Used to calculate whether a block will remain stationary or start sliding.

Difference Between Limiting Friction and Kinetic Friction

Limiting Friction: The maximum value of static friction when the body is about to move.
Kinetic Friction: The constant frictional force acting after the body has started moving.

Generally,

because once motion starts, there is less interlocking between surface irregularities.

Practical Example

Suppose a block weighing 100 N is placed on a horizontal table. If it starts moving when a force of 25 N is applied, the limiting friction is 25 N.
The coefficient of friction can be found as,

This means the frictional resistance is 25% of the body’s normal reaction.

Relation Between Limiting Friction and Angle of Friction

The angle of friction (φ) is related to limiting friction as:

Thus, the angle of friction is the measure of limiting friction in terms of geometry. A higher coefficient of friction corresponds to a greater angle of friction.

Conclusion

In conclusion, limiting friction is the maximum frictional resistance that can act between two surfaces before motion begins. It represents the boundary between static and kinetic friction and depends on the normal reaction, surface texture, and coefficient of friction. Mathematically, . Understanding limiting friction is essential for designing stable structures, ensuring safety in motion systems, and improving efficiency in mechanical components that involve contact and motion.