Short Answer:

The angle of friction is the angle made between the resultant of the limiting frictional force and the normal reaction acting on a body. It represents the maximum inclination of the resultant force before the body starts to move. The angle of friction depends on the coefficient of friction between the two surfaces. Mathematically,

where φ is the angle of friction and μ is the coefficient of friction. It helps in understanding how friction resists motion between two contacting surfaces.

Detailed Explanation :

Angle of Friction

In Engineering Mechanics, when a body is resting on a surface, two forces act at the point of contact — the normal reaction (N) perpendicular to the surface and the frictional force (F) tangential to it. The resultant of these two forces makes an angle with the normal reaction. This angle is known as the angle of friction.

The concept of the angle of friction is very important because it provides a geometric way to represent the frictional resistance between two surfaces. It helps engineers in the design and analysis of machines, wedges, screws, belts, and inclined planes, where friction plays a crucial role.

Definition

The angle of friction is defined as:
“The angle between the resultant of the limiting frictional force and the normal reaction acting on a body.”

In simple words, when the applied force increases and reaches the value of limiting friction, the body is just about to move. At this condition, the frictional force and the normal reaction together form a resultant force, which makes an angle φ with the normal reaction. This angle φ is called the angle of friction.

Mathematically,

where,

φ = Angle of friction
F = Limiting frictional force
N = Normal reaction

Since the ratio F/N is the coefficient of friction (μ),

Explanation of Angle of Friction

When a body rests on a horizontal surface, it is subjected to two main forces:

Normal Reaction (N) — acts perpendicular to the surface, and
Frictional Force (F) — acts tangentially to the surface, opposing motion.

If the body is in equilibrium (not moving), these two forces balance the applied force. When the applied force increases gradually, the frictional force also increases up to a certain maximum value, called limiting friction.

At the limiting condition, if we combine F and N, their resultant force R will make an angle φ with the normal reaction N. This angle represents how strong the frictional resistance is between the two surfaces. The greater the coefficient of friction, the larger will be the angle of friction.

Hence,

If μ is small (smooth surfaces), φ will be small.
If μ is large (rough surfaces), φ will be large.

Graphical Representation

Consider a body resting on a surface:

The normal reaction (N) acts vertically upward.
The frictional force (F) acts horizontally along the surface, opposing the motion.
The resultant R of these two forces is obtained by vector addition, and it makes an angle φ with the normal reaction N.

As the applied force increases, the angle φ also increases until the body is about to move. At this stage, φ reaches its maximum value, known as the angle of limiting friction.

Relation Between Angle of Friction and Coefficient of Friction

Let,

F = Limiting frictional force
N = Normal reaction
R = Resultant of F and N

From the geometry of forces,

Since the ratio of F/N represents the coefficient of friction (μ),

Therefore,

This relationship shows that the angle of friction is entirely dependent on the coefficient of friction. It also provides a way to determine μ experimentally by measuring φ.

Physical Meaning

The angle of friction gives a clear understanding of how much a surface resists motion:

A larger angle of friction means higher resistance to sliding (rough surfaces).
A smaller angle of friction means lower resistance to sliding (smooth surfaces).

For example:

For μ = 0.577,

This means that if a surface has a coefficient of friction of 0.577, the resultant force makes a 30° angle with the normal.

Angle of Limiting Friction

The angle of limiting friction is the maximum value of the angle of friction at the point where motion is about to start. Beyond this angle, the frictional force cannot increase, and the body will begin to slide.

Thus,

It means that when the resultant of the normal reaction and frictional force makes an angle equal to the angle of limiting friction with the normal, the body is on the verge of motion.

Practical Applications

Inclined Planes:
Helps in determining the maximum slope of a plane on which an object will rest without sliding.
Design of Screws and Wedges:
The angle of friction determines the self-locking condition of screws and wedges.
Belt and Rope Drives:
Used in calculating frictional resistance between the belt and pulley.
Brakes and Clutches:
Helps in designing surfaces that provide sufficient friction for safe operation.
Structural Stability:
Used to prevent slipping in retaining walls and earth slopes.

Example

If the coefficient of friction between a box and the floor is 0.5, find the angle of friction.

Hence, the angle of friction is 26.57°.

Importance in Engineering

The concept of the angle of friction helps engineers:

Determine whether a body will slide or remain stable.
Design machines and components that either use or minimize friction.
Calculate safe angles for slopes, ladders, and conveyors.
Evaluate energy losses due to friction in mechanical systems.

Conclusion

In conclusion, the angle of friction is the angle made by the resultant of the limiting frictional force and the normal reaction acting on a body. It provides a geometric interpretation of the resistance between two surfaces in contact. The relationship tan φ = μ shows that the angle of friction depends directly on the coefficient of friction. A higher coefficient results in a larger angle, indicating rougher surfaces. The concept of angle of friction is vital in engineering to ensure stability, prevent slipping, and design efficient mechanical systems.