**3.2. LIMITING FORCE OF FRICTION AND DEFINITIONS OF CERTAIN TERMS**

For defining the terms like co-efficient of friction *(µ) *and angle of friction (ɸ), consider a solid body placed on a horizontal plane surface as shown in Fig. 3.l.

Let W = Weight of body acting through C.G. downward,

*R *= Normal reaction of body acting through C -, G. upward,

*P *= Force acting on the body through C.G. and parallel to the horizontal surface.

If *P *is small, the body will not move as the force of friction acting on the body in the direction opposite to *P *will be more than *P. *But if the magnitude of *P *goes on increasing, a stage comes, when the solid body is on the point of motion. At this stage, the force of friction acting on the body is called *limiting force of friction. *The limiting force of friction is denoted by *F.*

Resolving the forces on the body vertically and horizontally we get

R = W

F = P

If the magnitude of *P *is further increased the body will start moving. The force of friction, acting on the body when the body is moving, is called *kinetic friction.*

**3.2.1. Co-efficient of Friction (µ).** It is defined as the ratio of the limiting force or friction *(F) *to the normal reaction (R) between· two bodies. It is denoted by the symbol 1.1. Thus

**3.2.2. Angle of Friction ****(µ).** It is defined as the angle made by the resultant of the

normal reaction *(R) *and the limiting force of friction *(F) *with the normal reaction *(R). *It is denoted by ɸ. Fig. 3.2 shows a solid body resting on a rough horizontal plane.

Let *S *= Resultant of the normal reaction *(R) *and limiting force of friction *(F)*

Then angle of friction = ɸ

= Angle between S and *R*

From Fig. 3.2, we have

Thus the tangent of the angle of friction is equal to the co-efficient of friction.

A block of weight W is placed on a rough horizontal plane surface as shown in Fig. 3.3 and a force *P *is applied at an angle θ with the horizontal such that the block just tends to move.

Let R = Normal reaction

µ = Co-efficient of friction

F = Force of friction

= µR

In this case the normal reaction *R *will not be equal to weight of the body. The normal reaction is obtained by resolving the forces on the block horizontally and

vertically. The force *P *is resolved in two components *i.e., *p cosθ the horizontal direction *and P *sinθ in the vertical direction.

Resolving forces on the block horizontally, we get

F = P cos θ

or µR = P cos θ

Resolving forces on the block vertically, we get

R +P sin θ = W

R = W – P sin θ … (ii)

From equation *(ii), *it is clear that normal reaction is not equal to the weight of the block.

If in equation *(ii), *the values of W, *P *and θ are known, then value of normal reaction *(R) *can be obtained. This value of* R *can be substituted in equation *(i) *to determine the value of coefficient of friction µ.

**Note**. (i) The force of friction is always equal to µR *(i.e., **F *= µR).

(ii) The normal reaction (R) is not equal to the weight of the body always.

**3.2.3. Cone of Friction**. It is defined as the right circular cone with vertex at the point of contact of the two bodies (or surfaces), axis in the direction of normal reaction *(R) *and semi-vertical angle equal to angle of friction (41). Fig. 3.4 shows the cone of friction in which,

O = Point of contact between two bodies

R = Normal reaction and also axis of the cone of friction

ɸ = Angle of friction.

**3.2.4. Types of Friction.** The friction is divided into following two types depending upon the nature of the two surfaces in contact:

1.Static friction, and

2.Dynamic friction.

If the two surfaces, which are in contact, are at rest, the force experienced by one surface is called static friction. But if one surface starts moving and the other is at rest, the force experienced by the moving surface is called dynamic friction. If between the two surfaces, no lubrication (oil or grease) is used, the friction, that exists between two surface is called *‘Solid Friction’ *or *‘Dry Friction’. *Solid friction or Dry friction is also known as *Coulomb friction. *

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