# Frictional force

Friction is define as the opposing force that try to resist the speed of the body.It is produced due to contact of two body or more with each other.

# Friction :

When we try to slide a body on a surface, the motion of the body is opposed by a force called the force of friction. The frictional force arises due to intermolecular interaction.

When a body is at rest on a surface and no external force is applied, no friction exists. In this case the normal reaction R balances the weight Mg i.e. for vertical equilibrium R = Mg and frictional force =0

No friction

When an external force (F) is applied to move the body and the body does not move, then the frictional force acts opposite to applied force F and is equal to the applied force i.e., F – f =0 frictional force ,f = applied force F. When the body remains at rest, the frictional force is called the static friction. Static friction is self adjusting force.

Static friction

When the external force F is increased, a stage comes when the body is just at the verge of moving. At this stage the force of friction is maximum and is called the limiting friction. It is found experimentally that Limiting frictional force

$f_s = {\mu}_s Mg$

where ${\mu}_s$ is called the coefficient of static friction. It is a dimensionless constant, but depends on the nature of surfaces in contact.

Thus the static friction is self adjusting force and upto the body is at rest,

the static friction = applied force.

Limiting friction

## Kinetic friction :

When the applied force is increased further (beyond the limiting frictional force), the body begins to move, then the force opposing the motion is called the kinetic or sliding friction. The kinetic friction is less than the limiting friction. The force of kinetic friction $f_k = {\mu}_k$ R where ${\mu}_k$ is called the coefficient of kinetic friction.

${\mu}_k < {\mu}s$

The kinetic frictional force in between two surfaces remains unchanged, whether the body moves with constant velocity or acceleration.

## Graph between applied force and frictional force :

If a graph is plotted between applied force and frictional force, then it is in the form of a straight line. Which tells that the

Frictional force = applied force.

Graph of applied force and frictional force

When the applied force increases beyond a certain value, the frictional force drops suddenly and becomes equal to ${\mu}_k$ R which remains constant.

## Rolling friction :

When a body rolls on the surface, the resistance offered by the surface is called Rolling friction. Rolling friction is less than static friction or kinetic friction. Rolling frictional coefficient is denoted by

${\mu}_R < {\mu}_K < {\mu}_s$

### Angle of Friction :

When a block of mass M is placed on a horizontal surface and pulled to the right by an external force, equal to limiting frictional force; then the resultant of normal reaction $\overrightarrow{R}$ and limiting frictional force  $\overrightarrow{F}_s is \overrightarrow{S}$ .The angle made by the resultant force $\overrightarrow{S}$ with the normal is called the angle of friction.

Angle of friction

$tan \lambda = \dfrac{f_s}{R}$  …………………….(1)

But coefficient of static friction

${\mu}_s = \dfrac{F}{R}$

As

$F = f_s$

$\therefore {\mu}_s = \dfrac{f_s}{R}$     ……………………..(2)

Comparing (1) and (2),

$\mu = \tan \lambda \textstyle{ and } \lambda = (tan^{-1} {\mu}_s)$

### Frictional Force on a bicycle in motion:

#### Wheel rotated about its axis :

When a wheel is rotated about its axis without sliding, frictional force acting on it is the rolling friction and it acts opposite to the direction of motion of its points of contact with the surface.For example in fig. wheel rotates clockwise, the frictional force acting on it will be forward.

#### (ii) When the bicycle is pedalled :

The force exerted on the rear wheel through the pedal –chain axle system is in the backward direction, therefore the force of friction on the rear wheel is in forward direction. The front wheel of cycle moves by itself in the forward direction, therefore the force of friction on the front—wheel is in the backward direction.

#### (iii) When the bicycle is not pedalled :

No external force is being exerted; both wheels move by itself in the forward direction, and so the frictional force on both the wheels is in the backward direction.
Body on Inclined plane ; Angle of Repose

Angle of repose

Suppose a body is placed on the inclined plane of slope angle $\alpha$, then the forces acting on the body are

(i)                  Its weight Mg (vertically downward),

(ii)                Normal reaction R

(iii)               Frictional force acting parallel to the plane upward

If we resolve the weight Mg, normal and parallel to the plane, then for equilibrium perpendicular to plane

$R = Mg \cos \alpha$  ……………………………..(1)

and for equilibrium parallel to plane.

$Mg \sin \alpha$  ……………………………….(2)

i.e. if body is at rest on inclined plane

frictional force, f = $Mg \sin \alpha$

If angle $\alpha$ of inclined plane increases, then Mg sin$\alpha$  increases and hence frictional force also increases. At one stage the body is just at the verge of sliding; at this stage, the frictional force is maximum and the slope angle of plane $\alpha$ is called the angle of repose. Thus angle of repose is the angle of inclined plane such that the block placed on it just starts sliding.

Thus

$f_s = Mg sin \alpha$

Also

$f_s = {\mu}_s R = {\mu}_s Mg cos \alpha$

or

$Mg sin \alpha = {\mu}_s Mg cos \alpha \rightarrow {\mu}_s = tan \alpha \rightarrow \alpha = tan^{-1} ({\mu}_s)$

As

$\lambda = tan^{-1} ({\mu}_s)$

So

$\alpha = \lambda$

Clearly angle of repose $\alpha$ is equal to its angle of friction $\lambda$

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