Friction is the opposite force that is set up between the surface of contact, when one body slides or rolls or tends to do so on the surface of another body.

It may be pointed out that frictional force comes into play not only when a solid body moves over another solid surface but a solid body encounters it, while it moves through a liquid or a gas. Friction in liquids and gasses is termed as viscosity. Further, friction is altogether different from inertia. The term inertia refers to the fact that a body continues to be in its state of rest or of uniform motion in the absence of an external applied force. Even a small net external force is capable of producing an acceleration in the motion of the body in spite of inertia. On the other hand, the term friction refers to some opposing force that comes into play, whenever a body tends to move or moves over a surface.

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As defined above frictional force is the resistance offered by minutely projecting particles of a body when it moves over another body. Frictional force has a remarkable property of adjusting itself in magnitude to the force producing or tending to produce the motion so that motion is prevented.

However, there is a limit beyond which the magnitude of this force cannot increase. If the applied force is more than this limit, there will be movement of one body over the other.

This limiting value of frictional force when the motion is impending, is known as Limiting Friction. It may be noted that when the applied force is less than the limiting friction, the body remains at rest and such frictional force is called Static Friction, which will be having any value between zero and the limiting friction. If the value of applied force exceeds the limiting friction, the body starts moving over the other body and the frictional resistance experienced by the body while moving is known as Dynamic Friction. Dynamic friction is found to be less than limiting friction.

STATIC FRICTION

Consider a body of mass M lying at rest on a horizontal table. As such, the weight Mg of the block is balanced by the normal reaction (R) of the table. Suppose that a force F is applied on the body along horizontally, which is small enough to move it [Fig below]. However, according to Newton's second law, the body must move with an acceleration,

$$a=\frac{F}{M}$$

Now, as the body remains at rest, it implies that an opposing force equal to the applied force must have come into play resulting in zero net force on the body. This force acts parallel to the surface of the body in contact with the table in a direction opposite to that of the applied force and is called static friction. It is denoted by $$F_{s}$$

Thus, static friction is the opposing force that is set up between the surfaces of contact of the two bodies, when one body tends to slide over the surface of another body.

The following points may be noted about the static friction.

1. The static friction depends on the nature of surfaces of the two bodies in contact.

2. The static friction does not exist by itself. When there is no applied force, there is no static friction. It comes into play only when the applied force tends to move the body.

3. The static friction opposes the impending motion.

4. The static friction is a self-adjusting force.

The term impending motion means the motion that would take place under the effect of the applied force, if friction were absent.{alertInfo}

Again consider that a body is lying on a horizontal surface. Give a gentle push to the body, so that it remains at rest. It implies that the applied force on the body has been exactly balanced by the static frictional force acting at the interface of the body and the horizontal surface. Now, increase the push on the body a little. The body may still remain at rest. Obviously, the force of friction has also increased and has become equal and opposite to the applied force. It shows that the force of friction always adjusts itself equal to the applied force i.e. as much force of friction comes into play as is just necessary to prevent the body from moving. Hence, the static friction is a self-adjusting force.

There is, however, a limit upto which the force of friction can increase with the increase in applied force. A stage comes, when the body just starts sliding.

The maximum value of the force of friction which comes into play before a body just begins to slide over the surface of another body is called the limiting value of static friction.

5. The limiting value of static friction is independent of the area of the surfaces of the two bodies in contact. It is directly proportional to the normal reaction i.e.

$$\left ( F_{s} \right )_{max} ∝ R$$

$$or \left ( F_{s} \right )_{max}=\mu _{s}R,$$

where the constant of proportionality is called the coefficient of static friction. Its value depends on the nature of surfaces of the two bodies in contact.

$$Obviously, F_{s}\leq \mu _{s}R$$

KINETIC FRICTION

We know that when the applied force on a body is small, it may not move. But set as the applied motion force [Fig below].The becomes force is greater of friction than the acting force between limiting friction, two surfaces, the body when one surface is in relative motion over the other surface, is called sliding or kinetic friction. It is denoted by $$F_{k}$$

Thus, kinetic friction is the opposing force that is set up between the surfaces of contact of the two bodies, when one body is in relative motion over the surface of another body.

The following points may be noted about the static friction.

1. The kinetic friction depends on the nature of surfaces of the two bodies relative motion.

2. The kinetic friction is independent of the area of the surfaces of the two bodies in contact.

3. The kinetic friction is nearly independent of the velocity, with which the body moves.

4. The kinetic friction is always less than the limiting static friction.

Fig above shows the variation of static and kinetic friction with the applied force.The part OA of the graph which is a straight line inclined equally to the two axes shows the self adjusting nature of the force of friction. The body remains at rest till the applied force does not exceed OL and likewise AL represents maximum value of static friction i.e. limiting static friction $$\left ( F_{s} \right )_{max}$$ Once the body starts moving, the force of friction drops to a value BM, slightly less than limiting static friction. Thus, BM represents the kinetic friction $$F_{k}.$$

5. It is directly proportional to the normal reaction i.e.

$$F_{k} ∝ R$$

$$F_{k}= μ_{k} R,$$

where the constant of proportionality $$μ_{k}$$ is called the coefficient of kinetic friction. Its value depends on the nature of surfaces of the two bodies in contact and is always less than the coefficient of static friction.

6. After the relative motion has begun, the acceleration of the body,

$$a=\frac{F-F_{k}}{M}$$

where F is the applied force and M, the mass of the body.

7. When the body moves with a constant velocity

$$F_{k}=F$$

8. When the applied force has been removed, the acceleration of the body,

$$a=-\frac{F_{k}}{M}$$

The negative value of the acceleration indicates  that the body comes to stop due to kinetic friction on removing the applied force.

LAWS OF LIMITING STATIC FRICTION

The limiting static friction obeys the following laws, which are based on experimental observations only :

1. The value of the limiting static friction depends upon the nature of the two surfaces in contact and their state of roughness.

2. The force of friction is tangential (parallel) to the two surfaces in contact and acts opposite to the direction in which the body would start moving on applying the force.

3. The value of limiting static friction between two given surfaces is directly proportional to the normal reaction between the two surfaces.

Consider that a body is lying on a horizontal surface [Fig above]. If R is normal reaction and F, the limiting static friction* (the value of applied force, when the body just begins to slide), then

F ∝ R

F=μ R

The constant of proportionality u is known as the coefficient of static friction.

Thus,  $$\mu =\frac{F}{R}=\frac{limiting friction}{normal reaction}$$

Therefore, the coefficient of static friction is defined as the ratio of limiting friction to the normal reaction.

4. The value of limiting friction for any two given surfaces is independent of the shape or area of the surfaces in contact so long as the normal reaction remains the same.

The laws of limiting friction are also applicable to kinetic friction. As the kinetic friction is quite smaller than the limiting friction, the coefficient of kinetic friction given by

$$\mu_{k} =\frac{F}{R}=\frac{kinetic friction}{normal reaction}$$

is also much smaller than the coefficient of static friction.

Therefore, the coefficient of kinetic friction is defined as the ratio of kinetic friction to the normal reaction.

VERIFY THE LAWS OF LIMITING STATIC FRICTION

The laws of static friction can be verified by making use of an inclined plane apparatus and the experimental arrangement as shown in Fig below

1. To verify that limiting static friction is directly proportional to the normal reaction (F ∝ R) : Place the inclined plane apparatus horizontally on the table. Place a rectangular wooden block A over its surface and connect it to a scale pan with a thread after passing it over the pulley fixed to the edge of the inclined plane apparatus. From a weight box, start adding weights to the scale pan and at the same time, keep on tapping the surface of the inclined plane gently. The weights are added, till the block just begins to slide. Since the block has just started sliding, the applied force P (weight of the pan+ weights placed in the pan) must be just equal to the force of limiting friction F between the block and the surface of the inclined plane. Obviously, the normal reaction R is equal to the weight W of the block.

Now, place another identical block B over the block A as shown in Fig below. Again repeat the experiment by adding weights to the scale pan, till the system of the two blocks just begins to slide. It will be found that now the applied force will be 2 P i.e. twice as that in the first observation. Obviously, the normal reaction R' is equal to the weight 2 W.

It verifies that the limiting static friction is directly proportional to the normal reaction.

2. To verify that limiting static friction is independent of area of the surfaces in contact : Now join the two blocks with a piece of thread and place them over the inclined plane apparatus as shown in Fig below. It may be noted that the normal reaction for each block is W i.e. weight of a block. Repeat the experiment as before by adding weights to the scale pan, till the system of the two blocks just begins to slide. It will be found that the applied force required to make the two blocks to just slide is P i.e. same as that for a single block.

It verifies that the limiting static friction is independent of the area of the surfaces in contact, so long as the normal reaction remains the same.

3. To verify that limiting static friction depends on the nature of the surfaces in contact : If the experiment is repeated with a metallic block (in place of wooden block), it will be found that the applied force required to make it just slide is smaller than that required in the case of wooden block.

It verifies that the limiting static friction depends on the nature of surfaces.

ANGLE OF FRICTION

The angle of friction can be defined as angle which the resultant of limiting friction and normal reaction makes with the normal reaction.

In fig below, the resultant of limiting friction and normal reaction R make an angle α with the normal reaction. Therefore, by definition, α is the angle of friction.

It follows that

$$tan \alpha =\frac{BC}{OB}=\frac{OA}{OB}$$

$$tan \alpha =\frac{F}{R}$$

But $$\frac{F}{R}=μ$$, the coefficient of static friction

tanα= μ     ...(i)

Hence, coefficient of static friction is equal to tangent of angle of friction.

ANGLE OF REPOSE

The angle of repose is defined as the angle of the inclined plane at which a body placed on it just begins to slide.

Consider an inclined plane, whose inclination with horizontal is gradually increased, till the body placed on its surface just begins to slide down. If θ is the inclination at which the body just begins to slide down, then θ is called the angle of repose [Fig below].

The body is under the action of the following forces :

(i) The weight M g of the body acting vertically downwards.

(ii) The limiting friction F acting along the inclined plane in the upward direction. In magnitude, it is equal to the component of the weight M g acting along the inclined plane i.e.

F= Mg sin θ       ...(ii)

The normal reaction R acting at right angle to the inclined plane in the upward direction. It is equal to the component of the weight M g acting perpendicular to the inclined plane i.e.

R = Mg cos θ     ...(iii)

Dividing the equation (ii) by (iii), we have

$$\frac{F}{R}=\frac{Mg sin θ }{Mg cos θ }=tan θ$$

$$\frac{F}{R}=\mu$$

tan θ = μ      ...(iv)

Therefore, the coefficient of static friction is equal to the tangent of the angle of repose.

From the equations i (from angle of friction) and equation (iv), it follows that

θ = μ

i.e. angle of repose is equal to angle of friction.

Work Done Against Friction

Let's find the work done against friction in following two cases:

1 Along a horizontal surface. Consider that a block of mass M is lying over a rough horizontal surface(fig below). Let  $$\mu _{k}$$ be the coefficient of kinematics friction between two surfaces is in contact. The force of friction between the block and horizontal surface is given by

$$F= \mu _{k} R= \mu _{k}Mg$$        ( ∵ R= Mg)

To move the block without acceleration, the force (P) required will be just equal to the force of friction i.e.

$$P = F =\mu _{k}Mg$$

If S is the distance moved, then work done is given by

W = P x S = $$\mu _{k}Mg S$$

2. Along an inclined plane. Consider that a block of mass M is to be moved along an inclined plane AB of length S inclined at an angle θ to the horizontal. The weight of the block can be resolved into two components :

(i) M g sin θ parallel to the inclined plane and

(ii) M g cos θ perpendicular to the inclined plane.

Therefore, normal reaction, R = M g cos θ

(a) When the block is moved up the inclined plane. When the block is moved up the inclined plane [Fig below], the force of friction,

$$F = \mu _{k}R = \mu _{k}Mg cos θ$$

acts parallel to the inclined plane and in downward direction. Therefore, in order to move the block up the inclined plane,the required force is given by

$$P = M g sin θ + F = M g sin θ + \mu _{k}Mg cos θ$$

$$or P = Mg (sin θ + \mu _{k} cos θ)$$

Hence, work done to move the block up the inclined plane,

W = P x S = $$Mg (sin θ + \mu _{k} cos θ) S$$

(b) When the block is moved down the inclined plane. When the block is moved down the inclined plane [Fig below], the force of friction,

$$F = \mu _{k} R = \mu _{k} M g cos θ$$

acts parallel  to the incline plane and in upward direction. Therefore, in order to move the block down the inclined plane, the required forece is given by

$$P = F - M g sin θ = \mu _{k}Mg cos θ - M g sin θ$$

$$P = M g (\mu _{k}cos θ - sin θ)$$

Hence,  work done to move the block down the incline plane,

W= P x S = $$Mg(\mu _{k} cos θ - sin θ)S$$

ROLLING FRICTION

In principle, if a body (such as a sphere, cylinder or a ring) rolls without slipping on a horizontal surface, it will not experience any friction. It is because the contact between the body and the horizontal surface is just at one point. Therefore, in absence of friction, the body should roll with constant velocity. However, in practice, it does not happen so. In order to keep the body rolling, some force has to be applied. It implies that some friction is there, which opposes the rolling motion and this friction is called rolling friction.

The coefficient of rolling friction $$(\mu _{r} )$$ between two surfaces is found to be much smaller than the coefficient of sliding friction $$(\mu _{k})$$ for the same two surfaces.

Cause of rolling friction. To explain the cause of rolling friction, consider a wheel rolling on a horizontal surface [Fig above]. When the wheel moves on the level surface, it causes a little depression and a small 'hump' is created just ahead of it. As a result, the contact between the wheel and the surface occurs over a finite area i.e. it no longer remains a point contact. In climbing up the hump, the wheel encounters some opposition to the motion, which is called the rolling friction. In fact, the wheel continuously climbs up such humps due to the depressions caused on the level surface due to rolling of the wheel.

The cause of rolling friction can be explained in another way also. When the wheel climbs up the depression, the contact force on the wheel acts along an oblique direction. The component of the contact force parallel to the level surface opposes the motion and is called the rolling friction

CONVERT SLIDING FRICTION INTO ROLLING FRICTION

It is found that it is much easier to move a heavy load from one place to another by placing it over a cart with wheels than to move it by sliding it over the surface. It is because of the fact that rolling friction between two surfaces is much smaller than the sliding friction between them. It is for this reason that wheel has been considered as one of the greatest inventions. In practice, as far as possible, sliding friction is converted into rolling friction by making use of wheels and bearings.

A ball bearing consists of two coaxial cylinders, between which hard steel balls are arranged as shown in Fig above. The axle attached to the vehicle fits tightly into the inner cylinder, while the wheel is put in firm contact with the outer cylinder. When the axle rotates, say in clockwise direction, the steel balls rotate in anticlockwise direction and thus allowing the outer cylinder and the wheels to rotate in anticlockwise direction. As the two cylinders have rolling motion relative to each other, friction gets reduced to a large extent.

Friction is a non-conservative force and it always opposes the motion. However, it has advantages as well as disadvantages. For this reason, friction is called a necessary evil.

1. Friction helps us to walk. When we press the ground with our foot backwards, we receive a forward reaction due to friction between the foot and the ground. Had there been no friction, the foot will simply slip away.

2. Friction helps us to write on the blackboard or on a paper. As we write, the chalk particles stay on the board due to friction.

3. Friction helps us to tie knots in strings and ropes. The knots will untie readily in the absence of friction.

4. Brakes make use of friction to stop the vehicles. Special high friction materials are used for the brakes of automobiles.

5. For driving vehicles on the road, a certain minimum amount of friction is necessary. Sometimes, it proves quite dangerous to drive on a slippery road.

6. The sand is sometimes thrown on the snow covered tracks or on the rails of locomotives after rains to allow the forward motion without the slipping of the wheels.

7. Treading of tyres of vehicles is done in order to increase friction.

8. The chains are attached to the wheels of vehicles plying on icy roads in order to increase friction.

9. Friction helps to transmit power from the motors and engines to other machines by making use of belts and clutches.

1. The friction causes unnecessary wear and tear of the machinery.

2. Due to friction between the moving parts of a machine, heat is produced, which in a way affects the working as well as the life of the machine.

3. A part of useful energy is dissipated in overcoming the friction.

4. A part of the fuel in the engines and other vehicles is used up to overcome friction.

METHODS TO REDUCE FRICTION

Since friction has disadvantages also, sometimes we have to devise the means to reduce it. Following are a few methods to reduce the friction :

1. By proper selection of materials. The friction depends upon the nature of materials. For example, friction between concrete and iron is higher than that between rubber and concrete. So, tires are made of rubber. Friction may also be reduced by lining the moving parts of machines with the materials of low coefficient of friction.

2. By polishing. Polishing a surface means depositing a fine layer of a suitable material on the surface of a body. This material fills up the space between the projections on the surface and makes it smooth. Thus, interlocking between the body and the surface decreases and the surface offers less friction to a body moving over it. Another method by which irregularities of the surfaces can be decreased is rubbing and it helps to reduce the friction.

3. By lubrication. Oil or grease, when put between the two surfaces in contact, spreads, forming a thin layer between them. This avoids direct contact of solid surfaces. In fact, by spreading the lubricants, the dry friction is converted into fluid friction, which is comparatively of lesser magnitude. The lubrication reduces friction as well as reduces the heating of the moving parts. Flow of compressed air is also used as lubricant. It reduces the friction between the moving parts by acting as an elastic cushion. In this manner, it not only reduces friction but also prevents dust particles from collecting over the moving parts. The compressed air as a lubricant has another advantage that it takes away the heat, which is produced between the moving parts.

4. By air cushion. If air is blown so as to maintain a thin cushion of air between the two moving solid surfaces, friction between the surfaces decreases.

5. By converting sliding friction into rolling friction. The friction during rolling becomes much smaller than during sliding. The wheels and ball bearings are used to reduce friction as they convert sliding friction into rolling friction.

6. By streamlining. It implies giving typical shape (sharp in the front) to the high speed vehicles. The fluid friction decreases due to streamlining. It is for this reason that airplanes, jets, etc are given such typical shapes at the front.

QUIZ

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Q. What is the unit of coefficient of limiting friction ?

Ans. It has no unit.

Q. What is the relation between angle of repose and angle of friction ?

Ans. Angle of friction and angle of repose are equal.

Q. Why is friction a non-conservative force ?

Ans. It is because, work done against friction along a closed path is non-zero.

Q. What happens to limiting friction, when a wooden block is moved with increasing speed on a horizontal surface ?

Ans. The limiting friction decreases as the wooden block is moved with increasing speed on the horizontal surface.

Q. Why are tyres made of rubber and not of iron ?

Ans. It is because, the coefficient of friction between rubber and concrete (material of the road) is less than that between iron and the road.

Q. Why are wheels made circular ?

Ans. The rolling friction is less than the sliding friction. The wheels are made circular so as to convert the sliding friction into the rolling friction.

Q. It is easier to roll a barrel than to slide it along the road. Why ?

Ans. The rolling friction is lesser as compared to the sliding friction.

Q. What happens to the fluid friction, as speed of the object moving through it is increased ?

Ans. The fluid friction increases, as the speed of the object moving through it, is increased.