FRICTION

If a very gentle force in the horizontal direction is given to an object at rest on the table, it does not move. It is because of the opposing force exerted by the surface on the object which resists its motion. This force is called the frictional force which always opposes the relative motion between an object and the surface where it is placed. If the force applied is increased, the object moves after a certain limit.

Relative motion: when a force parallel to the surface is applied on the object, the force tries to move the object with respect to the surface. This ‘relative motion’ is opposed

Figure 3.22 Frictional force

by the surface by exerting a frictional force on the object in a direction opposite to applied force. Frictional force always acts on the object parallel to the surface on which the object is placed. There are two kinds of friction namely 1) Static friction and 2) Kinetic friction.

Static friction is the force which opposes the initiation of motion of an object on the surface. When the object is at rest on the surface, only two forces act on it. They are the downward gravitational force and upward normal force. The resultant of these two forces on the object is zero. As a result the object is at rest as shown in Figure 3.23.

If some external force Fext is applied on the object parallel to the surface on which the object is at rest, the surface exerts exactly an equal and opposite force on the object to resist its motion and tries to keep the object at rest. It implies that external force and frictional force are exactly equal and opposite. Therefore, no motion parallel to the surface takes place. But if the external force is increased above a particular limit, the surface cannot provide sufficient opposing frictional force to balance the external force on the object. Then the object starts to slide. This is the maximal static friction that can be exerted by the surface. Experimentally, it is found that the magnitude of static frictional force fs satisfies the following empirical relation.

where µs is the coefficient of static friction. It depends on the nature of the surfaces in contact. N is normal force exerted by the surface on the body and sometimes it is equal to mg. But it need not be equal to mg always.

Equation (3.27) implies that the force of static friction can take any value from zero to µs N.

If the object is at rest and no external force is applied on the object, the static friction acting on the object is zero ( fs = 0).

If the object is at rest, and there is an external force applied parallel to the surface, then the force of static friction acting on the object is exactly equal to the external force applied on the object ( f F s ext = ). But still the static friction fs is less than µs N.

When object begins to slide, the static friction ( fs ) acting on the object attains maximum,

The static and kinetic frictions (which we discuss later) depend on the normal force acting on the object. If the object is pressed hard on the surface then the normal force acting on the object will increase. As a consequence it is more difficult to move the object. This is shown in Figure 3.23 (a) and (b). The static friction does not depend upon the area of contact.

Fig 3.23 Static friction and kinetic friction (a) Easier to move (b) Harder to move

EXAMPLE 3.17

Consider an object of mass 2 kg resting on the floor. The coefficient of static friction between the object and the floor is s  0.8. What force must be applied on the object to move it?

Solution

Since the object is at rest, the gravitational force experienced by an object is balanced by normal force exerted by floor.

$$ N = mg $$

The maximum static frictional force fs max =   s s N m  g

Therefore to move the object the external force should be greater than maximum static friction.

EXAMPLE 3.18

Consider an object of mass 50 kg at rest on the floor. A Force of 5 N is applied on the object but it does not move. What is the frictional force that acts on the object?

Solution

When the object is at rest, the external force and the static frictional force are equal and opposite.

The magnitudes of these two forces are equal, f F s ext

Therefore, the static frictional force acting on the object is

The direction of this frictional force is opposite to the direction of Fext.

EXAMPLE 3.19

Two bodies of masses 7 kg and 5 kg are connected by a light string passing over a smooth pulley at the edge of the table as shown in the figure. The coefficient of static friction between the surfaces (body and table) is 0.9. Will the mass m1 = 7 kg on the surface move? If not what value of m2 should be used so that mass 7 kg begins to slide on the table?

Solution

As shown in the figure, there are four forces acting on the mass m1

a) Downward gravitational force along the negative y-axis (m1 g)

b) Upward normal force along the positive y axis (N)

c) Tension force due to mass m2 along the positive x axis

d) Frictional force along the negative x axis Since the mass m1 has no vertical motion, m1 g = N

To determine whether the mass m1 moves on the surface, calculate the maximum static friction exerted by the table on the mass m1 . If the tension on the mass m1 is equal to or greater than this maximum static friction, the object will move.

The tension T m2 g = 5 x 9.8 = 49 N

The tension acting on the mass m1 is less than the maximum static friction. So the mass m1 will not move.

To move the mass m1 , T fs max > where T = m2 g

If the mass m2 is greater than 6.3 kg then the mass m1 will begin to slide. Note that if there is no friction on the surface, the mass m1 will move even when m2 is just 1 kg.

The values of coefficient of static friction for pairs of materials are presented in Table 3.1. Note that the ice and ice pair have very low coefficient of static friction. This means a block of ice can move easily over another block of ice.

If the external force acting on the object is greater than maximum static friction, the objects begin to slide. When an object slides, the surface exerts a frictional force called kinetic friction  fk (also called sliding friction or dynamic friction). To move an object at constant velocity we must apply a force which is equal in magnitude and opposite to the direction of kinetic friction.

Figure 3.24 Kinetic friction

Experimentally it was found that the magnitude of kinetic friction satisfies the relation

where µk is the coefficient of kinetic friction and N the normal force exerted by the surface on the object,

This implies that starting of a motion is more difficult than maintaining it. The salient features of static and kinetic friction are given in Table 3.2.

The variation of both static and kinetic frictional forces with external applied force is graphically shown in Figure 3.25.

Figure 3.25 Variation of static and kinetic frictional forces with external applied force

The Figure 3.25 shows that static friction increases linearly with external applied force till it reaches the maximum. If the object begins to move then the kinetic friction is slightly lesser than the maximum static friction. Note that the kinetic friction is constant and it is independent of applied force.

Note

The relation f N s s   is not a vector relation. This is because the normal force N and fs are not in the same direction even though fs is equal to µs times the normal force. This is also true in the case of kinetic friction.

When a body is pushed at an arbitrary angle θ 0 2 to       , the applied force F can be resolved into two components as F sinθ parallel to the surface and F cosθ perpendicular to the surface as shown in Figure 3.26. The total downward force acting on the body is mg + Fcosθ. It implies that the normal force acting on the body increases. Since there is no acceleration along the vertical direction the normal force N is equal to

As a result the maximal static friction also increases and is equal to

Equation (3.30) shows that a greater force needs to be applied to push the object into motion.

Figure 3.26 An object is pushed at an angle θ

When an object is pulled at an angle θ, the applied force is resolved into two components as shown in Figure 3.27. The total downward force acting on the object is

Figure 3.27 An object is pulled at an angle θ

Equation (3.31) shows that the normal force is less than Npush. From equations (3.29) and (3.31), it is easier to pull an object than to push to make it move.

The angle of friction is defined as the angle between the normal force (N) and the resultant force (R) of normal force and maximum friction force ( fs max )

Figure 3.28 Angle of Friction

In Figure 3.28 the resultant force is

But from the frictional relation, the object begins to slide when f N

From equations (3.32) and (3.33) the coefficient of static friction is

The coefficient of static friction is equal to tangent of the angle of friction

Consider an inclined plane on which an object is placed, as shown in Figure 3.29. Let the angle which this plane makes with the horizontal be θ . For small angles of θ , the object may not slide down. As θ is increased, for a particular value of θ , the object begins to slide down. This value is called angle of repose. Hence, the angle of repose is the angle of inclined plane with the horizontal such that an object placed on it begins to slide.

Figure 3.29 Angle of repose

Let us consider the various forces in action here. The gravitational force mg is resolved into components parallel (mg sinθ) and perpendicular (mg cosθ) to the inclined plane.

The component of force parallel to the inclined plane (mg sinθ) tries to move the object down.

The component of force perpendicular to the inclined plane (mg cosθ) is balanced by the Normal force (N).

When the object just begins to move, the static friction attains its maximum value

his friction also satisfies the relation

Dividing equations (3.35) and (3.36), we get

From the definition of angle of friction, we also know that

in which θ is the angle of friction.

Thus the angle of repose is the same as angle of friction. But the difference is that the angle of repose refers to inclined surfaces and the angle of friction is applicable to any type of surface.

EXAMPLE 3.20

A block of mass m slides down the plane inclined at an angle 60° with an acceleration g 2 . Find the coefficient of kinetic friction?

Solution

Kinetic friction comes to play as the block is moving on the surface.

The forces acting on the mass are the normal force perpendicular to surface, downward gravitational force and kinetic friction fk along the surface.

Along the x-direction

But a = g/2

There is no motion along the y-direction as normal force is exactly balanced by the mg cosθ.

1. Antlions make sand traps in such a way that when an insect enters the edge of the trap, it starts to slide towards the bottom where the antilon hide itself. The angle of inclination of sand trap is made to be equal to angle of repose. It is shown in the Figure 3.30.

Figure 3.30 Sand trap of antlions

2. Children are fond of playing on sliding board (Figure 3.31). Sliding will be easier when the angle of inclination of the board is greater than the angle of repose. At the same time if inclination angle is much larger than the angle of repose, the slider will reach the bottom at greater speed and get hurt.

Figure 3.31 Sliding board

ACTIVITY

Measuring the coefficients of friction

Take a hard bound note book and a coin. Keep the coin on the note book. The note book cover has to be in an inclined position as shown in the figure. Slowly increase the angle of inclination of the cover with respect to rest of the pages. When the angle of inclination reaches the angle of repose, the parallel component of gravitational force (mg sinθ ) to book surface becomes equal to the frictional force and the coin begins to slide down. Measure the angle of inclination and take the tangent of this angle. It gives the coefficient of static friction between the surface of the cover and coin. The same can be repeated with other objects such as an eraser in

order to observe that the coefficient of static friction differs from case to case.

Note

At the point of sliding tanθs = ms To measure the coefficient of kinetic friction, reduce the inclination of the book after it starts sliding, such that the coin/ eraser moves with uniform velocity. Now measure the angle from which coefficient of kinetic friction can be calculated as mK = tanθk Observe that   K S

The invention of the wheel plays a crucial role in human civilization. One of the important applications is suitcases with rolling on coasters. Rolling wheels makes it easier than carrying luggage. When an object moves on a surface, essentially it is sliding on it. But wheels move on the surface through rolling motion. In rolling motion when a wheel moves on a surface, the point of contact with surface is always at rest. Since the point of contact is at rest, there is no relative motion between the wheel and surface. Hence the frictional force is very less. At the same time if an object moves without a wheel, there is a relative motion between the object and the surface. As a result frictional force is larger. This makes it difficult to move the object. The Figure 3.32 shows the difference between rolling and kinetic friction.

Figure 3.32 Rolling and kinetic friction

Ideally in pure rolling, motion of the point of contact with the surface should be at rest, but in practice it is not so. Due to the elastic nature of the surface at the point of contact there will be some deformation on the object at this point on the wheel or surface as shown in Figure 3.33. Due to this deformation, there will be minimal friction between wheel and surface. It is called ‘rolling friction’. In fact, ‘rolling friction’ is much smaller than kinetic friction.

Figure 3.33 Rolling friction

Frictional force has both positive and negative effects. In some cases it is absolutely necessary. Walking is possible because of frictional force. Vehicles (bicycle, car) can move because of the frictional force between the tyre and the road. In the braking system, kinetic friction plays a major role. As we have already seen, the frictional force comes into effect whenever there is relative motion between two surfaces. In big machines used in industries, relative motion between different parts of the machine produce unwanted heat which reduces its efficiency. To reduce this kinetic friction lubricants are used as shown in Figure 3.34.

Figure 3.34 Reducing kinetic friction using lubricant

Ball bearings provides another effective way to reduce the kinetic friction (Figure 3.35) in machines. If ball bearings are fixed between two surfaces, during the relative motion only the rolling friction comes to effect and not kinetic friction. As we have seen earlier, the rolling friction is much smaller than kinetic friction; hence the machines are protected from wear and tear over the years.

Figure 3.35 Reducing kinetic friction using ball bearing

During the time of Newton and Galileo, frictional force was considered as one of the natural forces like gravitational force. But in the twentieth century, the understanding on atoms, electron and protons has changed the perspective. The frictional force is actually the electromagnetic force between the atoms on the two surfaces. Even well polished surfaces have irregularities on the surface at the microscopic level as seen in the Figure 3.36.

Points to Ponder

When you walk on the tiled floor where water is spilled, you are likely to slip. Why?

Figure 3.36 Irregularities on the surface at the microscopic level

Frictional force in the motion of a bicycle

When a bicycle moves in the forward direction, what is the direction of frictional force in the rear and front wheels?

When we pedal a bicycle, we try to push the surface backward and the velocity of point of contact in the rear wheel is backwards. So, the frictional force pushes the rear wheel to move forward. But as the front wheel is connected with a rigid support to the back wheel, the forward motion of back wheel pushes the front wheel in the forward direction. So, the frictional forces act backward. Remember both frictional forces correspond to only static friction and not kinetic friction. If the wheel slips then kinetic friction comes into effect. In addition to static friction, the rolling friction also acts on both wheels in the backward direction.

EXAMPLE 3.21

Solution

If an object moves with constant velocity, then it has no acceleration. According to Newton’s second law there is no net force acting on the object. The external force is balanced by the kinetic friction.

Note

It is not that ‘no force acts on the object’. In fact there are two forces acting on the object. Only the net force acting on the object is zero.