Concept of friction:

Component of normal reaction

We have seen that a ball rolling on a floor stops after some time. When we switch off the engine of a car, it stops after traveling some distance. Similarly when we apply brakes, our bicycle comes to rest after traveling some distance. The above examples show that some invisible force is opposing the motion of one body over the other. This opposing force is called friction. Friction is an opposing force that comes into play when one body actually moves (slides or rolls) or even tries to move over the surface of another body.

Cause of friction

Roughness of surfaces is the cause of friction.

When two bodies are in contact with each other, the irregularities on the surfaces get interlocked and oppose any relative motion.

Normal reaction

When a body of mass m is lying on a horizontal surface, it presses the surface due its weight. Contact force F_c = R in case no external force is acting on the body. Also in this case friction is zero and F_c is perpendicular to the surface. In the diagram below, R is the normal reaction.

Static and kinetic friction

The opposing force that comes into play when one body tends to move over the surface of another (but the actual motion has not yet started) is called static friction. Limiting friction is the maximum opposing force that comes into play when one body is at the verge of moving over the surface of another body. It is denoted by f_s, and is called maximum force of static friction or limiting friction.

Laws of friction

The following are the laws of limiting friction:

(a) The magnitude of the force of limiting friction f_s, is directly proportional to the normal reaction R.

(b) The direction of the force of limiting friction is always opposite to the applied force.

(c) It is independent of the apparent area of contact.

(d) It depends on the nature and material of the surfaces in contact.

For example, when two polished metal surfaces are in contact,  = 0.2

When these surfaces are lubricated,  gets reduced. Hence, it depends on the nature of the surfaces.

Rolling friction

Rolling friction is always less than dynamic and static friction. When a body rolls on a level track, the area of contact is very small. This causes a depression in the surface below. This causes rolling friction.

The velocity of the point of contact of the wheel with respect to the floor remains zero all the time. Thus, rolling wheel constantly climbs a hill and has to simultaneously get itself detached from the road. Rolling friction is also directly proportional to the normal reaction and is inversely proportional to the radius of rolling body. Combining the two, we get,

F is directly proportional to R/r

F = μx (R/r)

Where μx is called coefficient of rolling friction.

Advantages and Disadvantages of friction

Friction is a necessary evil. It is necessary because we cannot do any work without it. At the same time, it is also an evil because it involves unnecessary wastage of energy.

There are advantages of friction. It can be understood with the help of the following examples:

Walking will not be possible without friction.

Brakes of vehicles will not work without friction.

Writing on a blackboard or a piece of paper is possible only due to friction between the blackboard and the chalk or the paper and the pen. Cleaning with sand paper will not be possible without friction.

There are disadvantages of friction. Friction is an evil and this can be understood with the following   examples:
Extra energy is required to overcome the friction between moving parts.

ü       Friction causes wear and tear of different parts of machinery.

ü       Frictional forces result in the production of heat, which causes damage to the machinery.

Methods of reducing friction

By polishing, by lubrication, By streamlining, By using ball-bearings,

Angle of friction

It is the angle between normal Reaction and the resultant of the normal reaction and limiting frictional force. i.e. we have a body of mass m which is placed on a table and we say that the body and the surface of the table have a coefficient of friction m between them. (fig. 3.7)

If we apply a small force F, the body will not move. Let us gradually increase the force until the body starts moving. At one stage the applied force will be equal to the frictional force. The coefficient of static friction ms  = F/N where F is the applied force and N = mg is the normal force.

Newton’s third law of motion was discovered and formulated, during the investigation of the fact that in all experiments it appeared that “when ever a body exerts a force on a second body, the second body always exerts a force on the first one”

Let us visualize and understand this phenomena with an experiment.

Suppose, we throw a stone on a surface of good strength; and the surface is made of glass, one finds it broken (the surface). From here one concludes that a force was exerted by stone on the surface and consequently it was broken.

Now, the question is, did that surface also exert a force on the stone. Just to know about it let us change our throwing object from stone to an egg of almost equal mass. Now, one throws this egg on the same surface of good strength with the same throwing force which he used for the stone. What happens? Obviously with your daily experience you know that the egg will be broken (And the damage to the surface will not be visible due to egg’s spoiling the observation).

This is only possible if there was a force acting on the egg at the time it hit the surface. In fact we can now conclude that there is a mutual force acting on the contact point of the surface and the object thrown. The breaking of either one (or may be both) depends on their ability to absorb forces without getting damaged (that is their strength) so in precise words:

Þ  To every action there is always opposite and equal reaction, it is equivalent to say that mutual actions of two bodies upon each other are always equal and directed to contrary parts.

Þ  The most important fact to notice here is that these oppositely directed equal action and reaction can never balance or cancel each other because they always act, on two different point (broadly on two different objects) For balancing any two forces the first requirement is that they should act on one and the same object. (or point, if object can be treated as a point mass, which is a common practice).

Illustrations

1.      Suppose in figure 3.2 we put one more block of 5 kg mass adjacent to 10 kg and a force of 150 N acts as shown in the figure 3.3, then find the forces acting on the interface.

Sol.   The combined acceleration of the two bodies when treated as one is

a= F/ (10+5) = 150 / 15 = 10m/sec²

So each one moves with a = 10m/sec² keeping their contact established.

Here you can feel that due to 150N force the body of 5kg feels as if it is being pushed by the 10 kg mass. There is a force acting on 5kg called R₁, to oppose it by third law this body exerts a force R₂ on 10 kg. The interface is as shown in figure 3.4.

Also, third law tells us that R₁ = R₂ in magnitude and is opposite in direction. (figure 3.5)

R₁ = R₂ = R

Here since 150 N force acts on the 10 kg mass and only R acts on the 5kg mass. For motion in 5 kg only R is responsible. We can write the initial equation as:

F = 150 = (10+5) a           i.e., 150 = 10a + 5a

Here 10a is force experienced by 10 kg mass and 5a is experienced by 5 kg mass.

R = 5a      a = 10m/sec² =>           R = 50 N

Net force experienced by 10kg block is (150 – R) = 10a

150 – R = 10 × 10 = 100 N            =>     R = 50 N

Therefore we get R = 50N for both blocks. Hence we find “action and reaction are equal and opposite”. Now net force on the body of 10 kg mass is 100 N & Net force on the body of 5kg mass is 50N and on the interface action and reaction are both equal and also are equal to force experienced by second body.

It states that rate of change of momentum of a body is equal to the force applied on it, in terms of the magnitude as well as in the sense of direction. Here the momentum is defined as the product of mass and velocity i.e. .

Therefore we can write mathematically.

F = d(mv)/ dt, if ‘m’ remains constant then

Thus acceleration is rate of change of velocity.

Since direction of a is same as F, we can write

F = ma , which is mathematically Newton’s second law of motion.

Here, if F = 0 then we find a = 0. This reminds us of first law of motion. That is, if net external force  is absent, then there will be no change in state of motion, that means its acceleration is zero.

Further we can extend second law of motion, (in fact its decomposition) to three mutually perpendicular directions as per our coordinate system.

If components in x,y, and z directions are Fx, Fy, & Fz respectively, the three accelerations produced when Fx, Fy, & Fz act simultaneously) in the body are ,  Now,

If we add three forces then resultant is called net external force.

Similarly

is called net acceleration produced in the body.

Illustrations

1.             In Figure 3.1. Let us have M = 10 kg and a new net external force in the direction as shown in figure 3.2 is 150 N. Find its acceleration.

Sol.

The discovery of the laws of motion was a dramatic moment in the history of science. Before Newton’s time, the motions of things like the planets were a mystery, but after Newton there was complete understanding.

The contribution of Newton was three laws:

Þ      The first law describes what happens when the body is left alone or is not disturbed.

Þ      The second law gives a specific way of determining how the velocity changes under different influences called forces.

Þ      The third law characterizes the basic nature of force.

OBJECTIVE

In this chapter we will study the three important laws of motion given by Newton. The study includes the dynamics of particle under influence of different forces.

After studying this chapter we will be able to answer the question like Why can’t we drive fast on icy road, in cricket how hard should a player hit the ball for a sixer or how does a small engine pull a large train and many other similar questions.

PREQUISITE

We know by experience that all bodies in nature interact in some way with one another.

Before actually going into details of the three Laws of motion we need to get familiar with  “force”. It was believed and has been accepted that “force” is an external or internal agent present to “influence” the natural state of motion of an object. So this is an influence (force) needed to change the natural state of body; that is of rest or of uniform motion.

Having this idea of force we can now readily move to the Newton’s first law of motion.

Newton’s first law of motion

It states that every object persists in its natural state of motion i.e. continues to be at rest or moves in a straight line with uniform (constant) velocity. (This is what is meant by natural state of motion); In the absence of a net external force acting (impressed) on it.

Mathematically, it is equivalent to say that for producing acceleration (that is for changing velocity) in a body, we need to have a net external force. (By net external force we mean vector sum of all the external forces acting on it)

It can be easily deduced from the statement of change in the state of motion. It is directly related to a frame of reference about which we have discussed earlier. To mark the point here, we can discover that by viewing objects from different frame of references the natural state of motion as perceived by different observers will be obviously different (can only be same if the frames are truly equivalent). Therefore, the change in state will also depend on the choice of reference frame. Finally, the amount of acceleration produced in a body (or change in velocity) will depend on our choice of reference frames.

A mass ‘M’ is lying (figure 3.1) on a table which is at rest (w.r.t. table). Explain its state with the help of Newton’s First Law of motion.

Since ‘M’ is lying on a table, there is no external force acting on it (forget about gravity just for the immediate discussion). As per Newton’s first law of motion it will keep on lying at rest with respect to table for infinite time.

Here, comes out a very important, intrinsic (that is inherent) property of a body which is that it retains its state of motionlessness (as well as of motion, if its in motion) which is termed as INERTIA of an object. This is present in all materialistic bodies in this universe.