Laws of motion || Newton's Laws of motion ||

                         1. Introduction

Let us first guess the answer based on our common

experience. To move a football at rest, someone must kick it.

To throw a stone upwards, one has to give it an upward

push. A breeze causes the branches of a tree to swing; a

strong wind can even move heavy objects. A boat moves in a

flowing river without anyone rowing it. Clearly, some external

agency is needed to provide force to move a body from rest.

Likewise, an external force is needed also to retard or stop

motion. You can stop a ball rolling down an inclined plane by

applying a force against the direction of its motion.

In these examples, the external agency of force (hands,

wind, stream, etc) is in contact with the object. This is not

always necessary. A stone released from the top of a building

accelerates downward due to the gravitational pull of the

earth. A bar magnet can attract an iron nail from a distance.

This shows that external agencies (e.g. gravitational and

magnetic forces ) can exert force on a body even from a

distance.

In short, a force is required to put a stationary body in

motion or stop a moving body, and some external agency is

needed to provide this force. The external agency may or may

not be in contact with the body.

                         2. Law of Inertia

Galileo studied motion of objects on an inclined
plane. Objects (i) moving down an inclined plane
accelerate, while those (ii) moving up retard.
(iii) Motion on a horizontal plane is an
intermediate situation. Galileo concluded that
an object moving on a frictionless horizontal
plane must neither have acceleration nor
retardation, i.e. it should move with constant
velocity. Another experiment by Galileo leading to the  same conclusion  involves a double inclined plane. 

A ball released from rest on one of the planes rolls

down and climbs up the other. If the planes are

smooth, the final height of the ball is nearly the

same as the initial height (a little less but never

greater). In the ideal situation, when friction is

absent, the final height of the ball is the same

as its initial height.

If the slope of the second plane is decreased

and the experiment repeated, the ball will still

reach the same height, but in doing so, it will

travel a longer distance. In the limiting case, when

the slope of the second plane is zero (i.e. is a

horizontal) the ball travels an infinite distance.

In other words, its motion never ceases. This is,

of course, an idealised situation. 

To summarise, if the net external force is zero,

a body at rest continues to remain at rest and a

body in motion continues to move with a uniform

velocity. This property of the body is called

inertia. Inertia means ‘resistance to change’.

A body does not change its state of rest or

uniform motion, unless an external force

compels it to change that state.

         3. Newton's first law of motion

Every body continues to be

 in its state  of rest or of

 uniform motion in a straight  line unless compelled by some external 

force to act otherwise. The state of rest or uniform linear motion both

imply zero acceleration. The first law of motion can,

therefore, be simply expressed as:

If the net external force on a body is zero, its

acceleration is zero. Acceleration can be non

zero only if there is a net external force on

the body.

Two kinds of situations are encountered in the

application of this law in practice. In some

examples, we know that the net external force

on the object is zero. In that case we can

conclude that the acceleration of the object is

zero. For example, a spaceship out in

interstellar space, far from all other objects and

with all its rockets turned off, has no net

external force acting on it. Its acceleration,

according to the first law, must be zero. If it is

in motion, it must continue to move with a

uniform velocity.

The property of inertia contained in the First

law is evident in many situations. Suppose we

are standing in a stationary bus and the driver

starts the bus suddenly. We get thrown

backward with a jerk. Why ? Our feet are in touch

with the floor. If there were no friction, we would

remain where we were, while the floor of the bus

would simply slip forward under our feet and the

back of the bus would hit us. However,

fortunately, there is some friction between the

feet and the floor. If the start is not too sudden,

i.e. if the acceleration is moderate, the frictional

force would be enough to accelerate our feet

along with the bus. But our body is not strictly

a rigid body. It is deformable, i.e. it allows some

relative displacement between different parts.

What this means is that while our feet go with

the bus, the rest of the body remains where it is

due to inertia. Relative to the bus, therefore, we

are thrown backward. As soon as that happens,

however, the muscular forces on the rest of the

body (by the feet) come into play to move the body

along with the bus. A similar thing happens

when the bus suddenly stops. Our feet stop due

to the friction which does not allow relative

motion between the feet and the floor of the bus.

But the rest of the body continues to move

forward due to inertia. We are thrown forward.

The restoring muscular forces again come into

play and bring the body to rest.

       4. Newton's Second law of motion

4.1 Momentum

Momentum of a body is defined to be the product

of its mass m and velocity v, and is denoted

by p:

                       p = m v 

Momentum is clearly a vector quantity. The

following common experiences indicate the

importance of this quantity for considering the

effect of force on motion.

• Suppose a light-weight vehicle (say a small

car) and a heavy weight vehicle (say a loaded

truck) are parked on a horizontal road. We all

know that a much greater force is needed to

push the truck than the car to bring them to

the same speed in same time. Similarly, a

greater opposing force is needed to stop a

heavy body than a light body in the same time,

if they are moving with the same speed.

• If two stones, one light and the other heavy,

are dropped from the top of a building, a

person on the ground will find it easier to catch

the light stone than the heavy stone. The

mass of a body is thus an important

parameter that determines the effect of force

on its motion.

• Speed is another important parameter to

consider. A bullet fired by a gun can easily

pierce human tissue before it stops, resulting

in casualty. The same bullet fired with

moderate speed will not cause much damage.

Thus for a given mass, the greater the speed,

the greater is the opposing force needed to stop

the body in a certain time. Taken together,

the product of mass and velocity, that is

momentum, is evidently a relevant variable

of motion. The greater the change in the

momentum in a given time, the greater is the

force that needs to be applied.

• A seasoned cricketer catches a cricket ball

coming in with great speed far more easily

than a novice, who can hurt his hands in the

act. One reason is that the cricketer allows a

longer time for his hands to stop the ball. As

you may have noticed, he draws in the hands

backward in the act of catching the 

 ball . The novice, on the other hand, keeps 

his hands fixed and tries to catch the ball

almost instantly. He needs to provide a much

greater force to stop the ball instantly, and this hurts. The conclusion is clear: force not

only depends on the change in momentum,

but also on how fast the change is brought

about. The same change in momentum

brought about in a shorter time needs a

greater applied force. In short, the greater the

rate of change of momentum, the greater is

the force.

These qualitative observations lead to the

second law of motion expressed by Newton as

follows :

The rate of change of momentum of a body is

directly proportional to the applied force and

takes place in the direction in which the force

acts.

Thus, if under the action of a force F for time

interval ∆t, the velocity of a body of mass m

changes from v to v + ∆v i.e. its initial momentum

p = m v changes by ∆= ∆ p v m . According to the

Second Law,

 or k

t t

∆ ∆ ∝ = ∆ ∆

p p F F

where k is a constant of proportionality. Taking

the limit ∆t → 0, the term ∆t

∆p becomes the

derivative or differential co-efficient of p with

respect to t, denoted by d

dt

p . Thus, 

t = p F 

For a body of fixed mass m,

 ( ) d d d

d d d

m m m

t t t = = = p v

v a 

i.e the Second Law can also be written as

 F = k m a

which shows that force is proportional to the

product of mass m and acceleration a.

The unit of force has not been defined so far.

In fact, we use to define the unit of force.

We, therefore, have the liberty to choose any

constant value for k. For simplicity, we choose

k = 1. The second law then is

a

p F m

t d

d

In SI unit force is one that causes an acceleration

of 1 m s-2 to a mass of 1 kg. This unit is known as

newton : 1 N = 1 kg m s-2.

       5. Newton's Third law of motion 

To every action, there is always an equal and

opposite reaction.The second law relates the external force on a

body to its acceleration. What is the origin of the

external force on the body ? What agency

provides the external force ? The simple answer

in Newtonian mechanics is that the external

force on a body always arises due to some other

body. Consider a pair of bodies A and B. B gives

rise to an external force on A. A natural question

is: Does A in turn give rise to an external force

on B ? In some examples, the answer seems

clear. If you press a coiled spring, the spring is

compressed by the force of your hand. The

compressed spring in turn exerts a force on your

hand and you can feel it. But what if the bodies

are not in contact ? The earth pulls a stone

downwards due to gravity. Does the stone exert

a force on the earth ? The answer is not obvious

since we hardly see the effect of the stone on the

earth. The answer according to Newton is: Yes,

the stone does exert an equal and opposite force

on the earth. We do not notice it since the earth

is very massive and the effect of a small force on

its motion is negligible.

Thus, according to Newtonian mechanics,

force never occurs singly in nature. Force is the

mutual interaction between two bodies. Forces always occur in pairs. Further, the mutual forces

between two bodies are always equal and

opposite. This idea was expressed by Newton in

the form of the third law of motion.

To every action, there is always an equal and

opposite reaction.

WORK, POWER and ENERGY

MOTION IN A PLANE

MOTION IN A STRAIGHT LINE

HIMANSHU CHAUDHARY