# Laws of Motion

## Aristotle's Fallacy

Aristotelian law of motion says, *An external force is required to keep a body in motion.*

From our practical experience we may tend to think that we constantly need to apply force on an object to keep it in motion. For example; a child continuously needs to pull his toy car to keep it moving. But we need to keep in mind that whenever an object is in motion an opposing force, i.e. friction comes into play. So the child needs to continuously keep pulling his toy car to overcome the force of friction. But when there is zero friction, uniform motion is possible without a need to continuously apply force. There lies the fallacy in Aristotle’s law.

### Law of Inertia

A body does not change its state of rest or of uniform motion unless an external force compels it to change that state. This property of the body is called inertia. The dictionary meaning of inertia is ‘resistance to change’.

The law of inertia was given by Galileo after his conducted the experiment with a frictionless incline and ball.

## Newton's First Law of Motion

The first law of motion was derived from Galileo’s law of inertia.

*Every object 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 of uniform motion both imply zero acceleration. In this context, the first law of motion can be expressed as follows:

*If the net external force on a body is zero, its acceleration is zero. Acceleration can be non-zero only in case of a net external force acting on the body.*

But in real life, we do not know all the forces present. So, we know that if an object has zero acceleration, we can infer that the net external force on the object must be zero. But we also know that gravity is everywhere. Particularly in the case of terrestrial phenomena, every object experiences gravitational force due to earth. Moreover, an object in motion experiences friction, viscous drag, etc. If an object on earth is at rest or in uniform motion, it does not happen because no external force is acting on it, but because various external forces cancel out each other; resulting in zero net external force.

## Newton's Second Law of Motion

While the first law refers to a case when the net external force on a body is zero, the second law of motion refers to the case when there is net external force acting on the body. This law correlates the net external force to the acceleration of the body.

### Momentum

The product of mass m and velocity v of a body is called momentum, and is denoted by p.

`p=mv`

It is clear that momentum varies directly as mass and velocity of object. So, a heavier object moving at a particular velocity will have greater momentum than a lighter object moving at same velocity. Similarly, an object with a higher velocity and given mass will have greater momentum than another object with equal mass but lower velocity. Same force for the same time causes the same change in momentum for different bodies.

The second law of motion is 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.*

If a force F is applied for time interval Δt, the velocity of a body of mass m changes from v to v + Δv. As a result, the initial momentum p = mv changes to Δp = mΔv. As per the second law of motion

`F∝(Δp)/(Δt)`

Or, `F=k(Δp)/(Δt)`

Where k is the constant of proportionality. Taking the limit Δt → 0,

`F=k(dp)/(dt)`

For a body of fixed mass m

`(dp)/(dt)=d/(dt)(mv)`

`=m(dv)/(dt)=ma`

So, the second law of motion can also be written as

`F=kma`

If k = 1, then

`F=ma`

In terms of SI unit, force is one that causes an acceleration of 1 m s^{-2} to a mass of 1 kg. The SI unit of force is Newton

1N = 1 kg m s^{-2}

#### Some Important Point about Second Law

(a)If F = 0 then a = 0. So the second law is consistent with the first law which says in case of net force being zero, acceleration on an object shall be zero.

(b) The second law of motion is a vector law. It is equivalent to three equations, one for each component of the vectors:

`F_x=(dp_x)/(dt)=ma_x`

`F_y=(dp_y)/(dt)=ma_y`

`F_z=(dp_z)/(dt)=ma_z`

This means that if a force is not parallel to the velocity of the body, it changes only the component of velocity along the direction of force. The component of velocity normal to the force remains unchanged. For example; in projectile motion under the vertical gravitational force, the horizontal component of velocity remains unchanged because it is normal to the gravitational force.

(c) The second law of motion is applicable to a single point particle. In case of single point particle, F stands for net external force on the particle and a stands for acceleration of the particle. The same law applies to a rigid body or even generally to a system of particles. In this case, F refers to the total external force on the system and ‘a’ refers to the acceleration of the system as a whole. Any internal forces in the system are not included in F.

(d) The second law of motion is a local relation. It means that force F at a point in space (location of the particle) at a certain instant of time is related to acceleration a at that point at that instance. In other words, acceleration here and now is determined by the force here and now, not by any history of the motion of the particle.

**Impulse:** A large force acting for a short time to produce a finite change in momentum is called impulsive force, and the change in momentum due to this force is called impulse.

Impulse = Force × time duration = change in momentum

## Newton's Third Law of Motion

*To every action, there is always an equal and opposite reaction.*

Forces always occur in pairs. Force on a body A by B is equal and opposite to the force on the body B by A.

There is no cause-effect relation implied in the third law. The force on A by B and the force on B by A, both act at the same instant. So any one of them may be called action and the other may be called reaction.

Action and reaction forces act on different bodies. Let us take a pair of bodies A and B. According to the third law

F_{AB} = - F_{BA}

If we are considering the motion of any one body (A or B), only one of the two forces is relevant. It is an error to add up the forces and say that the net force is zero.

But when we consider the system of two bodies as a whole, then F_{AB} and F_{BA} are internal forces of the system (A + B). In this case, they add up to give a null force. Thus, internal forces in a system of particles cancel away in pairs.