Force and Laws of Motion

LINEAR MOMENTUM

• Momentum of a body is the quantity of motion possessed by the body.
• The momentum of a body is defined as the product if its mass and velocity.
• It is denoted by .
• Linear momentum is represented as:

• If a body is at rest, its velocity would be zero and hence its momentum will also be zero.
• Momentum is a vector quantity.
• The SI unit of momentum is given as kilogram metres per second.
• Every moving object possesses momentum.

RELATION BETWEEN KINETIC ENERGY OF A BODY AND ITS LINEAR MOMENTUM

Linear momentum p = mv

Squaring both sides and dividing by 2 we get,

So if two bodies having different masses have same kinetic energy than the one with lighter mass has smaller momentum.

NEWTON’S SECOND LAW OF MOTION

Newton’s second law of motion states that: 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.

INTERPRETATIONS FROM NEWTON’S LAW

• According to the first law, we know that a body will continue to be in a state of rest or uniform motion until a net external force acts on it.
• Newton’s first law of motion gives a qualitative definition of force while the second law helps us to find its magnitude.
• Newton’s second law of motion gives us a relationship between ‘force’ and ‘acceleration’.
• Force is an interaction that changes the state of the body, i.e. from rest to motion and vice versa.

SCENARIO

Consider the following situation:

Observations from the situation:

• One force is acting towards the right and an equal force is acting towards left hence the net force on the body is zero. So the body remains at rest.
• A force acting on a body does not mean that the state of the body will necessarily change.
• Gravitational force, frictional force, normal force etc. are some common forces that we experience in our daily lives.
• Now if a net force acts on the body then velocity either increases or decreases.

ACCELERATION

The rate of change of velocity is called acceleration.
Greater the force, greater is the acceleration. This indicates some relation between force and acceleration which is given by Newton’s Second Law.

RELATION BETWEEN FORCE AND ACCELERATION

• According to the Newton’s Second Law:
  $\mathrm{F} = \mathrm{ma}$
• Observations:
  • So, the Newton’s second law gives us a method of measuring the force in terms of mass and acceleration.
  • Acceleration depends on the force , as well as the mass of the body on which it is applied.
    $\mathrm{a} = \mathrm{F} / \mathrm{m}$
• If same force is applied on two different bodies then one with smaller mass will have greater acceleration.
• The SI unit of force is newton.
• A newton is that force which when acting on a body of mass produces an acceleration of
  $1\mathrm{m}/{\mathrm{s}}^2$

   in it.

  $1 \mathrm{newton} = 1 \mathrm{kg} 1\mathrm{m}/{\mathrm{s}}^2$

HOW TO CALCULATE THE RATE OF CHANGE OF MOMENTUM?

• Newton’s Second law relates force with the rate of change of momentum.
• According to the law, force is directly proportional to the rate of change in momentum.
  $\mathrm{F}\propto ∆\mathrm{p}$
• If the net force acting on the system is zero then the momentum of the system remains conserved. In other words, the change in momentum of the system is zero.
• If F = 0 , then the momentum will also be zero according to the second law.

LAW OF CONSERVATION OF MOMENTUM FROM NEWTON’S THIRD LAW OF MOTION

Force exerted by B on  A is:

According to Newton’s third law of motion,

Action =  – Reaction

Total momentum before collision = Total momentum after collision.

Thus total momentum of the two bodies is conserved provided no external force acts on them. This proves the law of conservation of momentum.