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Momentum and Newton’s Second Law of Motion


The first law basically refers to the simple case of when the net external force on a body is zero. Now, Newton’s second law of motion tells us what happens when a force is actually exerted on a body. Newton introduced a new concept called momentum.

Momentum
Suppose we have two cars, a toy and a real car. Imagine that they are moving with the same velocity. We try to stop them by pulling at a rope tied to the back of each car. The effort (or force) required to stop the real car is obviously much more than that required to stop the toy car. The reason is that the real car has a greater mass (or inertia) than the toy car.

The force required to stop a moving object depends on its mass. It is a matter of common experience that the force required to stop a moving body depends also on its velocity.

Suppose the real car is moving very slowly; a small force will be needed to stop it. But if it is moving with a high velocity, a much greater force is needed to stop it.

The force required to stop a moving body depends on two factors: (i) its mass, and (ii) its velocity.

Newton defined momentum as the product of the mass and velocity of a body.

If p denotes the momentum of a body of mass m moving with a uniform velocity v,
P = mv ....(1)

Since the body is moving in a straight line, p is called the linear momentum which is different from angular momentum.

Momentum
p is a vector quantity since it is the product of a scalar m and a vector v. The direction of p is the same as that of v.

The SI unit of momentum is kg ms-1.

A body of mass 1 kg moving with a constant velocity of 1 ms-1 has a linear momentum of 1 kg ms-1.

Statement of the Second Law


The law states that, the rate of change of momentum of a body is directly proportional to the applied force and the change takes place in the direction in which the force acts.

Interpretation of the Second Law

Suppose a force F acts on a body of mass m. The velocity and hence the momentum of the body changes. The momentum is given by
p = mv
 
where v is the velocity at a certain instant of time.
 
Differentiating this equation with respect to time, we get

= ma
 
where a = is the acceleration produced. But according to Newton’s second law the rate of change of momentum is proportional to the impressed force F, i.e.
F ma
or
 
The effect of force on a body is to produce an acceleration which is directly proportional to the applied force and inversely proportional to the mass of the body, the acceleration taking place in the direction in which the force acts.
 
A physically meaningful and a quantitative definition of force is that it is something that produces an acceleration in a material body.
 

Alternative Statement of the Second Law

The acceleration produced by an unbalanced force acting on a body is directly proportional to the magnitude to the applied force and inversely proportional to the mass of the body; the acceleration taking place in the direction in which the force acts.
 
Unit of Force
F ma
F = k ma

where k is a constant of proportionality, whose numerical value will depend on the units used for F, m and a. For convenience, we choose a unit of force in such a way that a unit force produces a unit acceleration in a unit mass, i.e.
1 = k × 1 × 1
 
which gives k=1. If the unit is fixed in this manner,
F = ma                        (2)
 
In the SI system, the unit of force is the newton (symbol N). A newton is defined as that force which produces an acceleration of 1 ms-2 in a body of mass 1 kg. The dimensions of force are MLT-2.

The first law is a special case of the second law. If no net force acts on a body, the value of F = 0. Substituting in eq. (2), we have a = 0, i.e. the body is not accelerated.

In other words, the velocity of the body remains constant.

If the velocity is zero, it remains zero and the body stays at rest.

On the other hand, if the velocity is finite, say, 10 m s-1, it will remain 10 m s-1.

Thus, if no net force acts on a body, its state of rest or of uniform motion in a straight line remains unchanged.

But this is the first law of motion. Thus, the first law is a special case of the second law because it can be directly deduced from the second law by putting F= 0.
 




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