# Newton's Second Law Of Motion

Force is equal to the rate of change of momentum. For a constant mass, force equals mass times acceleration.

Newton’s second law of motion pertains to the behaviour of objects for which all existing forces are not balanced. The second law informs us that the acceleration of an object depends on two variables – the net force acting on the body and the mass of the body. The acceleration of the body is directly proportional to force and inversely proportional to the mass. As a result, when the force acting on a body is increased, the acceleration increases. Likewise, when the mass of the body is increased, the acceleration decreases.

Newton’s second law can be formally stated as,

The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.

This statement is expressed in equation form as,

a = Fnet/m

The above equation is often rearranged to a more familiar form as shown below. The net force is equated to the product of the mass times the acceleration.

### Component Form of Newton’s Second Law

According to Newton’s second law, the net force of an object is an influence of the environment, acceleration is the object’s response, and the strength of an object is inversely proportional to the mass of the object. So it can be said that larger the mass, smaller is the acceleration.

The vector equation of Newton’s second law of motion can be written as three component equations as follows:

 $\vec{F_{x}}=m\vec{a}_{x}$ $\vec{F_{y}}=m\vec{a}_{y}$ $\vec{F_{z}}=m\vec{a}_{z}$

The second law is a description of how a body responds mechanically to its environment. The influence of the environment is the net force, the body’s response is the acceleration, and the strength of the response is inversely proportional to the mass m. The larger the mass of an object, the smaller its response (its acceleration) to the influence of the environment (a given net force). Therefore, a body’s mass is a measure of its inertia.

### Application of Second Law

The application of the second law of motion can be seen in identifying the amount of force needed to make an object move or to make it stop. Following are a few examples that we have listed to help you understand this point:

1. Kicking a ball
When we kick a ball we exert force in a specific direction, which is the direction in which it will travel. In addition, the stronger the ball is kicked, the stronger the force we put on it and the further away it will travel.
2. Pushing a cart
It is easier to push an empty cart in a supermarket than it is to push a loaded one. More mass requires more force to accelerate
3. Two people walking
Among the two people walking, if one is heavier than the other then the one weighing heavier will walk slower because the acceleration of the person weighing lighter is greater.

Understand the Laws of Motion and the concepts behind these theories by watching this intriguing video.

## Newton’s Second Law and Momentum

Newton’s second law in terms of momentum is stated as

The instantaneous rate at which an object’s momentum changes is equal to the net force acting on the body.

Following is the vector equation for Newton’s second law of momentum:

 $\vec{F}_{net}=\frac{d\vec{p}}{dt}$

Newton described momentum as quantity of motion which is a way of combining velocity of an object and its mass.

Momentum $\vec{p}$ is defined as the product of the mass of an object and its velocity which is given as:

 $\vec{p}=m\vec{v}$

Since velocity is a vector quantity, momentum too is a vector quantity.

The value of net force after substituting for momentum is given as:

$\vec{F}_{net}=m\frac{d(\vec{v})}{dt}=m\vec{a}$

Therefore, above is the Newton’s second law of motion in terms of momentum.

## Newton’s Second Law Examples

### Example 1:

If there is a block of mass 2kg, and a force of 5N is acting on it in the positive x-direction, and a force of 3N in the negative x-direction, then what would be its acceleration?

To calculate its acceleration, we first have to calculate the net force acting on it.

$F_{net}$ = 5N – 3N = 2N

Mass = 2kg

∴ Acceleration = $\frac 22$  = 1 m/$s^2$

### Example 2:

How much horizontal net force is required to accelerate a 1000 kg car at 4 m/s2?
Solution:
Newton’s 2nd Law relates an object’s mass, the net force on it, and its acceleration:
Therefore, we can find the force as follows:
Fnet = ma
Substituting the values, we get
1000 kg × 4 m/s2 = 4000 N
Therefore, the horizontal net force is required to accelerate a 1000 kg car at 4 m/s2 is 4000 N.

Newton’s second law is applied in daily life to a great extent. For instance, in Formula One racing, the engineers try to keep the mass of cars as low as possible. Low mass will imply more acceleration, and the more the acceleration, the chances to win the race are higher.