Newton’s Second Law of Motion: Explanation and Application

Newton’s Second Law of Motion: Newton’s Second Law of Motion is one of the three laws of motion given by Sir Isaac Newton. Newton’s Laws of Motion deal with force and its effects. Newton’s Second Law is going to help us with the calculation of the net force on an object and its acceleration. The formula of Newton’s Second Law of Motion is F=m×a.

We come across the applications of Newton’s second law of motion examples in our day-to-day life. We can find the 2nd law of motion’s applications in seat belts, cricket, in the car, etc. In this article, we will study Newton’s Second Law of Motion in detail. Continue reading to learn more!

What is Newton’s Second Law of Motion?

Newton’s Second Law of Motion is one of the three laws of motion that govern the motion of bodies. So let’s first know all the three laws of motion, then proceed with the law in detail. Newton’s Laws of Motion are as stated below:

Newton’s First Law of Motion: According to this law, a body continues to be in its state of rest or of uniform motion along a straight line unless it is acted upon by some net unbalanced external force to change the state

Newton’ Second Law of Motion: According to this law, the rate of change of linear momentum of a body is directly proportional to the external force applied on the body, and this change takes place always in the direction of the force applied. In simple terms, it means that a larger force will change the momentum of an object more quickly as compared to a smaller force.

Newton’s Third Law of Motion: According to this law, to every action, there is always an equal and opposite reaction. Hence, Newton’s First Law of Motion describes force, Newton’s Second Law of Motion gives the way to calculate the force, and Newton’s Third Law of Motion relate forces between two interacting bodies.

Newton’s Second Law of Motion Examples, Formula, Explanation

Newton’s 2nd Law of Motion can be well understood by first knowing the concept of linear momentum.
Linear Momentum: The linear momentum of a body describes the quantity of motion possessed by a body. It is defined as the product of the mass and the velocity of the body. The mass of the body is represented by the symbol \(m\), and the velocity of the body is represented by the symbol \(v\). So, the linear momentum \(\left(p \right)\) is given by the formula,
\(p = m \times v\)
The linear momentum is a vector quantity and is measured in \({\text{kg}}\, { {\text{s}}^ { – 1}}\) in SI system of units and in \({\text{g}}\, {\text{cm}}\, {{\text{s}}^ { {\text{-1}}}}\) in CGS system of units.
Let us consider two bodies, one lighter and one heavier than the other. Let the bodies be at rest. Now, equal forces are applied to each of them for the same interval of time. We will see that the lighter body will pick up more speed than the heavier body, but in the end, if we check the momentum of both bodies, we will see that both have the same momentum. This happens because of two reasons.

1. Equal forces were acting on both the bodies, which means the rate of change of momentum of each body was the same.
2. The time interval for which the force was acting was the same for both the bodies.

It can also be concluded that the higher the magnitude of force, the higher is the change in momentum of the body. Also, the change in momentum is in the direction of the force.

Thus, from Newton’s Second Law of Motion, we infer the following:

1. The same force will produce the same rate of change of momentum even in the bodies with different masses.
2. The higher the magnitude of the force, the higher is the rate of change of momentum of the body.
3. The rate of change of the momentum of the body occurs in the direction of the force.

Newton’s Second Law of Motion Statement

Newton’s Second Law of Motion is stated as under:
“The rate of change of Linear Momentum of a body is directly proportional to the external force applied on the body, and this change takes place always in the direction of the applied force.” Thus, Newton’s Second Law of Motion connects the change of Linear Momentum of the body with force applied to it and gives the direction in which the change occurs.

Mathematical Formulation of Newton’s Second Law of Motion

Newton’s Second Law of Motion gives the following relation:

\(F \propto \frac{{{p_f} – {p_i}}}{t}\)

Here \(F\) is the applied force
\({p_i}\) is the initial momentum
\({p_f}\) is the final momentum and
\(t\) is the time for which the force acts on the body to bring the change in momentum of the body

But we know that,

\({p_i} = mu\) and \({p_f} – mv\)

where,
\(m\) is the mass of the body
\(u\) is the initial velocity of the body
\(v\) is the final velocity of the body and
So, the above relation of force becomes

\(F \propto \frac{ {mv – mu}}{t}\)
\(\therefore \,\,F \propto \frac{{m\left({v – u} \right)}}{t}\)

But we know that,

\(\frac{{v – u}}{t} = a\)

where \(a\) is the acceleration produced in the body due to the applied force
\(\therefore \,\,F \propto ma\)

Removing the proportionality sign by inducing a constant \(k\) we get,
\(F = kma\)
Here \(k\) is the constant of proportionality. The value of \(k\) depends on the units adopted for measuring the force.
Both in SI and CGS systems of units, the unit of force is chosen in such a way that \(k = 1\). Putting this value of \(k\) in the above equation, we get,

\(F = ma\)

This is the mathematical form of Newton’s Second Law of Motion.
This can be summarised in the below diagram:

Applications of Newton’s Second Law of Motion

Newton’s Second Law of Motion examples of many situations in our day-to-day life. Some of them are as under:

1. We use seat belts in the car. This is a safety feature to protect us from getting injured when sudden breaks are applied or during an accident. Due to seat belts, the change of momentum that happens in a very short interval of time is made to change in a larger interval of time, thereby reducing the impact of force on us.
2. While catching the ball, the fielder in the cricket ground takes his hands a little lower. This is done to increase the time and reduce the impact of force to avoid getting injured.
3. The firefighters use a net to catch the person jumping from a height. By using a net, the time is increased for the change in momentum so as to reduce the impact of force on the person jumping on the net.

In all the above cases, the time is increased so as to reduce the rate of change of momentum. During an activity, momentum will change, say from a certain value to zero, which cannot be altered by us. But we can increase the time taken to change the momentum in order to reduce the impact of force and avoid injuries or damages.
We hope that from this article, you could get some idea about Newton’s Second Law of Motion, its statement, derivation of its formula, its usage in solving numerical problems and its applications.

Solved Examples on Newton’s Second Law of Motion

Q.1. What force would be needed to produce an acceleration of \(2\, {\text{m}} { {\text{s}}^ { {\text{-2}}}}\) on an object of mass \(5\,{\text{kg}}\)?
Ans: Given that,
The acceleration to be produced in the object is \(a = 2\, {\text{m}} {{\text{s}}^{ {\text{-2}}}}\)
The mass of the object is \(m = 5\, {\text{kg}}\)
The required force is \(F = m \times a = 5 \times 2 = 10 {\text{N}}\)
Thus, \(10 {\text{N}}\) of force is required to produce an acceleration of \(2\, {\text{m}} { {\text{s}}^ { {\text{-2}}}}\) on an object of mass \(5\,{\text{kg}}\).

Q.2. What is the acceleration produced by a force of \(20\, {\text{N}}\) exerted on a body of mass \(2\,{\text{kg}}\)?
Ans:
Given that,
The force applied to the body is \(F = 20\, {\text{N}}\)
The mass of the body is \(m = 2\, {\text{kg}}\)
The acceleration produced is \(a = \frac{F}{n} = \frac{{20}}{2} = 10\,{\text{m}}{{\text{s}}^{{\text{-2}}}}\)
Thus, a force of \(20\, {\text{N}}\) exerted on a body of mass \(m = 2\, {\text{kg}}\) produces an acceleration of \(10\,{\text{m}}{{\text{s}}^{{\text{-2}}}}\).

Summary

Newton’s Second Law of Motion states that the rate of change of linear momentum of a body is directly proportional to the external force applied on the body, and this change takes place always in the direction of the force applied. Newton’s Second Law of Motion states that the greater the magnitude of force, the greater is the change in momentum of the body. Newton’s Second Law of Motion can be formulated as \(F \propto \frac{{{p_f} – {p_i}}}{t}\). The formula of Newton’s Second Law of Motion is \(F = m \times a\).

FAQs on Newton’s Second Law of Motion

Q.1: Which physical quantity does Newton’s Second Law of Motion define?
Ans: Force is the physical quantity that is defined by Newton’s Second Law of Motion.

Q.2: What is the statement of Newton’s Second Law of Motion?
Ans: The statement of Newton’s Second Law of Motion is, “The rate of change of linear momentum of a body is directly proportional to the applied external force, and this change occurs in the direction of the force.”

Q.3: What is the formula of Newton’s Second Law of Motion?
Ans: The formula of Newton’s Second Law of Motion is \(F = m \times a\).

Q.4: What is Linear Momentum?
Ans: Linear Momentum is the product of the mass of the body and its velocity.

Q.5: What is the SI unit of Linear Momentum?
Ans: The SI unit of Linear Momentum is \({\text{kg}}\,{\text{m}}{{\text{s}}^{{\text{-1}}}}\).

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