Newton's Second Law of Motion: Definition, Formula Derivation, Working Principle, Applications

In Simple words second law of motion states that the external force applied to the body leads to a change in the momentum of the body. Quantitatively, the external force applied to the body will be equal to the rate of change in momentum of the body.

The second law of motion explains the changes in the object or its state (position/rest) after applying an external force to the system. A force can also lead to a change in shape. However, considering we are dealing with rigid objects, this can not happen.

So, what are the changes possible due to an external force? Let’s understand the various conceptual aspects of the second law of motion.

Net Force on the System: When we apply an external force, it creates an imbalance in internal forces in the system. The net force changes in the system and becomes non-zero. This means that the vector sum of all forces acting within a system is not zero.

$${\vec{F}}_{\text{net}}\mathrm{\ne }\vec{0}$$

This non-zero net force produces a change in velocity known as acceleration. The simplest version of this laws is known in this form: " The Net force will be equal to the product of the mass and the acceleration produced by the force."

$$\vec{F}=m\vec{a}$$

Second Law of Motion and Momentum: Relation between Mass, Force, and Acceleration

The Non-Zero force provides the object an acceleration. The magnitude of acceleration depends on two variables: Mass (how much inertia it holds) and Force (magnitude of force applied).

• Acceleration and Mass: Let’s say, if a boy throws 2 rocks, weighing 100g and 1 kg respectively, we know the 100g rock will have more acceleration and will go faster. The acceleration produced due the external force will be inversely proportional to the mass of the object $$a \propto \frac{1}{m}$$ $$a\propto \frac{1}{m}$$
• It can be simply said that the higher the mass the lower the acceleration will be.
• Acceleration and Force: In the case of force, it's quite the opposite. If we take the example of a small boy and a full-grown man and ask them to throw a 100 g rock, what will happen? The man will throw the rock with more acceleration, meaning the rock will go faster and farther than the boy's throw. This indicates that the acceleration produced by an external force will be directly proportional to the amount of force applied to the object. $$\mathrm{Acceleration}\propto \mathrm{Force}$$

Meaning “ The stronger the force the higher the acceleration will be.”

 Momentum and Force: The acceleration due to the eternal force can also be explained as a change in momentum. Momentum is nothing but the vector product of the mass of the object and its velocity, and it simply tells us the amount of impact of a moving object. Momentum is depicted using P, So;

$$\mathrm{<br>}\vec{p}=\mathrm{mass.}velocity$$

$$\mathrm{<br>}\vec{p}=m<br>\vec{v}<br>$$

When a body is subjected to an external force, it gets an acceleration and in other words, its velocity changes (increases/decreases). Which subsequently changes momentum. However, it changes momentum only for a duration till the force is applied.

Which means we can say “ The Net force applied to an object will be equal to the rate of change of momentum of the object.”

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

Impulse, Force, and Momentum: Furthermore, this law also explains Impulse for a large force applied on an object in a very small time, like hitting a ball with a bat. Impulse is a product of the force applied to the object and the time duration during which the force was applied. The impulse is always equal to the change in momentum of the object.

$$\vec{J} = \vec{F} \cdot \Delta t = \Delta \vec{p}$$

Importance of Newton’s Second Law of Motion and Momentum

The Laws of Motion chapter of class 11 physics is one of the foundations and cores of mechanics. The second law of motion helps you understand the usage and application of force not only in mechanics but also in upcoming chapters like electrostatics, friction, fluid mechanics, and more.

The second law of motion is very important for analytical and problem-solving technique for JEE Mains and JEE Advanced aspirants. It not only helps in JEE but also for many medical and other exams like CUET and NDA. This law is particularly important for NEET aspirants; many a time, conceptual and numerical questions are directly asked from this topic.

Class 11 NCERT Physics textbook also offers a simple definition, given below, for Newton’s second law of motion.

“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.”

This definition gives you an explanation for the effects of external force in a system. It summarizes that there will be a change in the magnitude and direction of the momentum of the object due to the effect of external force. Also, remember that both the direction and the magnitude of force will be responsible for the change.

Let’s take a few assumptions for better understanding;

• The object is rigid, meaning it will not change its shape under the application of force.
• The body initially has a net-zero internal force, and it is in either a state of rest or uniform motion. $$\vec{F}_{\text{internal}} = 0,\quad v = 0 \text{ or constant} \Rightarrow a = 0$$
• The body has mass m and velocity v, and initially possesses momentum P= mv

Now, to establish a relation between mass, acceleration, and force, the initial momentum P is constant, without external force.

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

If we apply an external force to the body with mass $m$, the body changes velocity from $\vec{u}$to $\vec{v}$in time $t$, then the change in momentum is:

$$\Delta \vec{p} = m (\vec{v} - \vec{u})$$

$$\mathrm{\Delta }\vec{p}={\vec{p}}_{\text{final}}-{\vec{p}}_{\text{initial}}=m\vec{v}-m\vec{u}$$

According to Newton's second law of Motion, the change in momentum is equal to the applied external force.

$$\vec{F}_{\text{net}} = \frac{\Delta \vec{p}}{t} = m \frac{(\vec{v} - \vec{u})}{t} = m \vec{a}$$force as the derivative of momentum;

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

We know, If $m$is constant, we get the famous formula.

$${\vec{F}}_{\text{net}}=m\cdot \frac{d\vec{v}}{dt}=m\vec{a}$$

Therefore, the formula of force in tems of mass and acceleration;

$$\boxed{{\displaystyle {\vec{F}}_{\text{net}}=m\vec{a}}}$$

The formulas given below will provide the foundation for your JEE and NEET preparation. These formulas are written in the required order and will help immensely in numerical problem solving. 

Equilibrium Conditions

• Static equlibrium: $\vec{F}_{\text{net}} = 0$, $\vec{a} = 0$
• Dynamic equilibrium (constant velocity): $\vec{F}_{\text{net}} = 0$, $\vec{v}=\text{constant}$

Fundamental Formulae

• Second Law in Mathematical Form $${\vec{F}}_{\text{net}}=m\vec{a}$$
•  Momentum and Force Relation (Vector Form) $$\vec{F}=\frac{d\vec{p}}{dt}\text{(where }\vec{p}=m\vec{v}\text{)}$$
• For constant mass $$\vec{F}=m\frac{d\vec{v}}{dt}=m\vec{a}$$
• Change in Momentum $$\mathrm{\Delta }\vec{p}=m(\vec{v}-\vec{u})$$
• Impulse $$\vec{J}=\vec{F}\cdot \mathrm{\Delta }t=\mathrm{\Delta }\vec{p}$$

• Average Force $$\vec{F}_{\text{avg}} = \frac{\Delta \vec{p}}{\Delta t}$$
• For Varying Force (Integral Form) $$\vec{J}={\int }_{t_1}^{t_2}\vec{F}(t)dt$$

Attached or Connected Body Problems

Block on Inclined Plane

• Net force on a frictionless incline $$F = mg \sin\theta \Rightarrow a = g \sin\theta$$
• Net force on an inclined plane with friction constant µ $$F\; =mg\sin \theta -\mu mg\cos \theta \Rightarrow a=g(\sin \theta -\mu \cos \theta )$$

Tension in Strings – Connected Bodies

• Acceleration of  the system when two masses $m_1$ and $m_2$ connected by a string on a smooth surface: $$a = \frac{F}{m_1 + m_2}$$
• Tension in the strings:

$$T\; =m_1\cdot a$$

Frictionless Pulley System- Connected Bodies

• Acceleration for masses $m_1$ and $m_2$ hanging via pully $$a = \frac{(m_2 - m_1)g}{m_1 + m_2}$$
• Tension on the strings: $$T = \frac{2m_1 m_2 g}{m_1 + m_2}$$

Variable Mass System: Rocket Propulsion 

• Force on the propelling rocket

$$F = \frac{dp}{dt} = \frac{d}{dt}(mv) = m\frac{dv}{dt} + v\frac{dm}{dt}$$

Recoil Velocity (Conservation of Momentum)

• When a gun fires a bullet, the bullet conserves its momentum. $$M{v}_{\text{gun}}=-m{v}_{\text{bullet}}\Rightarrow v_{\text{gun}}=-\frac{m}{M}{v}_{\text{bullet}}$$

The second law of motion explains various natural and real-life phenomena. It explains the behaviour of an object when an external force is applied, and how the moving object changes velocity.

Real-life Applications

• Throwing a ball: We use muscular force to throw a ball of a constant mass. The amount of force decides how far the ball will go and what the velocity will be.
• Moving a stone tied to a rope in a circular path: We apply external muscular force to change its velocity in every instant so that it continues to revolve in the same circular path.
• Rocket propulsion: When a rocket starts its propulsion, it burns a huge amount of fuel. As soon as the fuel burns, the rocket becomes lighter and faster because of less mass and the external force created due to the fuel burning.

Application in JEE Main and NEET problems

NEET and JEE students have a good chance to encounter motion second law-based problems. Here are a few most important areas listed below;

• • Connected bodies and pulley system
  • Elevator-based problems
  • Motion of a charge between charged plates problems
  • Circular motion problems
  • Inclined plane and friction problems

Question: An external force of 100 N is applied to a block of mass 10 kg. Calculate its acceleration.

Given:

$m = 10 \, \text{kg}$

$F = 100 \, \text{N}$

Second law formula

$$F = ma$$Acceleration: $$a=\frac{F}{m}$$

Practice the below problems

1. A ball of mass 6 kg initially at rest is struck with a certain force. It reaches a velocity of 42 m/s in 0.3 s. Find the impulse.
2. A 200 g object moving at 30 m/s comes to rest in 5 seconds when the brakes are applied. Calculate the force applied by the brakes.
3. A rocket's mass reduces from 1000 kg to 700 kg in 6 seconds due to burning fuel. At the same time, its velocity increases from 100 m/s to 250 m/s. Find the net change in momentum.
4. A force of 400 N acts on a body for 0.8 seconds. What is the impulse imparted and the change in velocity if the object is of 20g?
5. Two blocks, 2kg and 3 kg, are connected by a massless spring. A force of 10 N is applied to the 3 kg block. Find the acceleration of the system and the tension in the spring.

You can check the frequently asked questions related to the 2nd law of motion,