Newton's Laws of Motion

Motion is something we usually observe in our daily life. To understand the nature of motion we study Newton's laws of motion. Sir Isaac Newton published three laws for the motion of moving objects in this universe, which appeared in his book Principia in 1687.

But these laws cannot be proven by any old theory of physics. As a result, these three laws create a new branch of physics called kinetics. So let's take a view of these three laws of motion. After that, I will uncover the details of these laws.

1st Law: An object continues to be in its state of rest or of uniform motion in a straight line unless an external force is applied to it.

2nd Law: The rate of change of momentum of a body is directly proportional to the force applied on it and this change in momentum takes place in the direction of the applied force.

3rd Law: To every action, there is an equal and opposite reaction.

Now, it's time to explain these laws. So let's start...

Newton's First Law of Motion

It is our common experience that a force has to be applied on an object to produce motion in it. For example, to move a cycle the cyclist has to apply force on the pedals. In a car, a petrol engine provides the force needed to move the car.

However, the observation of motion has been going on since ancient times. Before Galileo, thinkers like Aristotle and many others tried to understand the nature of motion. In 384 BC Aristotle gave his opinion on the nature of motion.

He said that if a body moves with uniform velocity, it needs something external to keep it moving, i.e a force is constantly needed to keep the body moving. If we stop providing this external force then the body stops moving.

This view was based on the general observation of daily life, that the object stops its motion when the external force is withdrawn. We notice it when we ride a bicycle. We constantly apply force on the pedals to keep the bicycle moving.

If we stop pedaling, after a while it stops. So we continuously supply the force to keep the bicycle moving. But this is a simple understanding of the nature of motion. Galileo Galilei an Italian astronomer did not approve of such a view of motion.

After Aristotle, Galileo studied the nature of motion and came to a very mature understanding of it. From his experiment, he found that no force is needed to continue the motion of a body. He said that if a body is set in motion it remains in motion if there is no force of friction opposing the motion.

So it is the frictional force that stops the object in motion. From our daily experience, we know that the bicycle does not stop at once as we stop pedaling but it moves a certain distance before coming to rest.

If we reduce the force of friction between the moving parts of the bicycle by proper oiling or greasing and if we ride on a smooth road, then the cycle travels a much longer distance after pedaling is stopped. If we completely remove the frictional force the bicycle will remain moving forever.

We completely understand that the moving object stops due to the force of friction. But what is the cause of motion?

Galileo said that it is inertia that keeps the object in motion. In the absence of frictional force the natural properties of matter, called inertia, keeps the object in motion forever.

So every object that is at rest or in motion, has a natural tendency to remain in that state unless an external force is applied to change its state of rest or motion. This is known as Galileo's law of inertia.

After that Newton studied his observations and put them, in the form of law, which is now known as newton's first law of motion.

According to Newton's first law of motion, an object continues to be in its state of rest, or of uniform motion, in a straight line unless an external force is applied to it.

After studying, the first law of motion, we know two things. The inertia of matter, and the concept of force.

Inertia of Matter

In the first law of motion, we see that the natural property of a stationary object is to remain stationary and the natural property of a moving object is to keep the object in motion.

A stationary object cannot move on its own and a moving object cannot stop on its own. In both cases, object is incapable of changing its state by itself. This incapability of the object is called inertia.

In another way, we can say that the property of matter by which an object at rest or in motion tends to remain in its state of rest or motion is called inertia. This is why newton's first law of motion is also called the Law of Inertia. The inertia of an object is related to the mass of the object, the greater the mass the greater its inertia.

Types of Inertia

There are three types of inertia - the inertia of rest, the inertia of motion, and the inertia of direction.

Inertia of rest: The tendency of an object at rest to remain at rest forever is called inertia of rest.

Probably, you noticed that when a car starts moving suddenly the seat passenger or the standing passenger leans back. It happens because of the inertia of rest.

When the vehicle is at rest the passenger's body is also at rest. As soon as the car starts moving the lower body of the passenger moves in line with the car while his upper body tries to stay at rest.

As a result, the passenger leans backward in the opposite direction of the vehicle. A similar case happens when a book is quickly pulled from the middle of a book arranged from the bottom to top.

the uppermost books come down but the column of the book does not break. This also happens for the inertia of rest.

Inertia of motion: The tendency of a moving object to remain in motion forever is called inertia of motion.

When a moving vehicle stops suddenly the rider leans forward. It happens because of the inertia of motion.

When the car is moving, the entire body of the rider is in motion with the car. As the car stops the lower body of the rider comes to rest but the upper body wants to move forward.

As a result, the rider leans forward. Another example of inertia of motion is - when an electric fan is turned off the fan does not stop immediately.

Due to the inertia of motion, the fan continues to spin for a while until the frictional force between the moving parts of the fan and the air resistance stops the fan completely.

Inertia of direction: The tendency of a moving object to remain in the same direction forever is called inertia of direction.

For example, In a car, the rider's body moves to the side when the car makes a sharp turn. If you observe you will find many more such examples of inertia in your daily life.

What is Force?

Definition: A force is an external cause that can change either the size and shape of an object or the state of rest or motion of the object

For example, if we press a soft rubber ball from two sides its shape changes, or on stretching a rubber string its length increases.

For this reason, sometimes force is simply defined as a push, or pull of an object.

Push: When a force moves an object away from something that is a push

Pull: When force brings an object closer that is a pull

Types of force

According to the application the force is classified into two categories - the contact force and the noncontact force.

Contact force:  Any kind of force that acts on objects when they are in physical contact is called contact force.

Such as reaction force, tension force, and frictional force are examples of contact forces.

When these forces are in action they act on the object with physical contact. That is why they are called contact forces.

Non-contact force: Any kind of force that acts on objects when they are not physically in contact is called non-contact force.

Such as gravitational force, electrostatic force, and magnetic force are examples of noncontact forces.

When these forces are in action they always act on an object from a distance. That is why, they are called noncontact forces.

We will know, more about all these forces, in another article. For now, we will only focus on the concept of force and its measurements.

As we already know from newton's first law of motion an object continues to be in its state of rest or of uniform motion in a straight line unless an external force is applied to it.

This clearly reveals the concept of force. But how can we measure it

We can measure force from newton's second law of motion. But before getting into newton's second law of motion, we need to discuss another concept, which is the concept of momentum.

Concept of Momentum

All objects have mass and if it is in motion it also has a velocity. The combination of this mass and velocity gives the object a dynamic property called momentum

The momentum of an object is equal to the product of its mass and velocity i.e if the mass of an object is m and its velocity is v then its momentum is mv

Let's understand the concept of momentum with an example. Suppose two similar trucks one loaded and the other empty are moving at the same speed. In this case due to the mass of the loaded truck its momentum is relatively high.

Now if two trucks have to be stopped at the same time more force must be applied to the loaded truck than the empty one.

Consider again that the two trucks have the same mass but different velocities. In this case the higher the speed of the truck the higher its momentum.

Now if we want to stop the truck at the same time more force must be applied to the faster truck than the slower one.

So we see that the amount of force required to stop an object in a given amount of time depends not only on its initial velocity but also on the combination of its mass and velocity i.e its momentum.

It is noticeable that the momentum of an object depends on both its mass and velocity.

Mass is a scalar quantity but velocity is a vector quantity, so momentum is also a vector quantity. It has both magnitude and direction. In the case of momentum, the direction of velocity is considered the direction of momentum.

Unit of momentum

If we talk about the unit of momentum then

The unit of momentum = The unit of mass × The unit of velocity

We know in the SI unit system the unit of mass is kg, and the unit of velocity is m/s. So the SI unit of momentum is kg.m/s. Similarly, we can write, the CGS unit of momentum, as g.cm/s

Dimension of momentum

If we talk about the dimension of momentum then

The dimension of momentum = The dimension of mass × the dimension of velocity

We know, the dimension of mass is M and the dimension of velocity is LT⁻¹. So the dimension of momentum is MLT⁻¹.

Law of Conservation of Linear Momentum

According to the law of conservation of linear momentum when two objects collide in an isolated system the total momentum before and after the collision remains equal.

Consider two balls of mass m₁ and m₂. Where the ball of mass of m₁ has a velocity of u₁ and the ball of mass of m₂ has a velocity of u₂.

Suppose they collide while moving in the same direction for which the ball of mass m₁ gains a velocity of v₁ and the ball of mass m₂ gains a velocity of v

Before collision, the total momentum of these two balls is m₁u₁ + m₂u₂. And after collision, the total momentum of these two balls is m₁v₁ + m₂v

If there is no external force acting on the ball i.e only interaction takes place then according to the law of conservation of linear momentum m₁u₁ + m₂u₂ = m₁v₁ + m₂v.

Newton's Second Law of Motion

Newton's second law state that the rate of change of momentum of a body is directly proportional to the force applied to it and this change in momentum takes place in the direction of the applied force

If an object of mass m has velocity v the momentum of the object is equal to mv

According to Newton's second law of motion if force F is applied to the object then

${\color{Red} F\propto \frac{d}{dt}(mv)}$

Here, d/dt is called the derivative which represents the rate of change of momentum. You can also represent the rate of change of momentum as, m(v-u)/t. Where u is the initial velocity and v is the final velocity of the object.

However now we need a constant to replace the proportionality symbol and place an equal sign there. This constant is taken here as k. So the formula is written as

${\color{Red} F=k\frac{d}{dt}(mv)}$

Since the mass of the object is constant the change happens only in velocity.

${\color{Red} \therefore \;F=km\frac{dv}{dt}}$

We also know the change in velocity is acceleration. Hence, we can write, the symbol of acceleration a in the place of dv/dt. Then the formula looks as

${\color{Red} F=kma}$

If we take unit force, unit mass, and unit acceleration then we get k = 1. Therefore the final formula looks as

${\color{Red} F=ma}$

Force is a vector quantity because it has both magnitude and direction. Newton's second law of motion gives not only the magnitude of force but also gives the direction of force.

According to this law the direction of the applied force is the direction of change of momentum or the direction of change of velocity, i.e the direction of acceleration.

Unit of Force

Now if we talk about the unit of force then

The unit of force = The unit of mass × The unit of acceleration

We know in the SI unit system the unit of mass is kg and the unit of acceleration is m/s². So the SI unit of force is kg.m/s². This unit of force is called newton (N). Similarly, we can write the CGS unit of force as g.cm/s². This unit is called dyne (dyn)

1 Newton: The force acting on an object of mass 1 kg causing an acceleration of 1 m/s² is called 1 Newton. As we defined 1 newton, similarly we can define 1 dyne as follows

1 dyne: The force acting on an object of mass 1 g causing an acceleration of 1 cm/s² is called 1 dyne. The relation between Newton and dyne is as follows

1 N = 10⁵ dyn

Dimension of force:

If we talk about the dimension of force then

The dimension of force = The dimension of mass × the dimension of acceleration

We know the dimension of mass is M and the dimension of acceleration is LT⁻². So the dimension of force is MLT⁻²

Verification of the first law from the second law

From newton's second law of motion, we find that F = ma. Now if no force is applied from outside, there is no change in the velocity of the object. The object continues to move with uniform velocity. If the object is at rest it remains at rest. This is newton's first law of motion.

Within newton's second law of motion lies the first law of motion. But newton mentions the first law of motion separately mainly to establish the concept of the inertial frame of reference.

Here I want to clarify that the inertial frame of reference is not related to the acceleration. It only relates to the object that is at rest or moving with uniform velocity

If acceleration is considered in the motion of the object then it is related to the Non-inertial frame of reference. But in this frame of reference, none of Newton's laws of motion are applicable.

I will deeply discuss the concept of the frame of reference in another article. For now, let's go into Newton's third law of motion.

But before getting into newton's third law of motion, we need to discuss the concept of the impulse of force, impulsive force, and the concept of action and reaction.

Impulse of Force

When a force of constant magnitude acts on an object for a duration of time the product of this force and the time of action is called the Impulse of Force

Consider, a constant force F acts on an object of mass m for time t. Then

The impulse of force = F × t

We know from newton's second law of motion F = ma (where a is the acceleration). If the velocity of the object changes from u to v due to the action of the force then the acceleration of the object a = (v-u)/t

∴ The impulse of Force = mv - mu

This relation clearly shows that the change in momentum of an object is equal to the Impulse of Force.

The Impulse of Force is also a vector quantity because it has both magnitude and direction.

Unit of Impulse of Force

According to the definition

The unit of Impulse of Force = The unit of force × The unit of time

Therefore the SI unit of the Impulse of Force is N.s. and the CGS unit is dyn.s

The Dimension of Impulse of Force

The dimension of impulse of force is similar to the dimension of momentum which is MLT⁻¹

We have understood the concept of impulse of force. Now we have to understand the concept of impulsive Force.

Impulsive Force

When a large amount of force acts on an object for a short period of time, then the force is called Impulsive Force

Example:

When a batsman plays a shot for six, a force acts on the ball for a short interval of time. We call that force an impulsive force.

When we hit a nail with a hammer, the hammer exerts a force on the nail for a very short period of time. Here the force exerts by the hammer is also an example of impulsive force.

Newton's Third Law of Motion

Statement: To every action, there is always an equal and opposite reaction. In this law, we see that two forces are mentioned which are action and reaction

If you don't know what action and reaction forces are then let me tell you that the force exerted on an object is called action and the force of equal magnitude exerted by the object opposite to the applied force is called reaction.

For example: when we walk we exert a force on the road with our legs. This is called action. The road also exerts the same force on the leg but in the opposite direction, for which we go forward. This is called reaction.

Another example is when a book is placed on a table the book exerts a downward force on the table due to its weight. At the same time, the table also exerts an upward force on the book known as the normal force.

Here this downward force due to the weight of the book is called action and the normal force acting by the table is called reaction.

Now Consider two objects A and B. If object A exerts a force FAB on object B. Then According to newton's third law of motion object B also exerts a counter force on object A. Which is equal and opposite to FAB

If the force exerted by object B on A is expressed by FBA. Then FAB = - FBA. If force FAB is called action then FBA is the reaction i.e action-reaction forces are always present together.

A single isolated force does not exist in nature. As long as the action lasts the reaction lasts. Without action, there is no reaction.

When we walk, swim, or hit a ball with a bat, we face these action and reaction forces. Similarly, there are many examples of action and reaction forces we find in our daily life.

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