## 1. Newton's Law

Introduction

We have discussed motion and its different kinds like uniform motion and non-uniform motion. We have discussed accelerated motion where the speed of the objects varies with time. A moving body moves faster under acceleration and could stop also.  One thing which we have not discussed there is Who governs all these motions?

We have seen that magnets attract iron kept at a distance without even physical contact with it. And the moon causes tides on the earth even from such a large distance. So we can conclude that there must be an external agency that governs all these and these agencies can even affect from a distance ( gravitational force and electromagnetic force).

Concept of force

From the previous discussion, we can conclude that an external agency is required to describe what governs different kinds of motion.

We call it force!. To stop a moving object, to start a body from rest, or change the speed of the moving body, all require a force.

Now the next question is that does a force is required to keep a body moving in uniform motion?

Aristotle said that a force is required to maintain the uniform motion of the body. His statement is based on the motion of the body we see in our daily life.

When a child throws a ball with some initial speed it eventually stops after moving some distance, also a car moving at a constant speed cannot maintain its motion when we turn off the engine of the car. So someone can conclude that force is necessary to even maintain the uniform motion. But this is not correct and thus this statement is called the Aristotle fallacy.

Force is a vector quantity whose unit is Newton.

Let us first try to understand what is the correct answer to the question: does a force is required to keep a body moving in uniform motion?

The answer is No !  Force is not required for the uniform motion of the object.

The ball which is moving comes to rest later due to an external force which is a friction force acting on it, in its opposite direction. So an external force is required to cancel the friction force to maintain the uniform motion of the object.

If there is no friction then there will be no force required to maintain the uniform motion of the body.

Inertia: resistance to change

Inertia is that property of any matter by virtue of which it always resists any change in its state.

If we are habitual to wake up late in the morning and suddenly we have to wake up early for work, both the body and the mind try to resist this change.

If we have our opinion on something and then we listen to someone else's opinion about the same thing which is different from ours, then our mind tries to stick to its own opinion rather than accepting the opinion of the other.

Inertia is basically everywhere. But in this chapter, we will restrict ourselves to the concept of mechanical inertia.

Mechanical inertia is the inertia of matter by virtue of which it resists any change in its motion or rest. This concept laid the formulation of Newton’s first law.

Newton’s First law

Statement: An object which is at rest will try to remain at rest and an object which is in uniform motion will continue to do so, until and unless an external force is applied to it. This is called Newton’s first law which is also known as the law of inertia.

• When no force is acting on the object then there will be zero acceleration then the object at rest will remain at rest and an object moving with uniform speed will continue moving with uniform speed.

But we all know there is gravity everywhere on the earth and also some opposing forces like friction and viscous drag (in fluids) are present everywhere. So how could it be possible to have zero force on any object?

• Since force is a vector quantity, a force in a particular direction can be canceled by another force of the same magnitude acting in opposite direction.
• So if we have the sum of all the forces acting on an object is zero. Then also we can apply Newton’s First law as there is no net force acting on the object.

Example: A car is moving with uniform speed on the road when the external force provided by the engine of the car is exactly equal to the frictional force acting between the road and the tires of the car during motion.

A book kept in the book remains at rest as the gravitational force by the earth is balanced by the normal force from the table in the opposite direction to the gravity.

I am putting this for fun just to help to remember Newton's First Law. An object at rest will remain at rest unless acted upon by another force.

Significance of the Newton’s first law

1.  When we are sitting on a bike and it suddenly starts we get a jerk in the backward direction. Similarly when a moving bike stops we experience a jerk forward. This is true for any vehicle.

Explanation: When we are sitting on a bike our lower body is in contact with the bike but the upper body is not. When the bike starts, the lower body moves with the bike but the upper body resists the change in the state of rest and thus experiences a jerk backward.

2. When we place a playing card over a glass and a coin on the car. When we push the card, the card goes away but the coin falls into the glass.

Explanation: Here force is applied only on the card and thus moves away but the coin will try to remain at rest due to its inertia and thus falls into the glass as soon as the card is pushed away.

3.  When we hold the trunk of the tree and try to shake it, fruits fall from it.

Explanation: when we shake the trunk, the whole tree starts to shake The little branch with which the fruit is connected also vibrates and the fruit will try to remain at rest due to its inertia and thus detaches from it and falls on the ground.

4. Newton's law of inertia is the law that tells us why we should wear seatbelts while driving.

5. Law of inertia tells us while you go flying over the handlebars if you stop the bicycle suddenly.

Momentum

The momentum of a body is defined as the product of the mass and velocity of that body. It is a vector quantity.

p= mv

Let’s first discuss some common experiences related to motion in our daily life.

It is easier to put a car into motion than a loaded truck. Similarly, it would require greater force to stop a loaded truck moving at the same speed as a car at the same time.

Two stones, one lighter and the other heavier are dropped from the same height then it will be easier to catch the lighter stone than to catch the heavy stone.

From the above two discussions, we conclude that mass is one parameter that determines the effect of force on the motion.

A bullet fired from a gun can pierce human flesh before it stops and hence causes casualty. But the same bullet when thrown with hands does not harm much. This is because the stone fired from the gun has a much larger velocity than the bullet thrown by hands. Here we conclude that velocity is also a parameter that determines the effect of force on the motion.

As mass and velocity both are important parameters to describe the effect of force on the motion. Therefore a physical quantity which is the product of both mass and velocity (momentum) is a relevant variable of the motion

We can say that the greater the change in the momentum of a body, the greater the force will be needed. This statement laid the basis for the formulation of Newton's second law.

Newton’s second law of motion

Newton’s second law is a quantitative description of the changes that a force can produce in the motion of a body. It states that the time rate of change of the momentum of a body is equal in both magnitude and direction to the force imposed on it.

Statement:  The rate of change of momentum of the body is directly proportional to the applied force and takes place in the direction in which force acts.

If a force F is applied on a body of mass ‘m’ for a time Δt, if the velocity of the body change from ‘v to v+Δv’

Change in momentum  Δ p = m ( v +Δv ) - mv=  mΔv

Rate of change of momentum = m Δv/ Δt

So according to newton’s second law  F α Δp/Δt

F α m Δv/Δt   ;  F= kma  ; here k=1       so,    F= ma

Here K= proportionality constant which is equal to 1 here.

So mathematically, F= ma represents Newton’s second law.

•  S.I.  unit of force is Newton . 1 N= 1 kg m s^(-2)
• 1 N is the force that produces an acceleration of 1 ms^(-2) of an object of mass 1kg.
• When F= 0  then acceleration is also zero. So we can say that Newton’s second law is consistent with Newton's first law.
• Now since F and acceleration is a vector quantity

​​​​​​​F= Fx i + Fy j + Fz k     and   a= ax i + ay j + az k

So using Newton’s second law we have

Fx=ax/m    ; Fy = ay/m   ;   Fz= az/m

• The second law of motion is applicable to a single particle. In the case of an extended object, we consider it equivalent to a point particle and all the forces are applied on a single point which is the center of mass.
• Any internal forces within the body itself are not included in the force.
• The second law of motion is a local equation. It means that Force F at a given point at an instant ‘t’ relates to the acceleration at that point in that instant.

The same force for the same time produces the same change in momentum for different bodies

Some examples of Newton’s Second law from daily life.

• Karate player breaking slabs of bricks

A karate player makes use of the second law of motion to perform the task of breaking a slab of bricks. Since, according to law, the force is proportional to the acceleration, the player tends to move his/her hands over the slab of bricks swiftly. This helps him/her to gain acceleration and produce a proportionate amount of force. The force is sufficient enough to break the bricks.

• It is easier to push an empty shopping cart than a full one because the full shopping cart has more mass than the empty one. This means that more force is required to push the shopping cart.​​​​​​​

• Two people walking: of the two walking people, if one is heavier than the other, the one who weighs the heaviest walks slower because the acceleration of the one who weighs the lighter is more.
• Kick the ball: When we kick the ball we exert force in a specific direction, which is the direction the ball will move. In addition, the more forcefully the ball is kicked, the more force we apply to it and the further away the ball is.

• Racing cars:  Reducing the weight of racing cars to increase their speed, engineers try to keep vehicle mass as low as possible, as a lower mass means more acceleration, and the higher the acceleration the greater the chances of winning the race.

• Objects falling under gravity: When an object falls in a free fall onto the ground, it accelerates because the force of gravity of the earth pulls it. The velocity of the object keeps on increasing as it falls and has its maximum value just before hitting the ground.

​​​​​​​​​​​​​​​​​​​​​​​​​​​​There are many examples that illustrate Newton's second law in our daily life.
Impulse
Sometimes a large force acts on a body for a very short instant of time and thus produces a finite momentum on the body. For example, when a ball hits the wall, it bounces back. The force on the ball acts for a very short duration yet the force is large enough to reverse the momentum of the ball. Another example could be when a ball hits the bat and bounces back.
F=ΔP/Δtwhen  Δt is very small, F will be large
Since the force is very large and the time duration is very small. It is difficult to take account of both so we talk about change in momentum in such cases. Change in momentum is called impulse
F=ΔP/Δt  ; F Δt= ΔP  = Impulse
A large force acts for a very small time producing a finite change in momentum called Impulsive force. This is just like any other force in the mechanics.
Impulse = Pf - Pi
Newton’s Third Law
Forces exist in two forms, either as a result of contact interactions, i.e., normal, tensional, frictional, and applied forces; or as a result of actions-at-a-distance interactions, existing in the form of electrical, electrical, and magnetic forces. In this law, Isaac Newton described any two objects that are interacting to be exerting mutual forces upon each other.

• If you punch the bench with your first with some force, your fist will also experience a force from the bench and it will get hurt.
• If you are reading this article while sitting on the chair, you are exerting a force on the chair and in return, the chair also exerts an equal force on you. These forces cancel each other in pairs and thus you are sitting comfortably on it.

“Forces come in pairs.”. The two equal forces exerted are of the same magnitude but in opposite directions, known as action and reaction forces.  This led to the foundation of Newton’s third law.

Statement of Newton’s third law.  To every action, there is an equal and opposite reaction.

In fact, the term action-reaction is a misnomer. There is nothing like one force is the cause and the other force is the effect. There is no cause-effect relation implied to the third law. Object A applies a force F on B and object B also applies a force F on A in the opposite direction at the same instant.

## A variety of action-reaction force pairs are evident in nature, and in our real life. Here are 7 applications of Newton’s third law of motion:

1. Walking: When you walk, you push the street; i.e., you apply an active force on the street’s ground, and the reaction force moves you forward.

1. Gun Firing: when someone fires a gun, the action force pulls the bullet outside the gun, and the reaction force pushes the gun backward.

1. Jumping from a boat: the action force is applied on the boat, and the reaction force pushes you to land. Parallelly, the action force pushes the boat backward.

1. Slapping: when you slap someone, your hand feels pain and so does the cheek of the victim. The pain in the cheek is due to action force, and the pain in the palm is due to reaction force.
2. Bouncing a ball: when a ball hits the ground, the ball applies an action force on the ground. The ground applies a reaction force and the ball bounces back.
3. Flight motion of a bird: the wings of the bird push air downwards as an active force, and the air pushes the bird upwards as a reaction force.
4. Swimming of a fish: the fish’s fins push water around it backward as an active force, and the water applies a reaction force by pushing the fins forward, thus the fish.

## 2. equilibrium of force and friction

Conservation of linear momentum and its application

The second law and the third law led to important consequences - the Law of conservation of momentum.

Statement: When the net force on a system is zero, the angular momentum of the system is conserved.

F=dP/dt  ;     if F=0    then dP/dt=0  which means P=constant.

Let’s take an example of a gun and bullet system.

The Gun exerts a force  F on the bullet, the bullet also exerts the same force F but in the opposite direction, so the net force on the gun-bullet system is zero. So total linear momentum of the system remains conserved

Thus in an isolated system, the mutual force between pairs of particles of the system can cause momentum change in individual particles. Since the mutual forces for each pair are equal and opposite, the momentum changes cancel in pairs and the total momentum remains unchanged. This law is called the law of conservation of momentum.

The important application of the conservation of linear momentum is that when there is a collision between two particles, their momentum before and after the collision is conserved. Momentum conservation holds for both elastic and inelastic collisions.

By conservation of momentum m1u1+m2u2=m1v1+m2v2

Another application is rocket propulsion

Rockets have a gas chamber at one end of it. From this chamber, gas ejected with enormous velocity. Before the ejection of the gas, the total momentum is zero. Due to the ejection of the gas from the rocket, the rocket gains a recoil velocity and acceleration in the opposite direction. This is because of the conservation of momentum.

Equilibrium of concurrent forces

Conservation of momentum holds when no force is acting on the system. Gravity and friction forces are such forces that are always there on earth. So in most practical cases, it is not true that no force is acting on the system.

But we can have the net force acting on the system be zero. When all the forces acting on the system balance each other then we can say that the net force on the system is zero and thus conservation of momentum holds for such cases too, this is called equilibrium of concurrent forces.

Here  F=0  which means F1+F2+F3+.. Fn=0

When there are two forces acting on the particle and net force on the particle is zero. We have F1+F2=0  so  F1= -F2

Similarly in the case of three concurrent forces F1, F2 and F3

We have  F1+F2+F3=0

(F1x i +F1y j+F1z k)+(F2x i+F2y j+F2z k)+(F3x i+F3y j+F3zk)=0

On comparing the components of I, j and k on both sides we have

The sum of the x-component of all the concurrent force=0

F1x+F2x+F3x =0

Similarly, the sum of y and z components of all the concurrent forces is zero.

F1y+F2y+F3z=0

F1z+F2z+F3z=0

Lami’s theorem : Equilibrium of three force

If three concurrent forces are acting on a body kept in equilibrium, then each force is proportional to the sine of the angle between the other two forces and the constant of proportionality is the same.

We have P/(sin α)=Q/sinβ=R/sinγ

While applying Lami’s theorem in the free body diagram of a body we draw the direction of force either towards or away from the point of concurrency.

Some common forces in Mechanics

There are various kinds of forces in Mechanics. We will discuss contact and non-contact forces here.

Non-contact force:  Does not require direct contact between the body and the agency of force. These forces do not come under the branch of mechanics. We will study them later. For example electrostatics force, magnetic force etc.

Contact force: which requires direct contact between the body and the agency. Some of the contact forces are Tension force, normal force, friction force, spring force, etc.

Tension force:  It is a tension in the string due to which it balances the madd tied to it. A tension force is a force developed in a rope, string, or cable when stretched under an applied force. This force is acted along the length of cable/rope in a direction that is opposite to the force applied to it.

Normal force: The normal force is the force that surfaces exert to prevent solid objects from passing through each other. If two surfaces are not in contact they can’t exert a normal force on each other.

For example: If a book is resting upon a surface, then the surface is exerting an upward force upon the book in order to support the weight of the book.

Friction force:  When two objects are in contact and one object moves or intends to move, then a force develops between the two surfaces called frictional force.

Force in the spring

It is the restoring force that restores the spring to its original length. Spring force is the force required or exerted to compress or stretch a spring. When an object applies a force to a spring, then the spring applies an equal and opposite force to the object.

Pseudo forces: When we observe any object with respect to an accelerated body then there is a force acting on the object which we are observing. This force is called pseudo force. This is an imaginary force.

For example:  If we are sitting in an accelerating car and observing the body outside the car, then a pseudo force will act on that body.

Static and kinetic friction

Have you noticed that when you push a box by applying a force  F and it doesn’t move where does this Force F go?

According to Newton’s first law if the object remains at rest then the net force on the body must be zero. The question is which force balances the applied force when the body doesn’t move?

The answer to this question is Friction force.

When we apply a force on the object at rest, there is friction between the object and the surface that opposes the applied force.

The tangential component of the force of interaction between two surfaces in contact is called friction. It leads to resistance against movement between the surfaces and can cause mechanical deformation and heating. Friction depends on the type of the contacting surfaces. It is high for rough and dry surfaces and low for wet and smooth ones.

When we roll a ball on the floor, it stops after some time due to the force of friction acting on it opposite to the motion.

Types of Friction:

Depending on whether the surfaces are at rest or in relative motion against each other, the friction divides into static and kinetic friction.

Kinetic friction: The retarding force between two objects in contact that are moving against each other is called Kinetic friction.

Kinetic friction = μ_k N

Where μ_k=coefficient of Kinetic friction, N=normal force

Kinetic friction remains constant between two surfaces, regardless of the relative speed of their movement. The coefficient of kinetic friction has a constant value for each pair of contacting surfaces (materials). For example it is 0.57 for steel / steel contact, 0.47 for steel / aluminum contact, etc.

Static friction: The force that has to be overcome in order to get something to move is called static friction.

where μ_s=coefficient of static friction, N= normal force

In order to make a stationary object move, we have to overcome the static friction force with an applied force. When a small force is applied to a nonmoving object, the static friction is of equal magnitude, but in the opposite direction to the applied force. When the force is being increased, at a certain point it reaches the maximum static friction value. At that point, the static friction is overcome and the object starts to move.

On solid surfaces, static friction occurs as a consequence of the surface roughness of the objects in contact. Its value depends on the type of the contacting surfaces. It is higher for rough and dry surfaces and lowers for wet and smooth ones.

The force necessary to induce motion is always bigger than the one necessary to continue the motion. So the kinetic friction coefficient is smaller than the static friction one.    μ_s  > μ_k

Static and kinetic friction in an inclined plane

This is the force that prevents an object, placed on a sloped surface, from sliding.

When we place an object on an inclined plane and it remains stationary, this happens due to status friction. The gravity force on the object down the slope is actually balanced by the static friction in the opposite direction.

Laws of friction

## What are the Laws of Friction?

There are five laws of friction and they are:

• The friction of the moving object is proportional and perpendicular to the normal force.
• The friction experienced by the object is dependent on the nature of the surface it is in contact with.
• Friction is independent of the area of contact as long as there is an area of contact.
• Kinetic friction is independent of velocity.
• The coefficient of static friction is greater than the coefficient of kinetic friction.

Rolling friction

Rolling friction occurs when a wheel, ball, or cylinder rolls freely over a surface, as in ball and roller bearings.

A rolling wheel requires a certain amount of friction so that the point of contact of the wheel with the surface will not slip. The amount of grip of the tires on the road which can be obtained for an auto tire is determined by the coefficient of static friction between the tire and the road. If the wheel is locked and sliding, the force of friction is determined by the coefficient of kinetic friction and is usually significantly less.

Assuming that a wheel is rolling without slipping, the surface friction does not work against the motion of the wheel and no energy is lost at that point. However, there is some loss of energy and some deceleration from friction for any real wheel, and this is sometimes referred to as rolling friction. It is partly friction at the axle and can be partly due to flexing of the wheel which will dissipate some energy.

## 3. Uniform circular motion

Dynamics of uniform circular motion

What is circular motion? A body moving along the circumference of the circle with a constant speed is said to be exhibiting a circular motion.

For example, A car moves in a circular track of radius ‘r’ with velocity ‘v’. Then the circumference of the track will be  2Πr and it T be the time taken by the car to complete one round.

T= circumference/ speed = 2Πr/v

The natural tendency of the body is to move uniformly along a straight line. There is a requirement for some additional force to move a body along a circle and that force is called centripetal force.

Centripetal force fc=mv^2/r , centripetal acceleration 'ac=v^2/r

The direction of centripetal force and acceleration is always towards the center of the circle.

We know that circular motion can be both uniform and non-uniform. If the tangential component of acceleration is absent, it will be uniform circular motion, and if the tangential component of acceleration is present, it will be non-uniform circular motion. In the case of non-uniform circular motion, the net acceleration of the particle is the resultant of radial acceleration and tangential acceleration.

In the case of uniform circular motion, the tangential acceleration is zero so the speed of the object moving along a circular path is constant. There is only centripetal acceleration (negative radial direction) present.

Centripetal force

Suppose you are in an inertial frame of reference and you are observing a particle in a circular motion. The net force on the particle must be non-zero according to the second law of motion since the particle has some acceleration. Let’s take the case of uniform circular motion. The speed of the particle is constant, and the acceleration of the particle towards the center is v2/r

If ‘m’ be the mass of the object then centripetal force will be given by

fc= m *ac = mv2/r

This force is directed towards the center and is therefore known as centripetal forceThe centripetal force is required to keep the object in a uniform circular motion. This is just the name given to this type of force and this centripetal force can arise from tension, friction, etc.

Example of circular motion :

When a moving car on the road takes a turn, the motion of the car on the turn is also an example of circular motion

When a car takes a turn on a level road, the kinetic friction between the road and the car provides the necessary centripetal force to the car so that it would move in a circular path near the turn.

In the figure shown above, we can clearly see that Normal force is balancing the weight of the car on the level road   N= mg

And the frictional force is providing the necessary centripetal force. If the speed of the car is such that the required centripetal force exceeds the static friction between the car and the road, then it will no longer be able to move in the circular path. So there is a speed limit to take a safe turn.

We have  N=mg    and friction force =μ mg

centripetal force  fc= mv^2/r

Since frictional force provides the centripetal force.

mv^2/r  ≤ μmg     for safe turn

maximum safe velocity  v=μrg

This is the maximum safe velocity of the car in a circular motion in a level rod.

The motion of a car in a circular path on a Banked road

Consider a vehicle of mass m moving with speed v on a banked road of radius R as shown in the diagram. Let θ be the angle of banking.  N is the normal reaction exerted on the vehicle by a banked road. Let f be the frictional force between the road and the tires of the vehicle.

In the free body diagram, we can conclude that the vertical component of Normal force ( Ncosθ) balanced  weight ( mg ) and fsinθ

And horizontal component (Nsinθ) and fcosθ provide the necessary centripetal force for the circular motion of a car on a banked road.