Part 1

Introduction : What is Science?

Science is a systematic understanding of natural phenomena in detail so that it can be predicted, controlled and modified. Science involves exploring, experimenting and speculating phenomena happening around us.

The word Science is derived from the Latin verb Scientia meaning ‘to know.

The scientific method is a way to gain knowledge in a systematic and in-depth way. It involves:

• • Systematic observation
  • Controlled experiments
  • Qualitative and Quantitative reasoning
  • Mathematical modeling
  • Prediction and verification (or falsification) of theories
  • Speculation or Prediction

Natural Sciences

Natural science is a branch of science concerned with the description, prediction, and understanding of natural phenomena, based on observational and empirical evidence. It consists of the following disciplines:

• Physics
• Chemistry
• Biology

What is physics?

Physics is the study of nature and laws. There are so many different events in nature that are taking place and we expect that all these different events in nature are taking place according to some basic law and revealing these laws of nature from the observed events is Physics.

Humans have always been curious about the world around them. The night sky with its bright celestial objects has fascinated humans since time immemorial. The regular repetitions of the day and night, the annual cycle of seasons, the eclipse, the tides, the volcanoes, the rainbows, the color of the sky during sunrise and sunset have always been a source of wonder. All these phenomena are explained by the laws of Physics.

Word Physics has been originated from the Greek word phusikḗ which means nature.

There are two principal kinds of approaches in Physics which are listed below:

1. Unification: This is a method including all of the phenomena in the world in the form of a group of universal laws in various domains and conditions. The law of gravitation will be applied both on a falling apple from a tree and the movement of planets around the sun can be considered as examples. Every electric and magnetic phenomenon will be controlled by Electromagnetism laws.

2. Reduction: This is a method for deriving characteristics of complex systems from the properties and interaction of their constituent parts. We can take an example that the temperature studied under thermodynamics can be also connected to the average kinetic energy of molecules in a system (kinetic theory).

Role of Mathematics in Physics : 

 Description of all-natural phenomena is made simple by the help of mathematics. Thus we can say that mathematics is the language of Physics. With the help of Mathematics, we explain and understand the basic laws of physics in a better way.

 

Scope of Physics

Scope of Physics will be wide since it covers quantities with length magnitudes as big as 10⁴⁰ m and more than that (astronomical studies of the universe) and as low as 10^-14 m  or less (study of the electrons, protons etc). In the same way, the time scale is ranging from 10^-22 to  10¹⁸ s and the mass is ranging from 10^-30 kg to 10⁵⁵ kg.

Physics can be categorized broadly into two kinds on the basis of its scope - Classical Physics and Modern Physics.

Basically, there are two domains of interest: macroscopic and microscopic. The macroscopic domain includes phenomena at the laboratory, terrestrial and astronomical scales. The microscopic domain includes atomic, molecular and nuclear phenomena.

Classical physics is a branch which is dealing with macroscopic phenomena while modern physics will be dealing with macroscopic phenomena.

Macroscopic Domain

The macroscopic domain includes phenomena at large scales like a laboratory, terrestrial and astronomical. It includes the following subjects:

1. Mechanics – It is based on Newton’s laws on motion and the laws of gravitation. It is concerned with the motion/equilibrium of particles, rigid and deformable bodies and the general system of particles.

Examples:  Propulsion of rocket by ejecting gases, Water/Sound waves, Equilibrium of bent rod under a load etc.

2. Electrodynamics – It deals with electric and magnetic phenomena associated with charged and magnetic bodies.

Examples:  motion of a current-carrying conductor in a magnetic field, the response of a circuit to an ac voltage (signal), the propagation of radio waves in the ionosphere etc.

3. Optics – It deals with phenomena involving light.

Examples,  Reflection and refraction of light, Dispersion of light through a prism, Colour exhibited by thin films etc.

4. Thermodynamics – It deals with systems in macroscopic equilibrium and changes in internal energy, temperature, entropy, etc. of systems under the application of external force or heat.

 Examples: Efficiency of heat engines, thermal expansion, Direction of physical and chemical process etc.

Microscopic domain

Microscopic Domain

The domain includes phenomena at minuscule scales like atomic, molecular and nuclear. It also deals with the interaction of probes like electrons, photons

and other elementary particles. Quantum theory has been developed to handle these phenomena.

Physics is exciting in many ways. To some people, the excitement comes from the elegance and universality of its basic theories, from the fact that a few basic concepts and laws can explain phenomena covering a large range of magnitude of physical quantities. To some others, the challenge in carrying out imaginative new experiments to unlock the secrets of nature, to verify or refute theories, is thrilling. Applied physics is equally demanding.

Factors responsible for the progress of Physics

• Quantitative analysis along with qualitative analysis.
• Application of universal laws in different contexts.
• Approximation approach (complex phenomena broken down into a collection of basic laws).
• Extracting and focusing on essential features of a phenomenon.

Hypothesis, Axiom and Models

A hypothesis is a supposition without assuming that it is true. It may not be proved but can be verified through a series of experiments.

Axiom is a self-evident truth that is accepted without controversy or question.

Model is a theory proposed to explain observed phenomena.

The assumption is the basis of physics, where a number of phenomena can be explained. These assumptions are made from experiments, observation and a lot of statistical data.

Physics-Technology and society

With the advancement of technology, human civilization also advanced.

For example, with technological advancement, steam engines are invented and then various industries are set up and an industrial revolution happens which in turn changed human civilization.

There is a complementing relationship between physics and technology. Sometimes physics gives rise to new technology and sometimes new technology gives rise to new physics.

For example, the advancement of the research of semiconductor materials which made it possible to make transistors, chips and integrated circuits which later contributed to the technological advancement of modern computers. In this example, Physics is giving rise to new technology.

Now with the advancement in technology of accelerators and detectors, it allows us to penetrate deeper in the atomic level and give rise to microphysics like nuclear physics, atomic physics, particle physics etc. In this example, the advancement in technology gave rise to new physics.

Technological Applications of Physics:

There are numerous examples in which Physics and its concepts paved the way to inventions as mentioned below.

• The steam engine was invented during the industrial revolution in the eighteenth century.
• Development of wireless communication after the discovery of the laws of electricity and magnetism.
• Neuron-induced fission of uranium, attempted by Hahn and Meitner in 1938, showed the formation of nuclear power reactors and nuclear weapons.
• Electricity has been produced from solar, wind, geothermal etc. energy.

Fundamental forces in Nature

Fundamental Forces in nature

The forces which we see in our day-to-day life like muscular, friction, forces due to compression and elongation of springs and strings, fluid and gas pressure, electric, magnetic, interatomic and intermolecular forces are derived forces as their originations are due to a few fundamental forces in nature.

A few fundamental forces are:

1. Gravitational Force: It is the force of mutual attraction between any two objects by virtue of their masses. It is a universal force as every object experiences this force due to every other object in the universe.

2. Electromagnetic Force: It is the force between charged particles. Charges at rest have electric attraction (between unlike charges) and repulsion (between like charges). Charges in motion produce magnetic force. Together they are called Electromagnetic Force.

3. Strong Nuclear Force: It is the attractive force between protons and neutrons in a nucleus.It is charge-independent and acts equally between a proton and a proton, a neutron and a neutron, and a proton and a neutron. Recent discoveries show that protons and neutrons are built of elementary particles, quarks.

4. Weak Nuclear Force: This force appears only in certain nuclear processes such as the β-decay of a nucleus. In β-decay, the nucleus emits an electron and an uncharged particle called the neutrino. This particle was first predicted by Wolfgang Pauli in 1931.

Below table shows the difference between the above forces.

Conserved Quantities:

Physics has provided laws for summarising the investigations and observations of the phenomena happening in the universe.

• Physical quantities will be held fixed with time and can be defined as conserved quantities. In the case of a body under external force, the kinetic and potential energy will be varying over time but the total mechanical energy (kinetic + potential) will be a constant.
• Conserved quantities will be scalar (Energy) or vector (Total linear momentum and total angular momentum)

Conservation Laws:

A conservation law can be defined as a hypothesis on the basis of observation and experiments which is not able to be proven. These are verifiable through experiments.

Law of Conservation of Energy:

• In accordance with the General Law of conservation of energy, the energies will be fixed over time and get transformed from one form to another.
• The law of conservation of energy will be applied to the whole universe and it has been considered that the total energy of the universe is fixed.

Nature develops symmetric results at a different time under similar conditions.

Law of Conservation of Mass:

It can be defined as a principle that is usable in the analysis of chemical reactions.

• Basically, a chemical reaction can be defined as a rearrangement of atoms among various molecules.
• The difference will be formed as heat and the reaction is exothermic when the total binding energy of the reacting molecules will be less than the total binding energy of the product molecules.
• The opposite will be correct for energy-absorbing reactions such as endothermic reactions.
• As the atoms are not destroyed, only just rearranged, the summation of the mass of the reactants will be identified as the total mass of the products in a chemical reaction.
• Mass will be in relation to energy through Einstein's theory,  E= mc2  where c will be the speed of light in vacuum.

Law of Conservation of Linear Momentum:

• Law of conservation of linear momentum can be defined as the symmetry of laws of nature with respect to translation in space.
• The law of gravitation is exactly identical on earth and moon even when the acceleration due to gravity on the moon is ⅙  than that on earth.

 Law of Conservation of Angular Momentum:

Isotropy of space means that no intrinsically preferred direction in space specifies the law of conservation of angular momentum.

Introduction:

Measurement of any physical quantity involves comparison with a certain basic, arbitrary chosen, Internationally accepted reference standard called units. Any measured quantity is expressed by a number and a unit. The units for the fundamental or base quantities are called fundamental or base units. The units of all other physical quantities can be expressed as combinations of base units are called derived units. A complete set of these units, both the base units and derived units, is known as the system of units.

Need for measurement

There is a need for measurement in physical quantity. Without it, how can you define things? By measuring a physical quantity we define the properties of a material which the physical quantity is associated with. For example, distance, speed, mass, pressure, force, momentum, and energy.

Think this yourself, what is going to be the meaning of the length of an object if we cannot measure it. We will then inform its length subjectively as ‘long’, ‘short', etc. That makes no sense anymore now.

A physical quantity is a property that can be quantified by the measurement of a material or system. A physical quantity may be represented as the combination of a unit and a numerical value. For example, by specifying methods for calculating them, we outline distance and time, while we describe average speed by stating that it is measured as traveled distance divided by travel time. Measures of physical quantities are represented in unit units, which are numerical meanings

Physical quantities haven't just existed before the dawn of mankind. We made them up to serve our own inquisitive purposes. We have discovered that there are certain properties in the world around us, which can be altered in proportions but not in nature.

Heavier. More. Higher. Stronger. These are all comparative terms. And to compare, we need a basis for measurements to take place. Because, without one, there is no logical way to distinguish between different levels or amounts of the same property.

Units of measurement

 There are two types of units of measurement.

• Fundamental/ Base units
• Derived units

Fundamental units: The fundamental units are the base units defined by the International System of Units. These units are not derived from any other unit, therefore they are called fundamental units. There are seven fundamental units.

Derived units. The quantities that are derived using the fundamental quantities are called derived quantities. The units that are used to measure these derived quantities are called derived units. Examples: Force, Velocity, Density, Heat, Power, Energy, Momentum, Acceleration. Some derived units are given in the table below.

Difference between fundamental and derived units

System of units:

In earlier times scientists of different countries were using different systems of units for measurement.  We have here four types of system of units, one of which is S.I units which is accepted globally.

System of units are classified mainly into four types:

 1. C.G.S. system: It stands for Centimetre-Gram-Second system. In this system, length, mass and time are measured in centimeters, grams and seconds respectively.

2. M.K.S. system: It stands for Metre-Kilogram-Second system. In this system, length, mass and time are measured in meters, kilograms and seconds respectively.

 3. F.P.S. system: It stands for Foot-Pound-Second system. In this system, length, mass and time are measured in the foot, pound and second respectively.

 4. S.I. system: It stands for System International. This system has replaced all other systems mentioned above. It has been internationally accepted and is being used all over the world. As the SI units use a decimal system, conversion within the system is very simple and convenient.

Introduction

In our day-to-day life, we see the motion of various objects around us like the motion of cars, planes, sun, moon and people and things around us.

We all have a vague idea of Motion. But in this text, we will discuss this topic in detail with all the important aspects of the motion.

When the motion is confined to one dimension in a straight line we call this type of motion rectilinear motion. We will discuss this type of motion in this chapter.

You will have an idea of the state of motion and rest and I will introduce a very important concept of the frame of reference to discuss the state of rest or motion. Then we will learn about the kinematics of rectilinear motion in this unit. In kinematic we describe motion without going into the cause of the motion.

Motion and rest

Suppose a man is sitting in a park, he sees trees and plants around him which are at rest according to him. Now another man sitting in his car passes near that park. The man sitting in the car sees all the trees and plants moving in the opposite direction of its motion. I am sure all of you must have experienced the same situation.

Now tell me who is right and who is wrong?

Do the plants and trees are at rest or in motion?

Actually here both are right.  To understand this let's first try to understand how we define something to be in motion or at rest.

The fact is that there is nothing like absolute rest or absolute motion in the whole universe.  One thing which is at rest according to one observer in one frame of reference may appear to be moving according to another observer in another frame of reference.

We define something to be in rest or motion with respect to some reference which is called a frame of reference.

With respect to the man sitting in the park, trees and plants in the park appear to be at rest as they are not changing their position with respect to the man sitting there.

But since the man in the car is itself moving so when he crosses through the park, trees and plants are left behind with respect to him and as he moves ahead with some speed, he notices that trees appear to be moving with the same speed but in opposite directions.

We define all the motions with respect to some frame of reference.

We have two types of frame of reference: Inertial and Noninertial

A body is said to be at rest if its position doesn't change with respect to its surroundings, whereas when the position of a body changes with respect to its surroundings it is said to be in motion. The state of rest or motion of a body is relative to each other.

Position, path length and displacement.

Position:  To describe the position of the object, we must know and be able to describe its position.

In physics, we specify a position with the help of a reference point and a set of three mutually perpendicular axes of the rectangular coordinate system.

In this chapter we are confined to one- dimension, so we need only one axis to specify the position.

In the above example Dorm is chosen as a reference point and the position of the cafeteria, physics block and library can be described with the reference of the dorm.

• The cafeteria is 1 unit left to the dorm  (x=-1)
• The physics department is 2 units right to the dorm ( x=2)
• The library is 3 units right to the dorm. (x=3)

Similarly in the example below the lamp post is taken as a reference point and the position of different people is described with respect to the reference point.

Path length and displacement

Path length or distance traveled is the length of the actual path taken by the object from the initial to the final position. In the figure given below, the red represents the actual path between the start and finish points. This is the distance traveled or path length. It is a scalar quantity.

Displacement is the shortest distance between the initial and final positions. The line represented in blue is the displacement. It is a vector quantity.

The difference between Distance and Displacement is as follows:

Uniform and non-uniform motion

The motion of the object is said to be a uniform motion if the object travels an equal distance in an equal interval of time.

In the above example, a car C travels 4 m every 2 seconds throughout its journey, here car C is in uniform motion.

The motion of the object is said to be non-uniform motion if it travels an unequal distance in an equal interval of time

In the above example car C travels 1 m in the first second and then 9 m and 15 m in another consecutive second. So in this example, a car travels an unequal distance in an equal interval of time.

The distance-time graph of the uniform and non-uniform motion is given below

Speed and velocity

Speed: It is the rate of change of position of an object. It refers to how fast an object is moving. For example, if a bike covers 20 meters in 1 second. Then its speed will be 20 m/s. It is a scalar quantity.

Velocity: In contrast with speed, velocity is a vector quantity and it also tells us about the rate of change of position of the object, but it is direction aware. The magnitude of velocity is speed. In other words, we can say that velocity is the speed in a given direction.

Even if the object is in motion, its velocity can be zero. This will happen when the object returns to its initial position and hence displacement is zero, so velocity will also be zero. But speed cannot be zero if the object is in motion.

Velocity=Displacement/Time

Like displacement can be negative, positive and zero, velocity can also be positive, negative and zero.

Average speed and velocity

The average speed of an object refers to the total distance it travels divided by the time which is elapsed. It is a scalar quantity.

Average  speed = (Total distance traveled)/(Total time elapsed)

If the object travels with speed  ‘x’ from A to B and returns from B to A with speed ‘y’. Then the average speed will be

Average speed =  2xy/(x+y)

Average velocity: Average velocity is the ratio of displacement to the time taken in the entire journey. The average velocity can be zero even if the average speed is not zero.

Instantaneous velocity

The instantaneous velocity of an object is the limit of the average velocity as the elapsed time approaches zero, or the derivative of x with respect to t.

Instantaneous velocity at any point in time is the slope of the tangent of the Distance -time graph at that point.

Scalar and Vector quantity

There are many physical quantities in Physics. Like pressure, density, force, speed, temperature, acceleration, etc. These quantities can be classified into two categories- scalar and vector quantities.

Scalar quantities are those which have only magnitude. For example, density, temperature, distance, speed, pressure, etc and vector quantities have both magnitude and direction associated with them like force, acceleration, velocity, etc.

This section will learn about vectors and their properties in detail.

Position and displacement vector

Position vector

A position vector is defined as a vector that symbolizes either the position or the location of any given point with reference to the origin. The direction of the position vector always points from the origin of the vector towards the given point.

For example, Suppose an object is placed at coordinate (2, 3,4 ) in a rectangular coordinate system. The position of this point can be expressed as position vector     OP= 2 i +3 j+4k

Displacement vector

The change in the position vector of an object is known as the displacement vector.

Suppose an object is at position A at time t=0 and after some time ‘t’ it reaches position B. Let the position of A is (x1, y1, z1)  and B is

(x2,y2,z2).

Position vector of A  = OA= (x1 i+y1 j+z1 k)

Position vector of B= OB= (x2  i+ y2 j+z2 k)

So the displacement of AB is given by displacement vector AB

AB= OB-OA = ( (x2-x1)i +(y2-y1)j+(z2-z1)k)

The displacement of an object can be defined as the vector distance between the initial point and the final point.

Suppose an object travels from point A to point B via curve path in pink. The displacement of the particle would be the vector line AB headed in the direction A to B. The direction of the displacement vector is always from the initial to the final position.

General vector and their notations

A vector has both magnitude and direction. The length of the vector represents its magnitude and the tip of the vector specifies its direction

In cartesian coordinates, we generally express a vector in 3-D in terms of its projection along the x, y and z-axis.

In the figure given above Vector A=Ax i+ Ay j+Az k.

Here Ax, Ay and Az is the projection of vector A in direction x, y and z.

Equality of vector

Two vectors are equal if all the components of the vector are equal. In other words, if both the magnitude and direction of the vector is equal then only two vectors are equal.

Negative of the vector: The vectors and their negatives have the same magnitude but opposite directions.

Vector addition: Always add vectors from tip to tail. Place the tip of the first vector next to the tail of the second vector. The Result is the sum of the first two vectors.

Vector subtraction: When we have to subtract two vectors : A- B = A + (-B) . Then first take the negative of the second vector and then add them according to the rule of vector addition.

Vector addition and subtraction in 1 dimension

Vector  addition subtraction in 2 dimension

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/Δt;  when  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.

How Is Newton’s Third Law of Motion Useful in Our Real Life?

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.

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

3. 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.

4. 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.
5. 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.
6. 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.
7. 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.

Work

Work is the product of the component of the force in the direction of the displacement and the magnitude of this displacement.

work, in physics measure of energy transfer that occurs when an object is moved over a distance by an external force at least part of which is applied in the direction of the displacement. If the force is constant, work may be computed by multiplying the length of the path by the component of the force acting along the path.

To express this concept mathematically, the work W is equal to the product of force and the distance d,

work done= f∙d= fdcosθ, If the force is being exerted at an angle θ to the displacement.

Work done on a body is equal to the increase in the energy of the body, for work transfers energy to the body. If, however, the applied force is opposite to the motion of the object, the work is considered to be negative, implying that energy is taken from the object.

Positive, negative and zero work done

The work done by a force on an object can be positive, negative, or zero, depending upon the direction of displacement of the object with respect to the force. For an object moving in the opposite direction to the direction of force, such as friction acting on an object moving in the forward direction, the work done due to the force of friction is negative.

Positive work: when the force is along the displacement or angle θ between force and displacement is acute.

W= fd cosθ >0  , when cosθ >0 , θ< 90.

Example of positive work:

1. Players kicking the football in the direction of motion.
2. A nurse moves the patient into a wheelchair.
3. A person riding a skateboard.
4. Vehicles on the road, moving forward.
5. Cutting the vegetables using the knife.
6. Lifting the chair and moving it in another direction.
7. Moving a box across the table.
8. Two children throwing a ball at each other.

Negative work: when the force is opposite to the displacement or angle θ  between force and displacement is obtuse.

W= fd cosθ < 0 , when cosθ  <0 ,  θ >90.

Example of negative work

(i) When a body is thrown upwards, gravity does the negative work. Since the gravitational force acts downwards but the displacement is upwards.

(ii) When we walk frictional force does the negative work since frictional force acts opposite to displacement.

(iii) For a liquid flowing, viscous force does the negative work since it acts opposite to the direction of the force.

(iv) On a see-saw, negative work is done since we apply the force downwards but the person sitting opposite to us is displaced upwards.

Zero work done:

An object experiences zero work when

• The angle of displacement is perpendicular to the direction of the force
•  when the force applied couldn’t produce motion.

Consider an example of a coolie lifting a mass on his head moving at an angle of 90˚ with respect to the force of gravity. Here, the work done by gravity on the object is zero.

w= fd cosθ= fd cos 90= 0

Work done by constant force

Work done by a constant force is defined as the distance moved

multiplied by the component of force in the direction of displacement.

The area under the graph of force and displacement gives the value of work done by the force.

Example of work done by a constant force

• Work done by gravity
• When we apply a constant force F on a book and it moves.
• Motion of ball falling toward the ground.

Work done by a variable force

Variable force occurs when the direction and amount of a force vary throughout the motion of a body. Magnetic force, spring force, and electrostatic force are examples of variable forces. The majority of the forces we experience in our daily lives are variable forces. By splitting displacement into tiny intervals, the work done by a variable force may be computed.

A force is said to perform work on a system if there is displacement in the system upon application of the force in the direction of the force. In the case of a variable force, integration is necessary to calculate the work done.

The work done by a constant force of magnitude F, as we know, that displaces an object by Δx can be given asL:

W = F.Δx

In the case of a variable force, work is calculated with the help of integration.

The work done by force can also be calculated from the graphical method. The area under the curve in the graph of force vs displacement will give the magnitude of work done by the force.

For example, in the case of a spring, the force acting upon any object attached to a horizontal spring can be given as:

Fs = -kx

Where,

• k is the spring constant
• x is the displacement of the object attached

We can see that this force is proportional to the displacement of the object from the equilibrium position, hence the force acting at each instant during the compression and extension of the spring will be different. Thus, the infinitesimally small contributions of work done during each instant are to be counted in order to calculate the total work done.

The integral is evaluated as:

Introduction

In the mechanics we have studied so far, we have assumed the object to be a point mass object, we actually neglected the finite size of the objects. For example, when we talked about the motion of a car on a road, we actually neglected the finite size of the car and actually treated it like a point mass which is certainly not true.

Almost everything which we encounter in our daily life is of finite size and in dealing with the motion of an extended body (of finite size) the idealized concept ( taking it as a point size ) is inadequate.

We must try to understand the motion of the extended body as a system of particles. We shall begin with the consideration of the motion of the system as a whole. The center of mass of a system of particles will be a key concept here.

A wide variety of problems on extended bodies can be solved by taking it as a rigid body. Ideally, a rigid body is a body with a perfectly definite and unchanging shape. The distances between all pairs of particles of such a body do not change.

Although this assumption is not always true, in many cases the deformation produced after the application of the force on a rigid body is so small that it can be neglected.

What are the kinds of Motions a rigid body can have?

We can have three types of motion in a rigid body.

1. Purely translational motion.
2. Translational + rotational motion.
3. Purely rotational motion.

Purely translational motion

In pure translational motion at any instant of time, all particles of the body have the same velocity.

Consider a rectangular block sliding on an inclined plane. We suppose that this block is made up of a system of particles. Suppose we pinpoint two particles P1 and P2 here. We will see that in the case of translational motion, both the particles move with the same speed in the same direction which means they have identical velocities.

Pure rotational motion

Pure Rotational motion is the type of motion about a fixed axis. All the particles constituting it undergoes circular motion about a common axis, then that type of motion is rotational motion.

The line or fixed axis about which the body is rotating is its axis of rotation A rotating body is said to be in the pure rotation if all the points at the same radius from the Center of rotation will have the same velocity.

Rotational plus translational motion- Rolling motion

The rolling motion is a combination of translational motion and rotational motion. For a body, the motion of the center of mass is the translational motion of the body. During the rolling motion of a body, the surfaces in contact get deformed a little temporarily.

During rolling motion, all the particles have different velocities. Like P1, P2, P3, and P4 in the above example of a ball rolling down the inclined plane has different velocities.

Rolling of the wheel is an example of rolling motion. In pure rolling motion (rolling without slipping), the point of contact is at rest.

Pure rolling motion = pure translational + pure rotational

Out of these three types of motion of the rigid body, we will discuss the rotational motion of the rigid body in detail in this chapter.

Rotational motion of the rigid body

The rotational motion of a rigid body can be of two types.

• Rotation about a fixed axis of rotation :

In rotation of a rigid body about a fixed axis, every particle of the body moves in a circle, which lies in a plane perpendicular to the axis and has its center on the axis. For example, rotating fan, merry-go-round, potter's wheel, etc.

In the figure given above on the right, there is a rigid body that is rotating anticlockwise about the Z axis as shown. The solid line shows the axis of rotations.

We take two particles P1, and P2 at distances r1 and r2 respectively from the axis of rotations.  Circles C1 and C2 give the path of the particle followed while rotating. These circles C1 and C2 lie in a plane perpendicular to the axis of rotation.

 Another particle P3 is taken at the axis of rotation so here r=0, this point will remain at rest when the whole of the rigid body would be rotating. For any particle on the axis like P3, r = 0. Any such particle remains stationary while the body rotates. This is expected since the axis of rotation is fixed.

• Precessional motion: Rotation about an axis in the rotation (rotating axis )

In some examples of rotation, however, the axis may not be fixed. A prominent example of this kind of rotation is a top spinning in place. We know from experience that the axis of such a spinning top moves around the vertical through its point of contact with the ground, sweeping out a cone as shown in the figure.

The motion of a rigid body which is not pivoted or fixed in some way is either a pure translation or a combination of translation and rotation. The motion of a rigid body which is pivoted or fixed in some way is rotation

Center of mass

The center of mass is a position defined relative to an object or system of objects. It is the average position of all the parts of the system, weighted according to their masses.

For simple rigid objects with uniform density, the center of mass is located at the centroid. For example, the center of mass of a uniform disc shape would be at its center. Sometimes the center of mass doesn't fall anywhere on the object. The center of mass of a ring for example is located at its center, where there isn't any material.

For more complicated shapes, we need a more general mathematical definition of the center of mass.

We shall first see what the center of mass of a system of particles is and then discuss its significance. For simplicity, we shall start with a two-particle system.   Suppose we have two objects of mass m1 and m2, which are located on the x-axis at distances x1 and x2 from the origin. The Center of mass must be somewhere in between the two masses and let's suppose it is at a distance x_cm from the origin.

So position of center of mass x_cm = (m1 x1+ m2 x2 )/(m1+m2) as shown in the figure below.

For masses in two-dimensional plane x-y. Suppose we have ‘n’ masses with mass m1, m2, m3 … up to n  which are at coordinates (x1, y1), (x2,y2), (x3, y3)... And so on.

Then x and y coordinates of the position of the center of mass Xcm and Ycm is given by,

And similarly, we can extend this concept to 3-dimensional discrete distribution of masses to calculate the position of the center of mass (Xcm, Ycm, Zcm) using the masses of discrete mass and their coordinates.

Above three equations can be combined into one equation using the notation of position vectors. Let r_i  be the position vector of i th particle and R be the position vector of the center of mass:

position vector of ith particle   ri= xi  i + yi j + zi k

Position vector of center of mass   R= X i +Y j +Z k

where  R=1/M ∑_(i=1)^n m_(i ) r_i

But most of the objects we have are objects with continuous mass distribution, so we have to do integration in place of summation to get the center of mass of the objects with continuous mass distribution.

We can do the same for the calculation of Ycm and Z cm.

Ycm = 1M0M y dm  ;  Zcm= 1M 0M z dm

Examples of calculation of Center of mass

Example 1: Two masses of 3 kg and 5 kg kept at origin and x= 4 cm respectively along the x-axis. Then find the position of the center of mass.

Example: Calculation of center of mass of a uniform rod.

Consider a uniform rod of length L and total mass M. Then the Center of mass of this will be at the middle of the rod  Xcm=L/2

The motion of the center of mass

In this section, we will learn about the physical significance of the concept of the center of mass. We consider an object of total mass M to be made of up h ‘n’ particles of masses m1, m2, and m3…..  At positions r1, r2, r3,... and so on. If R is the position of the center of mass.

Then   MR= m1 r2 + m2 r2 + m3 r3 .......       (1)

If we differentiate with respect to time both sides 

We have then   M ddt R= m1 ddtr1 + m2ddtr2+m3ddtr3 ......

We know that velocity v= ddtr

Then  MV= m1 v1 + m2 v2 + m3 v3 .......      (2)

where V is the velocity of the center of mass.

Differentiate eq 2  with respect to time again and use ddtv= a.  

MddtV  = m1 ddtv1 + m2ddtv2+m3ddtv3.....        

M A = m1 a1 +m 2 a2 +m3 a3......       (3)

where A is the acceleration of the center of mass.

Now using Newton’s second law  F= ma,

we can write MA=F1+F2+F3.....; Fext= F1+F2+F3...       (4) 

Thus, the total mass of a system of particles times the acceleration of its center of mass is the vector sum of all the forces acting on the system of particles.

Note in equation 4 when we talk of the force F1 on the first particle, it is not a single force, but the vector sum of all the forces on the first particle and so on for other particles.

 Fext= F1+F2+F3..     where MA= Fext, where Fext represents the sum of all external forces acting on the particles of the system.

The center of mass of a system of particles moves as if all the mass of the system was concentrated at the center of mass and all the external forces were applied at that point.

Instead of treating extended bodies as single particles as we have done in earlier chapters, we can now treat them as systems of particles. We can obtain the translational component of their motion, i.e. the motion of the center of mass of the system, by taking the mass of the whole system to be concentrated at the center of mass and all the external forces on the system to be acting at the center of mass.

The motion of an explosive after the explosion.

Suppose a bomb is thrown and it is following a parabolic path (projectile motion) due to external force gravity on it. Suppose it explodes mid-air and breaks into fragments. These fragments then move in different directions in such a way that their center of mass will remain following the previous parabolic path the projectile would have followed with no explosion.

Linear momentum for a system of particle

We know that the linear momentum of the particle is p=mv.  Newton’s second law for a single particle is given by F=dPdt.

Where F is the force of the particle.

For  ‘n’ no of particle total linear momentum is        P= p1 +p2+p3 +p4....+pn     and each of the momentum is written as      m1 v1 +m2 v2+ m3 v3 + m4 v4.......  up to nth  particle.  we know that the velocity of the center of mass is   V= 1M  i=1n mi vi so we have MV=i=1n mi vi.

Now comparing these equations we get    P= M V   . Therefore we can say that the total linear momentum for a system of particles is equal to the product of the total mass of the object and the velocity of the center of mass. Differentiating the above equation we get ,ddtP= M ddtV= M A   . here dv/dt  is acceleration of center of mass . MA is the external force. So we have ddtP= Fext.   

The above equation is nothing but Newton's second law to a system of particles. If the total external force acting on the system is zero.Fext=0  then adt  P=0 which means that  P= constant.

So whenever the total force acting on the system of particles is equal to zero the total linear momentum P of the system is constant or conserved. This is nothing but the law of conservation of linear momentum of a system of particles.

A fun thing to do

Below is the link of the simulation of the balancing act

Balancing act

Tips:

• Consider the middle of the balance as the origin and left scales as negative and the right scales as positive.
• Using the formula of the center of mass in 1 Dimension, try to balance it by trying to keep the center of mass at origin.

What can you do with the simulation?

• You can put various masses on the balance and try to balance both sides by either changing masses or by changing its positions on the balance.
• Once you put your masses, change the toggle button to remove the supports and see what happens to your balance.

Introduction

In our daily life, we all have noticed that everything that is thrown up will fall back to earth. Going uphill is a lot more tiring than going downhills and raindrops from clouds move towards the earth. There are a lot more such phenomena. An Italian scientist Galileo recognized every object experiences a constant acceleration toward the earth, irrespective of their masses.

The stars, moon and planets have been observed since ancient times. The motion of the moon around the earth and the motion of the earth around the sun were some phenomena that needed to be explained.

In early times it was believed that the earth is the center of the universe and everything revolves around the earth. This was known as the geocentric theory. This prevails for a very long time and at that time there was not much advancement in this subject. Later Galileo and other astronomers found that the Sun is the center and not the earth, and all the planets including the earth move around the Sun. This theory was known as the heliocentric theory. After the establishment of the heliocentric theory, the rapid advancement of various subjects of Science happens. Therefore Galileo is known as the Father of Modern Science.

In this chapter, we will discuss the laws that will explain all the phenomena discussed above.

Kepler’s Law

A nobleman Tycho Brahe from Denmark spent his entire life recording observations for the planets with naked eyes. His compiled data was analyzed later by his assistant Johannes Kepler. He could extract from the data three law’s called Kepler’s Laws.

Three Laws of Kepler

1. Law of the orbit: All Planets move around the sun in an elliptical orbit with the sun at one of its foci.

2. Law of Area: The line that joins any planet to the sun sweeps equal areas in equal intervals of time. This law comes from the observations that planets appear to move slower when they are farther from the sun than when they are nearer.

3. Law of period:  The Square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of the ellipse traced out by the planet.  Mathematically,   T²  α a³.

The graph between the Time period and their semi-major axis is drawn below

This law is also consistent with the conservation of angular momentum.

Since the angular momentum of the planet revolving around the sun at any point of time is conserved.

m v1 r1= m v2 r2    so we have  v1/v2= r2/r1

The universal law of Gravitation

Kepler’s laws were known to Newton and enabled him to make a great scientific leap in proposing his universal law of gravitation.

Statement of Universal law of gravitation:

Every object in the Universe attracts every other object with a force directed along the line of the center for the two objects that is proportional to the product of their masses and inversely proportional to the square of the separation between the two objects.

Where G= universal gravitational constant

The Universal Gravitational Law can explain almost anything, right from how an apple falls from a tree to why the moon revolves around the earth.

• The gravitational force is always attractive. Also the magnitude of gravitational force on mass 1 due to mass 2 is equal to the magnitude of the force on mass 2 due to mass 1.   | F₁₂|= | F₂₁| But their directions are opposite. This is consistent with the third law of the motion.
• This law refers to point masses whereas we deal with extended objects which have finite size, so we use the concept of center of mass here. For example, The force of attraction between a hollow spherical shell of uniform density and a point mass of the shell is just as if the entire mass is concentrated at the center of the shell.
• If we have a collection of point masses the force on any one of them is the vector sum of the gravitational force exerted by the other point masses. This is actually the principle of superposition.

Net force on m1 = force on m1 due to m2 + force on m1 due to m3

• The force of attraction due to a hollow spherical shell of uniform density, on a point mass situated inside it is zero. Qualitatively, we can again understand this result. Various regions of the spherical shell attract the point mass inside it in various directions. These forces cancel each other completely.

Acceleration Due to Gravity

When you throw something in the air, you notice that it goes up and then it comes down toward the earth and finally lands on the ground. The same thing will happen if you drop an object from some height. It will also land up on the ground. This happens because of the gravitational pull of the earth.

Now let's take two objects of mass m1 and m2 (where m1>m2). When we drop these masses from some height you will notice that both the masses hit the ground at the same time irrespective of their masses. So we can conclude from here that the gravitational pull of the earth on various objects near it is independent of the masses of the objects.

The conclusion from the above discussion is that earth pulls every object towards it with a force that is independent of the mass of the object. This pull is due to the gravitational force between the earth and the object.

Let the acceleration due to gravity on the object be ‘g’ , so gravitational force due to earth on an object of mass ‘m’ would be  ‘mg’ which is actually equal to the weight of the object.

Let the mass of the earth is me and the radius of the earth is R, gravitational force between the earth and the object near the earth would be

The value of g near the earth is approximately 9.8 m/s²

From the above-derived formula of acceleration due to gravity ‘g’ is it clear that it is independent of the mass of the object and only depends on the mass of the earth and the distance of the object and the center of the earth.

When the objects are near the earth when (h<<R), then we take the radius of earth ‘R’ as the distance between the earth and the object.

But when we move to a greater height from the earth or at a certain depth inside the earth then the value of g would be different. In the next section, we would see the variation of ‘g’ with height and depth.

Variation of g with height and depth

The value of g varies with altitude and depth from the surface of the earth. We will discuss the derivation of the variation of g with depth and height. After that, we will see from the results that the value of g is maximum at the surface of the earth and decreases with both altitude and depth.

Variation of acceleration due to gravity with depth.

Suppose we have to calculate ‘g’ at the depth‘d’ from the surface of the earth. Suppose M be the mass of the whole earth and M1 be the mass of the inner sphere of the earth below the depth d as shown in the figure.

While calculating the value of g at the depth‘d’ we have to consider the gravitational force due to inner mass M1 and not the mass of the whole earth.

The acceleration due to gravity at depth d is g_d, the mass of the inner sphere is M1 and the distance of it from the center of the earth is (R_e-d) so we can write the value of acceleration due to gravity at depth d as

   ….1

Now simplify the value of M1.

     

Now put the value of M1 in equation 1.

So we have finally the result.

The acceleration due to gravity varies with depth as )

Variation of acceleration due to gravity with height ‘h’

When we want to calculate the value of acceleration due to gravity at an altitude ‘h’ where h≈R Then when we calculate ‘g’ we have to consider the distance (R+h) instead of just R. As now ‘h’ is of the same order as R and hence cannot be neglected.

Acceleration due to gravity at height h g_o is given as

So now we have results,

Acceleration due to gravity varies with height as  

The graph given below concludes the above discussion. It is clear that the value of g is maximum at the surface of the earth and decreases with both altitude and depth.

A fun thing to do: Virtual lab

Below is the link to understand the universal law of gravitation.

Force of gravity lab

What can we do in this virtual lab simulation?

• We can change the masses of the objects
• We can change the distance between them

What can we observe?

• We can see the direction of the gravitational force on each body
• We can also see the magnitude of the gravitational force both in decimal and scientific notations.

Measurement :

In everyday experience, it is necessary to make measurements. Anytime one interacts with the environment around, he/she is making measurements of physical quantities. For this reason, it is important that measurements are available as friendly as possible for specialized and non-specialized people. Measurement is available if the measurement system is accessible and easy to use

Any mechanical quantity can be expressed in terms of three fundamental quantities, mass, length and time.

For example, speed is a length divided by time. Force is mass times acceleration and is, therefore, a mass times a distance divided by the square of a time.

Measurement of Mass

The mass of an object is defined by Newton’s Laws. It is the resistance offered by an object to acceleration. In the SI system, we use kilograms to measure mass. But large quantities of matter like the mass of a mountain or the Earth or stars or the entire Universe is measured indirectly by using Newton’s Law of gravitation or other such equations

What are 3 ways to measure mass?

Mass is the amount of matter in an object. A number of tools exist for measuring mass in different environments. These include balances and scales, measurement transducers, vibrating tube sensors, Newtonian mass measurement devices and the use of gravitational interaction between objects.

Measurement of Time

All our activities depend on time. For example, to know the duration of a journey, to meet a schedule at work, to know whether it's day or night, to know the heartbeat, to know the amount of time taken by the computer to perform an operation, etc. Hence it is very important to measure time.

Two natural periodic events which were used in ancient times to measure time were the occurrence of the full moon and sunrise

• The time from one full moon to the next full moon was a month.
• The time from one sunrise to the next sunrise was called a day.

To measure time is to measure the length of time. We know the following units of measuring time: second, minute, hour, day, week, fortnight, month, year, and century. There are various methods of measuring time in different parts of the world.

These instruments can be anything that exhibits two basic components:

(1) a regular, constant, or repetitive action to mark off equal increments of time.

(2) a means of keeping track of the increments of time and of displaying the result

Standard time

As a standard, the atomic standard of time is now used, which is measured by Cesium or Atomic clock. In a Cesium clock, a second is equal to 9,192,631,770 vibrations of radiation from the transition between two hyperfine levels of cesium-133 atoms.

Measurement of Length

Length is the term used for identifying the size of an object or distance from one point Length is a measure of how long an object is or the distance between two points. It is used for identifying the size of an object or distance from one point to another.

What is used to measure length?

Measuring length means measurement of the length of any object with the help of measuring tools like a ruler, measuring tape, etc.

Measurement of length is defined as the act of measuring the length of objects in some specified units which can be standard or non-standard.

Tools that can be used to measure length include rulers, vernier calipers, micrometer screw gauges, measuring tape and odometers. The most precise tool used to measure length are vernier calipers.

Parallax method: for measuring large distances.

To measure large distances between objects methods such as the echo method, the laser method, sonar method, radar method, triangulation method and Parallax method are used.

Parallax is the effect whereby the position or direction of an object appears to differ when viewed from different positions. When I say parallax it means that we are viewing the same object but from two different positions.

Example:  Let us suppose you have some object. Let's say you have a candle that is quite far from you. You first close your right eye and view the candle with your left eye.  Then you open your right eye and view the candle by closing your left eye and trying to view the same candle which is located at the same position.

You will find that there is a change in position. This is because you observed the same object from two different positions. So, the distance between these two different observation points is known as the basis and this phenomenon is known as parallax.

Measuring the distance of distant stars or planets with the parallax method.

Distance of Planet From Earth

We will see another example here now. Let us suppose we have to find the distance of a distant planet from Earth. We observe the planet from Earth from two different observation points. Let us say we take an observation point as A (taken during June), and the other observation point as B(taken during December). We observe the planet from these two points.

Since the object is very far off, this object can be treated as a small point and the distance between points A and B are the basis. The basis, in this case, is nothing but the diameter of the earth. If we know the angle, that is the angle subtended at this planet from both the observation points then we can find out the value of D.

tan P= R/d    here R=mean distance of the earth from sun= 1 A.U.

P is parallax angle  ; Distance d=1 A.U / tan P 

for small P    , tan P≈P  so  d= R/P = 1 A.U. /P

Accuracy and Precision of measuring instruments.

Accuracy is the degree of closeness to the true value. Precision is the degree to which an instrument or process will repeat the same value. In other words, accuracy is the degree of veracity while precision is the degree of reproducibility.

Precision is defined as the closeness between two or more measured values to each other. Suppose you weigh the same box five times and get close results like 3.1, 3.2, 3.22, 3.4, and 3.0 then your measurements are precise. Remember: Accuracy and Precision are two independent terms.

Accuracy refers to how closely the measured value of a quantity corresponds to its “true” value. Precision expresses the degree of reproducibility or agreement between repeated measurements. The more measurements you make and the better the precision, the smaller the error will be.

The top left image shows the target hit at high precision and accuracy. The top right image shows the target hit at a high accuracy but low precision. The bottom left image shows the target hit at a high precision but low accuracy. The bottom right image shows the target hit at low accuracy and low precision.

More Examples

• If the weather temperature reads 28 °C outside and it is 28 °C outside, then the measurement is said to be accurate. If the thermometer continuously registers the same temperature for several days, the measurement is also precise.
• If you take the measurement of the mass of a body of 20 kg and you get 17.4,17,17.3 and 17.1, your weighing scale is precise but not very accurate. If your scale gives you values of 19.8, 20.5, 21.0, and 19.6, it is more accurate than the first balance but not very precise.

 Some frequently asked questions related to Accuracy and precisions

1. Why are accuracy and precision important in measurement?

In order to get the most reliable results in a scientific inquiry, it is important to minimize bias and error, as well as to be precise and accurate in the collection of data. Both accuracy and precision have to do with how close a measurement is to its actual or true value.

2. How do you measure instrument accuracy?

The accuracy formula provides accuracy as a difference of error rate from 100%. To find accuracy we first need to calculate the error rate. And the error rate is the percentage value of the difference between the observed and the actual value, divided by the actual value

3. What is better accuracy or precision?

Precision is how close measure values are to each other, basically how many decimal places are at the end of a given measurement. Precision does matter. Accuracy is how close a measured value is to the true value. Accuracy matters too, but it's best when measurements are both precise and accurate.

4. Which of the following is the most accurate instrument for measuring length?

Screw gauge of least counts 0. 001 cm is the most precise instrument for measuring length as a device with a minimum least count is more suitable for measuring length.

5. What is the importance of accuracy?

Accuracy is to ensure that the information is correct and without any mistakes. Information accuracy is important because the lives of people depend on it like the medical information at the hospitals, so the information must be accurate.

6. What is the importance of precision?

Precision in scientific investigations is important in order to ensure we are getting the correct results. Since we typically use models or samples to represent something much bigger, small errors may be magnified into large errors during the experiment.

7. How can you improve accuracy?

 The accuracy can be improved through the experimental method if every single measurement is made more accurate, e.g. through the choice of equipment. Implementing a method that reduces systematic errors will improve accuracy.

8. How do accuracy and precision relate to significant figures?

Accuracy refers to how closely a measured value agrees with the correct value. Precision refers to how closely individual measurements agree with each other. In any measurement, the number of significant figures is critical.

9. What are the three ways to ensure that you are making an accurate and precise measurement?

What three steps can you take to ensure that your measurements are both accurate and precise? First, use a high-quality measurement tool. Next, measure carefully. Finally, repeat the measurement a few times.

Error in measurements

Sometimes we fail to know the exact measurement and the values vary giving rise to errors. In this article, let us learn about measurement, errors in measurement, types of errors and how to avoid the errors.

An error may be defined as the difference between the measured value and the actual value. For example, if the two operators use the same device or instrument for measurement. It is not necessary that both operators get similar results. The difference that occurs between both the measurements is referred to as an ERROR.

There are three types of errors that are classified on the basis of the source they arise from; They are:

• Gross Errors
• Random Errors
• Systematic Errors

Gross Error

This category basically takes into account human oversight and other mistakes while reading, recording, and readings. For example, the person taking the reading from the meter of the instrument he may read 23 as 28.

Random Error

Random errors are those errors, which occur irregularly and hence are random. These can arise due to random and unpredictable fluctuations in experimental conditions (Example: unpredictable fluctuations in temperature, voltage supply, mechanical vibrations of experimental set-ups, etc

Systematic Error

Systematic errors can be better understood if we divide them into subgroups; They are:

• Environmental Errors

This type of error arises in the measurement due to the effect of external conditions on the measurement. The external condition includes temperature, pressure, and humidity

• Observational Errors

These are the errors that arise due to an individual’s bias, lack of proper setting of the apparatus, or an individual’s carelessness in taking observations. The measurement errors also include wrong readings due to Parallax errors.

• Instrumental Errors

Instrumental Errors: These errors arise due to faulty construction and calibration of the measuring instruments. Such errors arise due to the hysteresis of the equipment or due to friction.

Errors  in Calculation

Different measures of errors include:

Absolute Error

The difference between the measured value of a quantity and its actual value gives the absolute error. It is the variation between the actual values and measured values. It is given by

Absolute error = |VA-VE|

Percent Error

It is another way of expressing the error in measurement. This calculation allows us to gauge how accurate a measured value is with respect to the true value. Percent error is given by the formula

Percentage error (%) = (VA-VE) / VE) x 100

Relative Error

The ratio of the absolute error to the accepted measurement gives the relative error. The relative error is given by the formula:

Relative Error = Absolute error / Actual value

Combination of Error

a) Error of a sum or a difference

When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities.

Z = A + B

We have by addition, Z ± ΔZ = (A ± ΔA) + (B ± ΔB).

The maximum possible error in Z

ΔZ = ΔA + ΔB

For the difference Z = A – B, we have

Z ± Δ Z = (A ± ΔA) – (B ± ΔB) = (A – B) ± ΔA ± ΔB

or, ± ΔZ = ± ΔA ± ΔB

The maximum value of the error ΔZ is again ΔA + ΔB.

(b) Error of a product or a quotient

When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers.

Suppose Z = AB and the measured values of A and B are A ± ΔA and B ± ΔB. Then

Z ± ΔZ = (A ± ΔA) (B ± ΔB) = AB ± B ΔA ± A ΔB ± ΔA ΔB.

Dividing LHS by Z and RHS by AB we have,

1 ± (ΔZ/Z) = 1 ± (ΔA/A) ± (ΔB/B) ± (ΔA/A)(ΔB/B).

Since ΔA and ΔB are small, we shall ignore their product.

Hence the maximum relative error

ΔZ/ Z = (ΔA/A) + (ΔB/B).

(c) Error in case of a measured quantity raised to a power

The relative error in a physical quantity raised to the power k is the k times the relative error in the individual quantity.

Suppose Z = A²,

Then,

ΔZ/Z = (ΔA/A) + (ΔA/A) = 2 (ΔA/A).

Hence, the relative error in A² is two times the error in A.

In general, if Z = (A^p B^q)/C^r

Then,

ΔZ/Z = p (ΔA/A) + q (ΔB/B) + r (ΔC/C).

How To Reduce Errors In Measurement

Keeping an eye on the procedure and following the below-listed points can help to reduce the error.

• Make sure the formulas used for measurement are correct.
• Cross-check the measured value of a quantity for improved accuracy.
• Use the instrument that has the highest precision.
• It is suggested to pilot test measuring instruments for better accuracy.
• Use multiple measures for the same construct.
• Note the measurements under controlled conditions.

Significant figures

Significant figures in the measured value of a physical quantity tell the number of digits in which we have confidence. Larger the number of significant figures obtained in a measurement, the greater the accuracy of the measurement. The reverse is also true.

Important Rules for counting significant figures

• All the non-zero digits are significant.

• All the zeros between two non-zero digits are significant, no matter where the decimal point is, if at all.

• If the number is less than 1, the zero(s) on the right of the decimal point but to the left of the first non-zero digit are not significant.

[In 0.00 2308, the underlined zeroes are not significant].

• The terminal or trailing zero(s) in a number without a decimal point are not significant.

[Thus 123 m = 12300 cm = 123000 mm has three significant figures, the trailing zero(s) being not significant.] However, you can also see the next observation.

• The trailing zero(s) in a number with a decimal point are significant.

[The numbers 3.500 or 0.06900 have four significant figures each.]

• For a number greater than 1, without any decimal, the trailing zero(s) are not significant.

• For a number with a decimal, the trailing zero(s) are significant.

Rules for Arithmetic Operations with Significant Figures

(1) In multiplication or division, the final result should retain as many significant figures as there are in the original number with the least significant figures.

(2) In addition or subtraction, the final result should retain as many decimal places as there are in the number with the least decimal places.

Rounding off the Uncertain Digits

(i) If the digit dropped is less than 5, then the preceding digit is left unchanged.

(ii) If the digit to be dropped is more than 5, then the preceding digit is raised by one.

(iii) If the digit to be dropped is 5 followed by digits other than zero, then the preceding digit is raised by one.

(iv) If the digit to be dropped is 5 or 5 followed by zeroes, then the preceding digit is left unchanged if it is even.

(v) If the digit to be dropped is 5 or 5 followed by zeroes then the preceding digit is raised by one if it is odd.

Selected prefixes used in metric system