The branch of physics which deals with the study of transformation of heat into other forms of energy and vice-versa is called thermodynamics. Thermodynamics is a macroscopic science. It deals with bulk systems and does not go into the molecular constitution of matter.
A collection of an extremely large number of atoms or molecules confined within certain boundaries such that it has a certain values of pressure (P), volume (V) and temperature (T) is called thermodynamic system.

THERMAL EQUILIBRIUM

A thermodynamic system is in an equilibrium state if the macroscopic variables such as pressure, volume, temperature, mass composition etc. that characterize the system do not change in time. In thermal equilibrium, the temperature of the two systems are equal.

ZEROTH LAW OF THERMODYNAMICS

This law identifies thermal equilibrium and introduces temperature as a tool for identifying f equilibrium. According to this law “If two systems are in thermal equilibrium with a third system then those two systems themselves are in equilibrium.”

HEAT, WORK AND INTERNAL ENERGY

Energy that is transferred between a system and its surroundings whenever there is a temperature difference between the system and its surroundings is called heat.
Work is said to be done if a body or a system moves through a certain distance in the direction of the applied force.
It is given as: dW = PdV, where P is the pressure of the gas in the cylinder.

FIRST LAW OF THERMODYNAMICS

The first law of thermodynamics is simply the general law of conservation of energy applied to any system.
According to this law, “the total heat energy change in any system is the sum of the internal energy change and the work done.”
When a certain quantity of heat dQ is subjected to a system, a part of it is used in increasing the internal energy by dU and a part is used in performing external work dW,
Hence- dQ = dU + dW
For gases, the specific heat capacity depends on the process or the conditions under which heat capacity transfer takes place. There are mainly two principal specific heat capacities for a gas. These are specific heat capacity at constant volume and specific heat capacity at constant pressure.
From the First Law of Thermodynamics we find a relation between two principal specific heats of an ideal gas. According to the relation
Cp-Cυ = R
Here Cp and Cυ are molar specific heats under constant pressure and constant volume condition respectively.
The specific heat capacity of a gas at constant pressure is greater than the specific heat capacity of the gas at constant volume i.e. Cp > Cυ.
Reason is that when heat is supplied to a gas at constant volume, no work would be done by the gas against the external pressure and all the energy is used to raise the temperature of the gas.
On the other hand when the heat is supplied to the gas at constant pressure, its volume increases and the heat energy supplied to it is used to increase the temperature of the gas as well as in doing the work against the external pressure.
The difference between the two specific heats is the thermal equivalent of the work done in expanding the gas against the external pressure.

THERMODYNAMIC STATE VARIABLES

Thermodynamic state variables of a system are the parameters which describe equilibrium states of the system.
For example, the equilibrium state of gas is completely specified by the values of pressure, volume, temperature, mass and composition.

EQUATION OF STATE

The equation of state represents the connection between the state variables of a system. For example, the those equation of state of an ideal/perfect gas in represented as
PV = μRT, where g is the number of moles of the gas and R is gas constant for one mole of the gas.
Thermodynamic state variables are of two kinds, extensive and intensive.
Extensive variables indicate the size of the system but intensive variables do not indicate the size.
Volume, mass, internal energy of a system are extensive variables but pressure, temperature and density are intensive variables.

THERMODYNAMIC PROCESSES

Any process in which the thermodynamic variables of a thermodynamic system change is known as thermodynamic process.

QUASI-STATIC PROCESSES

Processes that are sufficiently slow and do not involve accelerated motion of piston and/or large temperature gradient are quasi-static processes.
In this process, the change in pressure or change in volume or change in temperature of the system is very small.

ISOTHERMAL PROCESS

A change in pressure and volume of a gas without any change in its temperature, is called an isothermal change. In such a change, there is a free exchange of heat between the gas and its surroundings.

ISOTHERMAL PROCESS

ADIABATIC PROCESS

A process in which no exchange of heat energy takes place between the gas and the surroundings, is called an adiabatic process.
P-V Diagram
A graph representing the variation of pressure with the variation of volume is called the P-V diagram.
The work done by the thermodynamic system is equal to the area under the P-V diagram. It is given as

REVERSIBLE PROCESS

A process which can retrace so that the system passes through the same states is called a reversible process, otherwise it is irreversible.
Irreversibility arises mainly from two causes:
(i) Many processes like free expansion or an explosive chemical reaction take the system to non-equilibrium states.
(ii) Most processes involve friction, viscosity and other dissipative effects.

SECOND LAW OF THERMODYNAMICS

This principle which disallows certain phenomena consistent with the First law of thermodynamics is known as the second law of thermodynamics.
Following are the two statements of the second law of thermodynamics.

KEVIN-PLANK STATEMENT

It is impossible to construct an engine, operating in a cycle, to extract
heat from a hot body and convert it completely into work without leaving any change anywhere i.e., 100% conversion of heat into work is impossible.

CLAUSIUS STATEMENT

It is impossible for a self acting machine, operating in a cycle, unaided by any external energy to transfer heat from a cold body to a hot body. In other words heat can not flow itself from a colder body to a hotter body.
A heat engine is a device by which a system is made to undergo a cyclic process that results in conversion of heat to work.
Basically, a heat engine consists of:
(i) a heat source,
(ii) a heat sink, and
(iii) a working substance.

CARNOT'S ENGINE

He proposed a hypothetical engine working on a cyclic/reversible process operating between two temperatures.
Its efficiency is independent of the working substance and is given by, η=1-T2/T1 where T1 is the temperature of source and T2 is the temperature of sink.
According to Carnot’s theorem: (a) working between two given temperatures T1 and T2 of the hot and cold reservoirs respectively, no engine can have efficiency more than that of Carnot’s engine, and (b) the efficiency of the Carnot engine is independent of the nature of the working substance.

REFRIGERATOR

The process of removing heat from bodies colder than their surroundings is called refrigeration and the device doing so is called refrigerator.
In the refrigerator, heat is absorbed at low temperature and rejected at higher temperature with the help of external mechanical work. Thus, a refrigerator is a heat engine working backward and hence it is also called a heat pump.
Refrigerator works on the reverse process of the Carnot engine.
By the work done on the system, heat is extracted from low temperature sink T2 and passed on to high temperature source T1.
The coefficient of performance is given by

IMPORTANT TABLES

Heat is the form of energy transferred between two (or more) systems or a system and its surroundings by virtue of temperature difference.
The SI unit of heat energy transferred is expressed in joule (J).
In the CGS system, units of heat are calorie and kilocalorie (kcal).
1 cal = 4.186 J and 1 kcal = 1000 cal = 4186 J.
Temperature of a substance is a physical quantity which measures the degree of hotness or coldness of the substance.
The SI unit of temperature is kelvin (K) and °C is a commonly used unit of temperature.
A branch of science which deals with the measurement of temperature of a substance is known as thermometry.
A device used to measure the temperature of a body is called a thermometer.
A thermometer calibrated for a temperature scale is used to measure the value of given temperature on that scale.
For the measurement of temperature, two fixed reference points are selected.
The two convenient fixed reference points are the ice point and the steam point of water at standard pressure, which are known as freezing point and boiling point of water at standard pressure.
The two familiar temperature scales are the Fahrenheit temperature scale and the Celsius temperature scale.
The ice and steam point have values 32°F and 212°F respectively, on the Fahrenheit scale and 0°C and 100°C on the Celsius scale.
On the Fahrenheit scale, there are 180 equal intervals between two reference points, and on the Celsius scale, there are 100.
If tc and tF are temperature values of a body on Celsius temperature scale and Fahrenheit temperature scale respectively, then the relationship between Fahrenheit and Celsius temperature is given by

An ideal gas obeys the following law. That is PV = gRT, where P,V and T are the pressure, volume and temperature of the gas respectively, g is the number of moles in an ideal gas and R = 8.31 J mol-1 K-1 is known as the universal gas constant.
The equation, PV – gRT is known as the ideal gas equation.
The absolute minimum temperature for an ideal gas, inferred by extrapolating the straight line P – T graph is found to be – 273.15 °C and is designated as absolute zero.
Absolute temperature scale (T) and Celsius scale are related by
t° C = T – 273.15

THERMAL EXPANSION

The increase of size of a body due to the increase in the temperature is called thermal expansion. Three types of expansions can take place in solids viz. linear, superficial and volume expansion,
(i) Linear Expansion: The increase in the length of a solid on heating is called linear expansion. If the temperature of a rod of original length l is raised by a small amount Δt, its length increases by Δl. Then the linear expansion is given by: Δl = l ∞ Δt
where a is the coefficient of linear expansion of the given solid.
(ii) Superficial or Area Expansion: The increase in surface area of the solid on heating is called superficial expansion.
If A0 is the area of a solid at 0°C and A( its area at t°C then At = A0(l + βt) where β is known as the coefficient of superficial expansion. Unit of β is °C-1 or K-1.
(iii) Volume Expansion: The increase in volume of the solid on heating is called volume expansion. The change in the volume of a solid with a change in temperature Δt is given by Δv = Vγ Δt, where y is the coefficient of volume expansion.
The relation among coefficients of linear expansion (α), superficial expansion (β) and volume expansion (γ) is given as

For a given solid, the three coefficients of expansion α , β, γ are not constant. Their values depend on the temperature range.
Liquids have volume expansion only. If we do not take into account the expansion of solid containers, then the expansion of liquid is called apparent expansion.
On the other hand, if we take into account the expansion of solid too, it is referred to as the real expansion of liquid. It is found that γr = γa + γg, where γr= real expansion coefficient of liquid, γa = apparent expansion coefficient of liquid and γg = volume expansion coefficient of container vessel (glass).
Water exhibits an anomalous behaviour. It contracts on heating between 0 °C and 4 °C but expands on heating beyond 4 °C. Thus, specific volume of water is minimum at 4 °C or density of water is maximum at 4 °C. This property of water has an important environmental effect.

THERMAL STRESS

When a rod is held between two fixed supports and its temperature is increased, the fixed supports do not allow the rod to expand, which results in a stress which is called thermal stress. Thermal stress in the rod is given by

where Y is the Young’s modulus for the material of the rod, A is the area cross-section of the rod, A is the coefficient of linear expansion and F is the developed force in the rod.

THERMAL CAPACITY

The thermal capacity of a body is the quantity of heat required to raise the temperature of the whole of the body through a unit degree. It is measured in calorie per °C or joule per K. Dimensional formula of heat capacity is [ML2T -2K-1 ]. If Q be the amount of heat needed to produce a change in temperature (Δt) of the substance, then thermal capacity of the substance is given by

SPECIFIC HEAT CAPACITY

The specific heat capacity (also referred to as specific heat) of a substance is the amount of heat required to raise the temperature of a unit mass of substance through 1 °C. It is measured in cal g-1(°C)-1 or J kg-1 K-1.
The specific heat capacity of a substance is given by:

where m is mass of substance and Q is the heat required to change its temperature Δt.
Molar specific heat capacity of a substance is defined as the amount of heat required to raise the temperature of 1 mole of the substance by 1°C.
It is given by

The unit of molar specific heat capacity is J mol-1 K-1 in SI system and Cal mol-1 °C-1 in CGS system. The dimensional formula of molar specific heat capacity is [ML2T-2 K-1 mole-1].

CALORIMETRY

Calorimetry is concerned with the measurement of heat, the basic apparatus for this purpose being called the calorimeter.
When two bodies at different temperatures are ‘mixed’, heat ‘flows’ from the body at a higher temperature to the one at a lower temperature, until a common ‘equilibrium’ temperature is reached.
Assuming this ‘heat exchange’ to be confined to the two bodies alone (i.e, neglecting any heat loss to the surroundings) we have, from the law of energy conservation:
Heat gained by one body = heat lost by the other.
Transition of matter from one state (solid, liquid and gas) to another is called a change of state.
The change of state from solid to liquid is called melting and from liquid to solid is called fusion.
It is observed that the temperature remains constant until the entire amount of the solid substance melts i.e., both the solid and liquid states of the substance co-exist in thermal equilibrium during the change of state from solid to liquid.
The temperature at which a solid melts is called its melting point. The melting point of a solid is characteristic of the substance and depends on pressure also.
Melting of ice under increased pressure and refreezing on reducing the pressure is called regelation.
The change of state from liquid to vapour (or gas) is called vaporisation.
The temperature at which the liquid and vapour states of a substance co-exist is called its boiling point.
The change from solid state to vapour state without passing through the liquid state is called sublimation.

THE BASIC HEAT FORMULA

The heat Q required to raise the temperature of a mass m of a substance of specific heat capacity s through t degrees is given by i.e., Heat required = mass x specific heat x change in temperature.

LATENT HEAT

Latent heat of a substance is the amount of heat energy required to change the state of unit mass of the substance from solid to liquid or from liquid to gas/vapour without any change in temperature.
The latent heat of fusion (Lf) is the heat per unit mass required to change a substance from solid into liquid at the same temperature and pressure. The latent heat of evaporation (Lv) is the heat per unit mass required to change a substance from liquid to vapour state without change in temperature and pressure.

HEAT TRANSFER

Heat can be transferred from one place to another by three different methods, namely, conduction, convection and radiation.
Conduction usually takes place in solids, convection in liquids and gases, and no medium is required for radiation.

(i) Conduction

According to Maxwell, conduction is the flow of heat through an unequally heated body from places of higher temperature to those of lower temperature. Rate of heat transfer is given by

where K is called Thermal Conductivity and A is the area of cross-section.

(ii) Convection

Maxwell defines convection as the flow of heat by the motion of the hot body itself carrying its heat with it.

(iIi) Radiation

Radiation is the mode of heat transfer in which heat travels directly from one place to another without the agency of any intervening medium.
Thermal conductivity is defined as heat energy transferred in unit time from unit area having a unit difference in temperature over unit length. It is expressed in Js-1 m-1 °C-1 or W-1 K-1

THERMAL RESISTANCE

The thermal resistance of a body is a measure of its opposition to the flow of heat through it. It is defined as

NEWTON'S LAW OF COOLING

Newton’s law of cooling states that the rate of loss of heat of a body is directly proportional to the difference in temperature of the body and the surroundings, provided the difference in temperature is small, not more than 40 °C.

– ve sign implies that as time passes, temperature T decreases. When an object at temperature T1 is placed in a surrounding of temperature T2 the net energy radiated per second is,

BLACK BODY RADIATION

(i) Emissive Power:

The amount of heat energy rediated per unit area of the surface of a body, per unit time and per unit wavelength range is constant which is called as the ’emissive power’ (eλ) of the given surface, given temperature and wavelength. It’s S.I. The unit is Js-1 m-2 .

(ii) Absorptive Power:

When any radiation is incident over a surface of a body, a part of it gets reflected, a part of it gets refracted and the rest of it is absorbed by that surface. Therefore, the ‘absorptive power’ of a surface at a given temperature and for a given wavelength is the ratio of the heat energy absorbed by a surface to the total energy incident on it at a certain time. It is represented by (aλ). It has no unit as it is a ratio.

(iii) Perfect Black Body:

A body is said to be a perfect black body if its absorptivity is 1. It neither reflects or transmits but absorbs all the thermal radiations incident on it irrespetive of their wavelengths.

(iv) Wein’s Displacement Law :

This law states that as the temperature increases, the maximum value of the radiant energy emitted by the black body, moves towards shorter wavelengths.
Wein found that “The product of the peak wavelength ( λm) and the Kelvin temperature (T) of the black body should remain constant.”
λm x T= b , where b is constant known as Wein’s constant. Its value is 2.898 x 10-3 mk.

(v) Stefan’s Law :

This law states that the thermal radiation energy emitted per second from the surface of a black body is directly proportional to its surface area A and to the fourth power of its absolute temperature T.
Emission coefficient or degree of blackness of a body is represented by a dimensionless quantity ε, 0 < ε < 1. If ε = 1 then the body is perfectly black body. Hence,

(vi) The Solar Constant:

The average energy emitted from the surface of the sun, absorbed per unit area, per minute by the earth is constant which is called the solar constant which is represented by S whose value is 8.135 jm-2 min-1.

Fluids are the sustances which can flow e.g., liquids and gases. It does not possess a definite shape.
When an object is submerged in a liquid at rest, the fluid exerts a force on its surface normally. It is called the thrust of the liquid.

PRESSURE

The thrust experienced per unit area of the surface of a liquid at rest is called pressure.
When a liquid is in equilibrium, the force acting on its surface is perpendicular everywhere. The pressure is the same at the same horizontal level.
The pressure at any point in the liquid depends on the depth (h) below the surface, density of liquid and acceleration due to gravity.

PASCAL'S LAW

According to Pascal’s Law, the pressure applied to an enclosed liquid is transmitted undiminished to every portion of the liquid and the walls of the containing vessel.
Hydraulic system works on Pascal’s law. Force exerted to area ratio will be same at all crosssections.

Note: A large force is experienced in larger cross-section it a smaller force 4cross is applied in smaller by the relation section.
A column of height h of a liquid of density p exerts a pressure P given

If Pa be the atmospheric pressure then pressure in a liquid at a depth h from its free surface is given by P = Pa+ hρg. Relation is true for incompressible fluids only.
The gauge pressure (Pg), is the difference of the absolute pressure (P) and the atmospheric pressure (Pa).
Absolute pressure (P) = Gauge pressure (Pg) + Atmospheric pressure (Pa)
Pg=P-Pa

ARCHIMEDES PRINCIPLE

When a body is partially or completely immersed in a liquid, it loses some of its weight. The loss in weight of the body in the liquid is equal to the weight of the liquid displaced by the immersed
part of the body.
The upward force excerted by the liquid displaced when a body is immersed is called buoyancy. Due to this, there is apparent loss in the weight experienced by the body.

LAW OF FLOATATION

“A body floats in a liquid if the weight of the liquid displaced by the immersed portion of the body is equal to the weight of the body.”
When a body is immersed partially or wholly in a liquid, then the various forces acting on the body are (i) upward thrust (T) acting at the centre of buoyancy and whose magnitude is equal to the weight of the liquid displaced and (ii) the weight of the body (W) which acts vertically downward through its centre of gravity.
(a) When W > T, the body will sink in the liquid;
(b) When W = T, then the body will remain in equilibrium inside the liquid;
(c) When W < T, then the body will come upto the surface of the liquid in such a way that the weight of the liquid displaced due to its immersed portion equals the weight of the body. Thus the body will float with only a part of it immersed inside the liquid.
The flow of a liquid is said to be steady or stream line flow if such particle of the fluid passing through a given point travels along the same path and with same speed as the preceding particle passing through that very point.
If the liquid flows over a horizontal surface in the form of layers of different velocities, then the flow of the liquid is called laminar flow.
The flow of fluid in which velocity of all particles crossing a given point is not same and the motion of fluid becomes irregular or disordered is called turbulent flow.

LAW OF FLOATATION

According to the equation of continuity, if there is no fluid source or sink along the length of a pipe, then the mass of the fluid crossing any section of the pipe per unit time remains constant. i.e„ a1 v1ρ1 = a2v2 ρ2
For incompressible liquids (i.e., fluids) ρ1 = ρ2 and hence the equation is given as : a1v1=a2v2
It means that speed of flow of liquid is more where the pipe is narrower and speed of flow is less where the cross-section of the pipe is more.

ENERGY OF A LIQUID

A liquid can possess three types of energies:
kinetic energy, potential energy and pressure energy
The energy possessed by a liquid due to its motion is called kinetic energy i.e., 1/2mv2.
The potential energy of a liquid of mass m at a height h is given by P.E. = mgh
The energy possessed by a liquid by virtue of its pressure is called pressure energy.
Pressure energy of liquid in volume dV = PdV
Pressure energy per unit mass of the liquid:

BERNOULLI'S THEOREM

For an incompressible, non-viscous, irrotational liquid having streamlined flow, the sum of the pressure energy, kinetic energy and potential energy per unit mass is a constant i.e.,

For steady flow of a non-viscous fluid along a horizontal pipe, Bernoulli’s equation is simplified as

VISCOSITY

Viscosity is the property of the fluid (liquid or gas) by virtue of which an internal frictional force comes into play when the fluid is in motion in the form of layers having relative motion. It opposes the relative motion of the different layers.
Viscosity is also called fluid friction.
The viscous force directly depends on the area of the layer and the velocity gradient.

COEFFICIENT OF VISCOSITY

Coefficient of viscosity of a liquid is equal to the tangential force required to maintain a unit velocity gradient between two parallel layers of liquid each of area unity.

The SI unit of coefficient of viscosity is poiseuille (Pl) or Pa – s or Nm-2 s or kg m-1 s-1 . Dimensional formula of q is [ML-1T-1].

STOKE'S LAW

According to Stokes’ law the backward dragging force acting on a small spherical body of radius r moving with a velocity v through a viscous medius of coefficient of viscosity ή is given by
F = 6πήr

TERMINAL VELOCITY

It is the maximum constant velocity acquired by the body while falling freely in a viscous medium. This is attained when the apparent weight is compensated by the viscous force. It is given by:

where p be the density of the material of the body of radius r and o be the density of the medium.

POISEUILLE'S EQUATION

According to Poiseuille, if a pressure difference (P) is maintained across the two ends of a capillary tube of length ‘l’ and radius ‘r’, then the volume of liquid coming out of the tube per second is
directly proportional to the pressure difference (P).
directly proportional to the fourth power of radius (r) of the capillary tube.
inversely proportional to the coefficient of viscosity (ή) of the liquid.
inversely proportional to the length (i) of the capillary tube. It is given as

REYNOLD'S NUMBER

The Reynold number Re is a dimensionless number whose value gives an approximate idea whether the flow of a fluid will be streamline or turbulent. It is given by

where p = density of fluid flowing with a speed u, d stands for the diameter of the pipe and q is the viscosity of the fluid. Value of Re remains the same in any system of units.
It is observed that flow is streamline or laminar for Re <= 1000 and the flow is turbulent for Re >= 2000. The flow becomes unsteady for Re between 1000 and 2000. The critical value of Re , at which turbulence sets, is same for the geometrically similar flows.
Re may also be expressed as the ratio of inertial force (force due to inertia i.e., mass of moving fluid or due to inertia of obstacle in its path) to viscous force i.e.,

CRITICAL VELOCITY

The critical velocity is that velocity of liquid flow, upto which its flow is streamline and above which its flow becomes turbulent. It is given by

where K is a dimensionless constant, q is the coefficient of viscosity of the liquid, p is the density of the liquid and r is the radius of tube.

SURFACE TENSION

It is the property of the liquid by virtue of which the free surface of the liquid at rest tends to have minimum area and as such it behaves as a stretched elastic membrane.
The force acting per unit length of line drawn on the liquid surface and normal to it parallel to the surface is called the force of surface tension.
It is given by. The SI unit of surface tension is Nm-1 and its dimensional formula is [MT-2],

SURFACE ENERGY

Energy possessed by the surface of the liquid is called surface energy. Change in surface energy is the product of surface tension and change in surface area under constant temperature.
The height to which water rises in a capillary tube of radius r is given by

where T is the surface tension of the liquid and 0 is the angle of contact.
Due to surface tension there is excess pressure on the concave side of a surface film of a liquid over the convex side and is equal to 2T/r . For a soap bubble the excess pressure is 4T/r where r denotes the radius of the surface.

ANGLE OF CONTACT

The angle which the tangent to the liquid surface at the point of contact makes with the solid surface inside the liquid is called angle of contact.
Intermolecular force amongst molecules of the same material is called the force of cohesion. However, force amongst molecules of different materials is called the force of adhesion.

TORRICELLI'S THEOREM

According to this theorem, velocity of efflux i.e., the velocity with which the liquid flows out of an orifice {i.e., a narrow hole) is equal to that which a freely falling body would acquire in falling through a vertical distance equal to the depth of orifice below the free surface of liquid. The velocity is given by: V = √2gh

MAGNUS EFFECT

When a ball is given a spin when it is in a streamline of air molecules, it will follow a curved path which is convex towards the greater pressure side. This idea is the basis of the ball from spin bowlers getting a lift and areodynamics.

INTERMOLECULAR FORCE

In a solid, atoms and molecules are arranged in such a way that each molecule is acted upon by the forces due to the neighbouring molecules. These forces are known as intermolecular forces.

ELASTICITY

The property of the body to regain its original configuration (length, volume or shape) when the deforming forces are removed, is called elasticity.
The change in the shape or size of a body when external forces act on it is determined by the forces between its atoms or molecules. These short range atomic forces are called elastic forces.

PERFECTLY ELASTIC BODY

A body which regains its original configuration immediately and completely after the removal of deforming force from it, is called a perfectly elastic body. Quartz and phospher bronze are examples of nearly perfectly elastic bodies.

PLASTICITY

The inability of a body to return to its original size and shape even on removal of the deforming force is called plasticity and such a body is called a plastic body.

STRESS

Stress is defined as the ratio of the internal force F, produced when the substance is deformed, to the area A over which this force acts. In equilibrium, this force is equal in magnitude to the externally applied force.

TYPES OF STRESS

(i) Normal stress: It is defined as the restoring force per unit area perpendicular to the surface of the body. Normal stress is of two types: tensile stress and compressive stress.
(ii) Tangential stress: When the elastic restoring force or deforming force acts parallel to the surface area, the stress is called tangential stress.

STRAIN

It is defined as the ratio of the change in size or shape to the original size or shape. It has no dimensions, it is just a number.

TYPES OF STRAIN

(i) Longitudinal strain:

If the deforming force produces a change in length alone, the strain produced in the body is called longitudinal strain or tensile strain. It is given as:

(ii) Volumetric strain:

If the deforming force produces a change in volume alone, the strain produced in the body is called volumetric strain. It is given as:

(iii) Shear strain:

The maximum stress to which the body can regain its original status on the removal of the deforming force is called elastic limit.

HOOKE'S LAW

Hooke’s law states that, within elastic limits, the ratio of stress to the corresponding strain produced is a constant. This constant is called the modulus of elasticity. Thus

STRESS STRAIN CURVE

Stress strain curves are useful to understand the tensile strength of a given material.

YOUNG'S MODULUS

For a solid, in the form of a wire or a thin rod, Young’s modulus of elasticity within elastic limit is defined as the ratio of longitudinal stress to longitudinal strain. It is given as:

BULK MODULUS

Within the elastic limit the bulk modulus is defined as the ratio of longitudinal stress and volumetric strain.
It is given as:

– ve indicates that the volume variation and pressure variation always negate each other.
Reciprocal of bulk modulus is commonly referred to as the “compressibility”.
It is defined as the fractional change in volume per unit change in pressure.

SHEAR MODULUS OR MODULUS OF RIGIDITY

It is defined as the ratio of the tangential stress to the shear strain. Modulus of rigidity is given by

POISSON'S RATIO

The ratio of change in diameter (ΔD) to the original diameter (D) is called lateral strain.
The ratio of change in length (Δl) to the original length (l) is called longitudinal strain.
The ratio of lateral strain to the longitudinal strain is called Poisson’s ratio.

ELASTIC FATIGUE

It is the property of an elastic body by virtue of which its behaviour becomes less elastic under the action of repeated alternating deforming forces.

RELATIONS BETWEEN ELASTIC MODULI

For isotropic materials (i.e., materials having the same properties in all directions), only two of the three elastic constants are independent.
For example, Young’s modulus can be expressed in terms of the bulk and shear moduli.

BREAKING STRESS

The ultimate tensile strength of a material is the stress required to break a wire or a rod by pulling on it. The breaking stress of the material is the maximum stress which a material can withstand. Beyond this point breakage occurs.

IMPORTANT TABLES

KEPLER'S LAW OF PLANETARY MOTION

Johannes Kepler formulated three laws which describe planetary motion. They are as follows:
(i) Law of orbits. Each planet revolves around the sun in an elliptical orbit with the sun at one of the foci of the ellipse.
(ii) Law of areas. The speed of the planet varies in such a way that the radius vector drawn from the sun to the planet sweeps out equal areas in equal times.

NEWTON'S LAW OF GRAVITATION

Newton’s law of gravitation states that every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
The direction of the force is along the line joining the particles.
Universal constant of gravitation G is numerically equal to the force of attraction between two particles of unit mass each separated by unit distance.

IMPORTANT CHARACTERISTICS OF GRAVITATIONAL FORCE

(i) Gravitational force between two bodies is a central force i.e., it acts along the line joining the centres of the two interacting bodies.
(ii) Gravitational force between two bodies is independent of the nature of the intervening medium.
(iii) Gravitational force between two bodies does not depend upon the presence of other bodies.
(iv) It is valid for point objects and spherically symmetrical objects.
(v) Magnitude of force is extremely small.

PRINCIPLE OF SUPERPOSITION OF GRAVITATION

ACCELERATION DUE TO GRAVITY

The acceleration produced in a body on account of the force of gravity is known as acceleration due to gravity.
It is usually denoted by ‘g’. It is always towards the centre of Earth.
If a body of mass ‘m’ lying on the surface of the earth, the gravitational force acting on the body is given by

MASS AND MEAN DENSITY OF EARTH

Mass and Mean density of Earth is given in the following manner.

VARIATION OF ACCELERATION DUE TO GRAVITY

The value of acceleration due to gravity changes with height (i.e., altitude), depth, shape of the earth and rotation of earth about its own axis.
(a) Effect of Altitude. As one goes above the surface of Earth, the value of acceleration due to gravity gradually decreases. If gh be the value of acceleration due to gravity at a height h from the surface of Earth, then:

GRAVITATIONAL FIELD

The space around a body within which its gravitational force of attraction is experienced by other bodies is called a gravitational field.

INTENSITY OF GRAVITATIONAL FIELD

The intensity of the gravitational field of a body at a point in the field is defined as the force experienced by a body of unit mass placed at that point provided the presence of unit mass does not disturb the original gravitational field.

GRAVITATIONAL POTENTIAL

The gravitational potential at a point in the gravitational field of a body is defined as the amount of work done in bringing a body of unit mass from infinity to that point.
Gravitational potential at a point situated at a distance r from a body or particle of mass M is given by

GRAVITATIONAL POTENTIAL ENERGY

The work done in carrying a mass ‘m’ from infinity to a point at distance r is called gravitational potential energy. The gravitational potential energy of the system is given by

i.e., Gravitational potential energy = gravitational potential x mass of the body. It is a scalar quantity and measured in joules.

ESCAPE VELOCITY

The minimum velocity required to project a body vertically upward from the surface of earth so that it comes out of the gravitational field of earth is called escape velocity.

SATELLITE

A satellite is a body which is revolving continuously in an orbit around a comparatively much larger body.
The orbit may be either circular or elliptical. A man-made object revolving in an orbit around a planet is called an artificial satellite.

ORBITAL VELOCITY

Orbital velocity of a satellite is the minimum velocity required to put the satellite into a given orbit around earth.

GEOSTATIONARY SATELLITE

The satellite having the same time period of revolution as that of the earth is called a geostationary satellite. Such satellites should rotate in the equatorial plane from west to east.
The orbit of a geostationary satellite is called ‘parking orbit’. These satellites are used for communication purposes.
A geostationary satellite revolves around the earth in a circular orbit at a height of about 36,000 km from the surface of the earth.

IMPORTANT TABLES

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.

CENTRE OF MASS

For a system of particles, the centre of mass is defined as that point where the entire mass of the system is imagined to be concentrated, for consideration of its translational motion.
If all the external forces acting on the body/system of bodies were to be applied at the centre of mass, the state of rest/ motion of the body/system of bodies shall remain unaffected.
The centre of mass of a body or a system is its balancing point. The centre of mass of a two- particle system always lies on the line joining the two particles and is somewhere in between the particles.

MOTION OF CENTRE OF MASS

The centre of mass of a system of particles moves as if the entire mass of the system were concentrated at the centre of mass and all the external forces were applied at that point.
Velocity of centre of mass of a system of two particles, m1 and m2 with velocity v1 and v2 is given by,

If no external force acts on the body, then the centre of mass will have constant momentum. Its velocity is constant and acceleration is zero, i.eMVcm = constant.
Vector Product or Cross Product of two vectors

TORQUE

Torque is the moment of force. Torque acting on a particle is defined as the product of the magnitude of the force acting on the particle and the perpendicular distance of the application of force from the axis of rotation of the particle.

ANGULAR MOMENTUM

The angular momentum (or moment of momentum) about an axis of rotation is a vector quantity, whose magnitude is equal to the product of the magnitude of momentum and the perpendicular distance of the line of action of momentum from the axis of rotation and its direction is perpendicular to the plane containing the momentum and the perpendicular distance.

AXIS OF ROTATION

A rigid body is said to be rotating if every point mass that makes it up, describes a circular path of a different radius but the same angular speed. The circular paths of all the point masses have a common centre. A line passing through this common centre is the axis of rotation.
A rigid body is said to be in equilibrium if under the action of forces/torques, the body remains in its position of rest or of uniform motion.
For translational equilibrium, the vector sum of all the forces acting on a body must be zero. For rotational equilibrium, the vector sum of torques of all the forces acting on that body about the reference point must be zero. For complete equilibrium, both these conditions must be fulfilled.

COUPLE

Two equal and opposite forces acting on a body but having different lines of action, form a couple. The net force due to a couple is zero, but they exert a torque and produce rotational motion.

MOMENT OF INERTIA

The rotational inertia of a rigid body is referred to as its moment of inertia.
The moment of inertia of a body about an axis is defined as the sum of the products of the masses of the particles constituting the body and the square of their respective perpendicular distance from the axis.
It is given by,

RADIUS OF GYRATION

The distance of a point in a body from the axis of rotation, at which if whole of the mass of the body were supposed to be concentrated, its moment of inertia about the axis of rotation would be the same as that determined by the actual distribution of mass of the body is called radius of gyration.
If we consider that the whole mass of the body is concentrated at a distance K from the axis of rotation, then moment of inertia I can be expressed as I = MK2

THEOREM OF PARALLEL AXIS

According to this theorem, the moment of inertia I of a body about any axis is equal to its moment of inertia about a parallel axis through centre of mass, Icm, plus Ma2 where M is the mass of the body and V is the perpendicular distance between the axes, i.e., I = Icm + Ma2

THEOREM OF PERPENDICULAR AXIS

According to this theorem, the moment of inertia (I) of the body about a perpendicular axis is equal to the sum of moments of inertia of the body about two axes at right angles to each other in the plane of the body and intersecting at a point where the perpendicular axis passes, i.e.,

ROLLING MOTION

The combination of rotational motion and the translational motion of a rigid body is known as rolling motion.

LAW OF CONSERVATION OF ANGULAR MOMENTUM

According to the law of conservation of angular momentum, if there is no external couple acting, the total angular momentum of a rigid body or a system of particles is conserved.

IMPORTANT TABLES

Work is said to be done when a force applied on the body displaces the body through a certain distance in the direction of applied force. It is measured by the product of the force and the distance moved in the direction of the force, i.e., W = F-S

ENERGY

The energy of a body is its capacity to do work. Anything which is able to do work is said to possess energy. Energy is measured in the same unit as that of work, namely, Joule. Mechanical energy is of two types: Kinetic energy and Potential energy.

KINETIC ENERGY

The energy possessed by a body by virtue of its motion is known as its kinetic energy.
For an object of mass m and having a velocity v, the kinetic energy is given by:
K.E. or K = 1/2 mv 2

POTENTIAL ENERGY

The energy possessed by a body by virtue of its position or condition is known as its potential energy.
There are two common forms of potential energy: gravitational and elastic.
Gravitational potential energy of a body is the energy possessed by the body by virtue of its position above the surface of the earth. It is given by
(U)P.E. = mgh where, m —> mass of a body g —> acceleration due to gravity on the surface of earth. h —> height through which the body is raised.
When an elastic body is displaced from its equilibrium position, work is needed to be done against the restoring elastic force. The work done is stored up in the body in the form of its elastic potential energy.
If an elastic spring is stretched (or compressed) by a distance Y from its equilibrium position, then its elastic potential energy is given by
U= 1/2 kx2 where, k —> force constant of given spring

WORK-ENERGY THEOREM

According to the work-energy theorem, the work done by a force on a body is equal to the change in kinetic energy of the body.

THE LAW OF CONSERVATION OF ENERGY

According to the law of conservation of energy, the total energy of an isolated system does not change.

Energy may be transformed from one form to another but the total energy of an isolated system remains constant.
Energy can neither be created, nor destroyed.
Besides mechanical energy, the energy may manifest itself in many other forms.
Some of these forms are: thermal energy, electrical energy, chemical energy, visual light energy, nuclear energy etc.

EQUIVALENCE OF MASS AND ENERGY

According to Einstein, mass and energy are inter-convertible. That is, mass can be converted into energy and energy can be converted into mass.

COLLISION

Collision is defined as an isolated event in which two or more colliding bodies exert relatively strong forces on each other for a relatively short time
Collision is said to be one dimensional, if the colliding particles, move along the same straight line path both before as well as after the collision.
Collision between particles have been divided broadly into two types: (i) Elastic collisions (ii) Inelastic collisions

(i) Elastic Collision

A collision between two particles or bodies is said to be elastic if both the linear momentum and the kinetic energy of the system remain conserved. Example: Collisions between atomic particles, atoms, marble balls and billiard balls.

(ii) Inelastic Collision

A collision is said to be inelastic if the linear momentum of the system remains conserved but its kinetic energy is not conserved. Example: When we drop a ball of wet putty on to the floor then the collision between ball and floor is an inelastic collision.

Coefficient of Restitution or Coefficient of Resilience

Coefficient of restitution is defined as the ratio of relative velocity of separation after collision to the relative velocity of approach before collision.

NON-CONSERVATIVE FORCES

A force is said to be non-conservative if the work done in moving from one point to another depends upon the path followed.
Examples of non-conservative forces are :
(i) Force of friction (ii) Viscous force
Law of conservation of energy holds goods for both conservative and non-conservative forces.

IMPORTANT TABLES

Dynamics is the branch of physics in which we study the motion of a body by taking into consideration the cause i.e., force which produces the motion.

FORCE

Force is an external cause in the form of push or pull, which produces or tries to produce motion in a body at rest, or stops/tries to stop a moving body or changes/tries to change the direction of motion of the body.
The inherent property, with which a body resists any change in its state of motion, is called inertia. Heavier the body, the more inertia there is and lighter the body, lesser the inertia.
Law of inertia states that a body has the inability to change its state of rest or uniform motion (i.e., a motion with constant velocity) or direction of motion by itself.

NEWTON'S LAWS OF MOTION

Law 1: A body will remain at rest or continue to move with uniform velocity unless an external force is applied to it.
First law of motion is also referred to as the ‘Law of inertia’.
Law 2: When an external force is applied to a body of constant mass the force produces an acceleration, which is directly proportional to the force and inversely proportional to the mass of the body.
Law 3 : “To every action there is an equal and opposite reaction force”. When a body A exerts a force on another body B, B exerts an equal and opposite force on A.

LINEAR MOMENTUM

The linear momentum of a body is defined as the product of the mass of the body and its velocity.

IMPULSE

Forces acting for short duration are called impulsive forces. Impulse is defined as the product of force and the small time interval for which it acts.
It is given by:

Impulse of a force is a vector quantity and its SI unit is 1 Nm.
— If the force of an impulse is changing with time, then the impulse is measured by finding the area bound by force-time graph for that force.
— Impulse of a force for a given time is equal to the total change in momentum of the body during the given time.
Thus, we have

LAW OF CONSERVATION OF MOMENTUM

The total momentum of an isolated system of particles is conserved.
In other words, when no external force is applied to the system, its total momentum remains constant.

Recoiling of a gun, flight of rockets and jet planes are some simple applications of the law of conservation of linear momentum.

CONCURRENT FORCES AND EQUILIBRIUM

“A group of forces which are acting at one point are called concurrent forces.”
Concurrent forces are said to be in equilibrium if there is no change in the position of rest or the state of uniform motion of the body on which these concurrent forces are acting.
For concurrent forces to be in equilibrium, their resultant force must be zero.
In case of three concurrent forces acting in a plane, the body will be in equilibrium if these three forces may be completely represented by three sides of a triangle taken in order.
If the number of concurrent forces is more than three, then these forces must be represented by sides of a closed polygon in order for equilibrium.

COMMONLY USED FORCES

(i) Weight of a body:

It is the force with which earth attracts a body towards its centre. If M is the mass of the body and g is acceleration due to gravity, the weight of the body is Mg in a vertically downward direction.

(ii) Normal Force:

If two bodies are in contact a contact force arises, if the surface is smooth the direction of force is normal to the plane of contact. We call this force the Normal force.
Example. Let us consider a book resting on the table. It is acted upon by its weight in a vertically downward direction and is at rest. It means there is another force acting on the block in the opposite direction, which balances its weight. This force is provided by the table and we call it a normal force.

(iii) Tension in string:

Suppose a block is hanging from a string. Weight of the block is acting vertically downward but it is not moving, hence its weight is balanced by a force due to string. This force is called ‘Tension in string’.
Tension is a force in a stretched string. Its direction is taken along the string and away from the body under consideration.

SIMPLE PULLEY

Consider two bodies of masses m1 and m2 tied at the ends of an inextensible string, which passes over a light and frictionless pulley.
Let m1 > m2. The heavier body will move downwards and the lighter will move upwards. Let a be the common acceleration of the system of two bodies, which is given by

APPARENT WEIGHT AND ACTUAL WEIGHT

‘Apparent weight’ of a body is equal to its ‘actual weight’ if the body is either in a state of rest or in a state of uniform motion.
Apparent weight of a body for vertically upward accelerated motion is given as:
Apparent weight =Actual weight + Ma = M (g + a)
Apparent weight of a body for vertically downward accelerated motion is given as:
Apparent weight = Actual weight Ma = M (g – a).

FRICTION

The opposition to any relative motion between two surfaces in contact is referred to as friction. It arises because of the ‘inter-meshing’ of the surface irregularities of the two surfaces in contact.

STATIC AND DYNAMIC (KINETIC) FRICTION

The frictional forces between two surfaces in contact (i) before and (ii) after a relative motion between them has started, are referred to as static and dynamic friction respectively. Static friction is always a little more than dynamic friction.
The magnitude of kinetic frictional force is also proportional to normal force.

LIMITING FRICTIONAL FORCE

This frictional force acts when the body is about to move. This is the maximum frictional force that can exist at the contact surface. We calculate its value using laws of friction.

Laws of Friction:

(i) The magnitude of limiting frictional force is proportional to the normal force at the contact surface.
(ii) The magnitude of limiting frictional force is independent of the area of contact between the surfaces.

COEFFICIENT OF FORCE

The coefficient of friction (μ) between two surfaces is the ratio of their limiting frictional force to the normal force between them, i.e.,

ANGLE OF FORCE

It is the angle which the resultant of the force of limiting friction F and the normal reaction R makes with the direction of the normal reaction. If θ is the angle of friction, we have

ANGLE OF REPOSE

Angle of repose (α) is the angle of an inclined plane with the horizontal at which a body placed over it just begins to slide down without any acceleration.
Angle of repose is given by α = tan-1 (μ)

MOTION ON A ROUGH INCLINED PLANE

Suppose a motion up the plane takes place under the action of pull P acting parallel to the plane.

CENTRIPETAL FORCE

Centripetal force is the force required to move a body uniformly in a circle. This force acts along the radius and towards the centre of the circle.

CENTRIFUGAL FORCE

Centrifugal force is a force that arises when a body is actually moving along a circular path, by virtue of the tendency of the body to regain its natural straight line path.
The magnitude of centrifugal force is the same as that of centripetal force.

MOTION IN A VERTICAL CIRCLE

The motion of a particle in a horizontal circle is different from the motion in a vertical circle.
In horizontal circle, the motion is not affected by the acceleration due to gravity (g) whereas in the motion of vertical circle, the motion is not affected by the acceleration due to gravity (g) whereas in the motion of vertical circle, the value of ‘g’ plays an important role, the motion in this case does not remain uniform.
When the particle moves up from its lowest position P, its speed continuously decreases till it reaches the highest point of its circular path.
This is due to the work done against the force of gravity.
When the particle moves down the circle, its speed would keep on increasing.

IMPORTANT TABLES

Motion is one of the significant topics in physics.
Everything in the universe moves.
It might only be a small amount of movement and very-very slow, but movement does happen.
Even if you appear to be standing still, the Earth is moving around the sun, and the sun is moving around our galaxy.
“An object is said to be in motion if its position changes with time”.
The concept of motion is a re’ live one and a body that may be in motion relative to one reference system, may be at rest relative to another.
There are two branches in physics that examine the motion of an object.
(i) Kinematics: It describes the motion of objects, without looking at the cause of the motion.
(ii) Dynamics: It relates the motion of objects to the forces which cause them.

POINT OBJECT

If the length covered by the objects are very large in comparison to the size of the objects, the objects are considered point objects.

TOTAL PATH LENGTH (DISTANCE)

For a particle in motion the total length of the actual path traversed between initial and final positions of the particle is known as the ‘total path length’ or distance covered by it.

TYPES OF MOTION

In order to completely describe the motion of an object, we need to specify its position. For this, we need to know the position co-ordinates.
In some cases, three position co-ordinates are required, while in some cases two or one position co-ordinate is required.
Based on these, motion can be classified as:
(i) One dimensional motion. A particle moving along a straight-line or a path is said to undergo one dimensional motion. For example, motion of a train along a straight line, freely falling body under gravity etc.
(ii) Two dimensional motion. A particle moving in a plane is said to undergo two dimensional motion. For example, motion of a shell fired by a gun, carrom board coins etc.
(iii) Three dimensional motion. A particle moving in space is said to undergo three dimensional motion. For example, the motion of a kite in sky, motion of aeroplane etc.

DISPLACEMENT

Displacement of a particle in a given time is defined as the change in the position of a particle in a particular direction during that time. It is given by a vector drawn from its initial position to its final position.

FACTORS DISTINGUISHING DISPLACEMENT FROM DISTANCE

Displacement has direction. Distance does not have direction.
The magnitude of displacement can be both positive and negative.
Distance is always positive. It never decreases with time.

UNIFORM SPEED AND UNIFORM VELOCITY

UNIFORM SPEED

An object is said to move with uniform speed if it covers equal distances in equal intervals of time, however small these intervals of time may be.

UNIFORM VELOCITY

An object is said to move with uniform velocity if it covers equal displacements in equal intervals of time, however small these intervals of time may be.

VARIABLE SPEED AND VARIABLE VELOCITY

VARIABLE SPEED

An object is said to move with variable speed if it covers unequal distances in equal intervals of time, however small these intervals of time may be.

VARIABLE VELOCITY

An object is said to move with variable velocity if it covers unequal displacements in equal intervals of time, however small these intervals of time may be.

AVERAGE SPEED AND AVERAGE VELOCITY

AVERAGE SPEED

It is the ratio of total path length traversed and the corresponding time interval Or The average speed of an object is greater than or equal to the magnitude of the average velocity over a given time interval.

INSTANTANEOUS SPEED AND INSTANTANEOUS VELOCITY

INSTANTANEOUS SPEED

The speed of an object at an instant of time is called instantaneous speed. Or “Instantaneous speed is the limit of the average speed as the time interval becomes infinitesimally small”.

INSTANTANEOUS VELOCITY

The instantaneous velocity of a particle is the velocity at any instant of time or at any point of its path or “Instantaneous velocity or simply velocity is defined as the limit of the average velocity as the time interval Δt becomes infinitesimally small.”

ACCELERATION

The rate at which velocity changes is called acceleration.

UNIFORM ACCELERATION

If an object undergoes equal changes in velocity in equal time intervals it is called uniform acceleration.

AVERAGE ACCELERATION

It is the change in the velocity divided by the time-interval during which the change occurs.

INSTANTANEOUS ACCELERATION

It is defined as the limit of the average acceleration as the time-interval Δt goes to zero.

KINEMATICAL GRAPHS

The ‘displacement-time’ and the ‘velocity-time’ graphs of a particle are often used to provide us with a visual representation of the motion of a particle. The ‘shape’ of the graphs depends on the initial ‘co-ordinates’ and the ‘nature’ of the acceleration of the particle (Fig.)

The following general results are always valid
(i) The slope of the displacement-time graph at any instant gives the speed of the particle at that instant.
(ii) The slope of the velocity-time graph at any instant gives the magnitude of the acceleration of the particle at that instant.
(iii) The area enclosed by the velocity-time graph, the time-axis and the two coordinates at ,time instants t1 to t2 gives the distance moved by the particle in the time-interval from t1 to t2.
Equations of Motion for Uniformly Accelerated Motion
For uniformly accelerated motion, some simple equations can be derived that relate displacement (x), time taken (f), initial velocity (u), final velocity (v) and acceleration (a).
Following equation gives a relation between final and initial velocities v and u of an object moving with uniform acceleration a:
v = u + at

RELATIVE VELOCITY

Relative velocity of an object A with respect to another object B is the time rate at which object A changes its position with respect to object B.
The relative velocity of two objects moving in the same direction is the difference of the speeds of the objects.
The relative velocity of two objects moving in opposite direction is the sum of the speeds of the objects.

IMPORTANT TABLE

MEASUREMENT

The process of measurement is basically a comparison process.
To measure a physical quantity, we have to find out how many times a standard amount of that physical quantity is present in the quantity being measured.
The number thus obtained is known as the magnitude and the standard chosen is called the unit of the physical quantity.

UNIT

The unit of a physical quantity is an arbitrarily chosen standard which is widely accepted by society and in terms of which other quantities of similar nature may be measured.

STANDARD

The actual physical embodiment of the unit of a physical quantity is known as a standard of that physical quantity.
To express any measurement made we need the numerical value (n) and the unit (μ).
Measurement of physical quantity = Numerical value x Unit
For example: Length of a rod = 8 m
where 8 is the numerical value and m (metre) is the unit of length.

FUNDAMENTAL PHYSICAL QUANTITY/UNITS

It is an elementary physical quantity, which does not require any other physical quantity to express it.
It means it cannot be resolved further in terms of any other physical quantity.
It is also known as basic physical quantity.
The units of fundamental physical quantities are called fundamental units.
For example, in the M. K. S. system, Mass, Length and Time expressed in kilogram, metre and second respectively are fundamental units.

DERIVED PHYSICAL QUANTITY/UNITS

All those physical quantities, which can be derived from the combination of two or more fundamental quantities or can be expressed in terms of basic physical quantities, are called derived physical quantities.
The units of all other physical quantities, which car. obtained from fundamental units are called derived units.
For example, units of velocity, density and force are m/s, kg/m3, kg m/s2 respectively and they are examples of derived units.

SYSTEMS OF UNITS

Earlier three different unit systems were used in different countries.
These were CGS, FPS and MKS systems.
Now-a-days internationally SI system of units is followed.
In the SI unit system, seven quantities are taken as the base quantities.
(i) CGS System. Centimetre, Gram and Second are used to express length, mass and time respectively.
(ii) FPS System. Foot, pound and second are used to express length, mass and time respectively.
(iii) MKS System. Length is expressed in meters, mass is expressed in kilograms and time is expressed in seconds. Metre, kilogram and second are used to express length, mass and time respectively.
(iv) SI Units. Length, mass, time, electric current, thermodynamic temperature, Amount of substance and luminous intensity are expressed in metre, kilogram, second, ampere, kelvin, mole and candela respectively.

DEFINITIONS OF FUNDAMENTAL UNITS

SUPPLEMENTARY UNITS

Besides the above mentioned seven units, there are two supplementary base units.
These are :
(i) radian (rad) for angle, and
(ii) steradian (sr) for solid angle.

ADVANTAGES OF SI UNIT SYSTEM

The SI Unit System has following advantages over the other
Besides the above mentioned seven units, there are two supplementary base units. These are systems of units:
(i) It is internationally accepted,
(ii) It is a rational unit system,
(iii) It is a coherent unit system,
(iv) It is a metric system,
(v) It is closely related to CGS and MKS systems of units,
(vi) Uses the decimal system, hence is more user friendly.

OTHER IMPORTANT UNITS OF LENGTH

For measuring large distances e.g., distances of planets and stars etc., some bigger units of length such as ‘astronomical unit’, ‘light year’, parsec’ etc. are used.
The average separation between the Earth and the sun is called one astronomical unit. 1 AU = 1.496 x 1011 m.
The distance travelled by light in vacuum in one year is called the light year. 1 light year = 9.46 x 1015 m.
The distance at which an arc of length of one astronomical unit subtends an angle of one second at a point is called parsec. 1 parsec = 3.08 x 1016 m
Size of a tiny nucleus = 1 fermi = If = 10-15 m
Size of a tiny atom = 1 angstrom = 1A = 10-10 m

PARALLAX METHOD

This method is used to measure the distance of planets and stars from earth. If a distant object e.g., a planet or a star subtends parallax angle 0 on an arc of radius b (known as basis) on Earth, then distance of that distant object from the basis is given by

To estimate the size of atoms we can use electron microscopy and tunneling microscopy techniques.
Rutherford’s a-particle scattering experiment enables us to estimate the size of nuclei of different elements.
Pendulum clocks, mechanical watches (in which vibrations of a balance wheel are used) and quartz watches are commonly used to measure time.
Cesium atomic clocks can be used to measure time with an accuracy of 1 part in 1013 (or to a maximum discrepancy of 3 ps in a year).
The SI unit of mass is kilogram.
While dealing with atoms/ molecules and subatomic particles we define a unit known as “unified atomic mass unit” (1 u), where 1 u = 1.66 x 10-27 kg.

DIMENSIONS

The dimensions of a physical quantity are the powers to which the fundamental units of mass, length and time must be raised to represent the given physical quantity.

DIMENSIONAL FORMULA

The dimensional formula of a physical quantity is an expression telling us how and which of the fundamental quantities enter into the unit of that quantity.
It is customary to express the fundamental quantities by a capital letter, e.g., length (L), mass (AT), time (T), electric current (I), temperature (K) and luminous intensity (C). We write appropriate powers of these capital letters within square brackets to get the dimensional formula of any given physical quantity.

APPLICATIONS OF DIMENSIONS

The concept of dimensions and dimensional formulae are put to the following uses:
(i) Checking the results obtained
(ii) Conversion from one system of units to another
(iii) Deriving relationships between physical quantities
(iv) Scaling and studying of models.
The underlying principle for these uses is the principle of homogeneity of dimensions.
According to this principle, the ‘net’ dimensions of the various physical quantities on both sides of a permissible physical relation must be the same; also only dimensionally similar quantities can be added to or subtracted from each other.

LIMITATIONS OF DIMENSIONAL ANALYSIS

The method of dimensions has the following limitations:
(i) by this method the value of dimensionless constant cannot be calculated.
(ii) by this method the equation containing trigonometric, exponential and logarithmic terms cannot be analyzed.
(iii) if a physical quantity in mechanics depends on more than three factors, then relation among them cannot be established because we can have only three equations by equalizing the powers of M, L and T.
(iv) it doesn’t tell whether the quantity is vector or scalar.

SIGNIFICANT FIGURES

The significant figures are a measure of accuracy of a particular measurement of a physical quantity.
Significant figures in a measurement are those digits in a physical quantity that are known reliably plus the first digit which is uncertain.

THE RULES FOR DETERMINING THE NUMBER OF SIGNIFICANT FIGURES

(i) All non-zero digits are significant.
(ii) All zeros between non-zero digits are significant.
(iii) All zeroes to the right of the last non-zero digit are not significant in numbers without a decimal point.
(iv) All zeroes to the right of a decimal point and to the left of a non-zero digit are not significant.
(v) All zeroes to the right of a decimal point and to the right of a non-zero digit are significant.
(vi) In addition and subtraction, we should retain the least decimal place among the values operated, in the result.
(vii) In multiplication and division, we should express the result with the least number of significant figures as associated with the least precise number in operation.
(viii) If scientific notation is not used:
(a) For a number greater than 1, without any decimal, the trailing zeroes are not significant.
(b) For a number with a decimal, the trailing zeros are significant.

ERROR

The measured value of the physical quantity is usually different from its true value.
The result of every measurement by any measuring instrument is an approximate number, which contains some uncertainty. This uncertainty is called error.
Every calculated quantity, which is based on measured values, also has an error.

CAUSES OF ERRORS IN MEASUREMENT

Following are the causes of errors in measurement:
Least Count Error: The least count error is the error associated with the resolution of the instrument. Least count may not be sufficiently small. The maximum possible error is equal to the least count.
Instrumental Error: This is due to faulty calibration or change in conditions (e.g., thermal expansion of a measuring scale). An instrument may also have a zero error. A correction has to be applied.
Random Error: This is also called chance error. It gives different results for the same measurements taken repeatedly. These errors are assumed to follow the Gaussian law of normal distribution.
Accidental Error: This error gives too high or too low results. Measurements involving this error are not included in calculations.
Systematic Error: The systematic errors are those errors that tend to be in one direction, either positive or negative. Errors due to air buoyancy in weighing and radiation loss in calorimetry are systematic errors. They can be eliminated by manipulation.
Some of the sources of systematic errors are:
(i) intrumental error
(ii) imperfection in experimental technique or procedure
(iii) personal errors