Oscillations And Waves Class 11 Physics Revision Notes

Class 11 Physics students should refer to the following concepts and notes for Oscillations And Waves in standard 11. These exam notes for Grade 11 Physics will be very useful for upcoming class tests and examinations and help you to score good marks

Oscillations And Waves Notes Class 11 Physics

Oscillations and Waves

•Periodic Motion: A motion which repeats itself over and over again after a regular interval of time.

• Oscillatory Motion: A motion in which a body moves back and forth repeatedly about a fixed point.

• Periodic function: A function that repeats its value at regular intervals of its argument is called periodic function. The following sine and cosine functions are periodic with period T.

f(t) =[ sin 2π /T]   and      g(t) = [cos  2π /T]

These are called Harmonic Functions.

Note :- All Harmonic functions are periodic but all periodic functions are not harmonic.

One of the simplest periodic functions is given by

f(t) = A cos ωt       [ω = 2π/T]

If the argument of this function ωt is increased by an integral multiple of 2π radians, the value of the function remains the same. The function f(t) is then periodic and its period, T is given by

T = 2π /ω

Thus the function f(t) is periodic with period T

f(t) = f(t +T)

Linear combination of sine and cosine functions

f(t) = A sin ωt + B cos ωt

A periodic function with same period T is given as

A = D cos ø and B = D sin ø

f(t) = D sin (ωt + ø)

D = √ A² + B² and ø = tan^-1 x /α

• Simple Harmonic Motion (SHM): A particle is said to execute SHM if it moves to and fro about a mean position under the action of a restoring force which is directly proportional to its displacement from mean position and is always directed towards mean position. Restoring Force Displacement

F α x

F= -kx

Where ‘k’ is force constant.

• Amplitude: Maximum displacement of oscillating particle from its mean position.

xMax = + A

•Time Period: Time taken to complete one oscillation.

• Frequency: 1 /r . Unit of frequency is Hertz (Hz).

1 Hz = 1 S^-1

•  Angular Frequency:

= 2π /T = 2πν

S.I unit ω = rad S^-1

• Phase:

1. The Phase of Vibrating particle at any instant gives the state of the particle as regards its position and the direction of motion at that instant.

It is denoted by ø.

2. Initial phase or epoch: The phase of particle corresponding to time t = 0.

It is denoted by ø.

• Displacement in SHM :

  X=A cos( ωt+ ø0)

Where,X= Displacement,

A = Amplitude

ωt = Angular Frequency

ø0 = Initial Phase.

Chapter 13 Kinetic Theory Of Gases Hand written Notes Class 11 Physics

Chapter 13 Kinetic Theory Of Gases Notes Class 11 Physics

Kinetic Theory of Matter:-

(a) Solids:- It is the type of matter which has got fixed shape and volume. The force of attraction between any two molecules of a solid is very large.

(b) Liquids:- It is the type of matter which has got fixed volume but no fixed shape. Force of attraction between any two molecules is not that large as in case od solids.

(c) Gases:- It is the type of matter does not have any fixed shape or any fixed volume.

• Ideal Gas:- A ideal gas is one which has a zero size of molecule and zero force of interaction between its molecules.
• Ideal Gas Equation:- A relation between the pressure, volume and temperature of an ideal gas is called ideal gas equation.

PV/T = Constant  or PV = nRT

Here, n is the number of moles and R is the universal gas constant.

Gas Constant:-

(a) Universal gas constant (R):-

R= P₀ V₀/T₀

=8.311 J mol^-1K^-1

(b) Specific gas constant (r):- 

PV= (R/M) T = rT,              

Here,  r = R/M

• Real Gas:-The gases which show deviation from the ideal gas behavior are called real gas.
• Vander wall’s equation of state for a real gas:-

[P+(na/V)²?][V-nb] = nRT

Here n is the number of moles of gas.

Avogadro’s number (N):- Avogadro’s number (N), is the number of carbon atoms contained in 12 gram of carbon-12.

N = 6.023×10²³

(a) To calculate the mass of an atom/molecule:-

Mass of one atom = atomic weight (in gram)/N

Mass of one molecule = molecular weight (in gram)/N

(b) To calculate the number of atoms/molecules in a certain amount of substance:-

Number of atoms in m gram = (N/atomic weight)×m

Number of molecules in m gram = (N/molecular weight)×m

(c) Size of an atom:-

Volume of the atom, V = (4/3)πr³

Mass of the atom, m = A/N

Here, A is the atomic weight and N is the Avogadro’s number.

Radius, r =[3A/4πNρ]^1/3\

Here ρ is the density.

Gas laws:-

(a) Boyle’s law:- It states that the volume of a given amount of gas varies inversely as its pressure, provided its temperature is kept constant.

PV = Constant

(b) Charlers law or Gey Lussac’s law:- It states that volume of a given mass of a gas varies directly as its absolute temperature, provided its pressure is kept constant.

V/T= Constant

V–V₀/V₀t = 1/273 = γ_p

Here γ_p (=1/273) is called volume coefficient of gas at constant pressure.

Volume coefficient of a gas, at constant pressure, is defined as the change in volume per unit volume per degree centigrade rise of temperature.

(c) Gay Lussac’s law of pressure:- It states that pressure of a given mass of a gas varies directly as its absolute temperature provided the volume of the gas is kept constant.

P/T = P₀/T₀ or P – P₀/P₀t = 1/273 = γ_p

Here γ_p (=1/273) is called pressure coefficient of the gas at constant volume.

Pressure coefficient of a gas, at constant volume, is defined as the change in pressure per unit pressure per degree centigrade rise of temperature.

(d) Dalton’s law of partial pressures:-

Partial pressure of a gas or of saturated vapors is the pressure which it would exert if contained alone in the entire confined given space.

P= p₁+p₂+p₃+……..

nRT/V = p₁+p₂+p₃+……..

(e) Grahm’s law of diffusion:- Grahm’s law of diffusion states that  the rate of diffusion of gases varies inversely as the square root of the density of gases.

R∝1/√ρ    or R₁/R₂ =√ρ₂/ ρ₁

So, a lighter gas gets diffused quickly.

(f) Avogadro’s law:- It states that under similar conditions of pressure and temperature, equal volume of all gases contain equal number of molecules.

For m gram of gas, PV/T = nR = (m/M) R

• Pressure of a gas (P):- P = 1/3 (M/V) C² = 1/3 (ρ) C²
• Root mean square (r.m.s) velocity of the gas:- Root mean square velocity of a gas is the square root of the mean of the squares of the velocities of individual molecules.

C= √[c₁²+ c₂²+ c₃²+…..+ c_n²]/n = √3P/ ρ

• Pressure in terms of kinetic energy per unit volume:- The pressure of a gas is equal to two-third of kinetic energy per unit volume of the gas.

P= 2/3 E

• Kinetic interpretation of temperature:- Root mean square velocity of the molecules of a gas is proportional to the square root of its absolute temperature.

C= √3RT/M

Root mean square velocity of the molecules of a gas is proportional to the square root of its absolute temperature.

At, T=0, C=0

Thus, absolute zero is the temperature at which all molecular motion ceases.

• Kinetic energy per mole of gas:-

K.E. per gram mol of gas = ½ MC² = 3/2 RT

• Kinetic energy per gram of gas:-

½ C² = 3/2 rt

Here, ½ C² = kinetic energy per gram of the gas and r = gas constant for one gram of gas.

• Kinetic energy per molecule of the gas:-

Kinetic energy per molecule = ½ mC² = 3/2 kT

Here, k (Boltzmann constant) = R/N

Thus, K.E per molecule is independent of the mass of molecule. It only depends upon the absolute temperature of the gas.

• Regnault’s law:- P∝T
• Graham’s law of diffusion:-

R₁/R₂ = C₁/C₂ = √ρ₂/ ρ₁

Distribution of molecular speeds:-

(a) Number of molecules of gas possessing velocities between v and v+dv :-

(b) Number of molecules of gas possessing energy between u and u+dv:-

(c) Number of molecules of gas possessing momentum between p and p+dp :-

(d) Most probable speed:- It is the speed, possessed by the maximum number of molecules of a gas contained in an enclosure.

V_m= √[2kT/m]

(e) Average speed (V_av):- Average speed of the molecules of a gas is the arithmetic mean so the speeds of all the molecules.

V_av= √[8kT/πm]

(f) Root mean square speed (Vrms):- It is the square root of the mean of the squares of the individual speeds of the molecules of a gas.

V_rms = √[3kT/m]

• V_rms > V_av > V_m
• Degree of Freedom (n):- Degree of freedom, of a mechanical system, is defined as the number of possible independent ways, in which the position and configuration of the system may change.

In general, if N is the number of particles, not connected to each other, the degrees of freedom n of such a system will be,

n = 3N

If K is the number of constraints (restrictions), degree of freedom n of the system will be,

n = 3N –K

Degree of freedom of a gas molecule:-

(a) Mono-atomic gas:- Degree of freedom of monoatomic molecule, n = 3

(b) Di-atomic gas:-

At very low temperature (0-250 K):- Degree of freedom, n = 3

At medium temperature (250 K – 750 K):- Degree of freedom, n = 5 (Translational = 3, Rotational = 2)

At high temperature (Beyond 750 K):- Degree of freedom, n = 6 (Translational = 3, Rotational = 2, Vibratory =1), For calculation purposes, n = 7

• Law of equipartition of energy:- In any dynamical system, in thermal equilibrium, the total energy is divided equally among all the degrees of freedom and energy per molecule per degree of freedom is ½ kT.

E = ½ kT

• Mean Energy:- K.E of one mole of gas is known as mean energy or internal energy of the gas and is denoted by U.

U = n/2 RT

Here n is the degree of freedom of the gas.

(a) Mono-atomic gas(n = 3):- U = 3/2 RT

(b) Diatomic gas:-

At low temperature (n=3):- U = 3/2 RT

At medium temperature (n=5):- U = 5/2 RT

At high temperature (n=7):- U = 7/2 RT

Relation between ratio of specific heat capacities (γ) and degree of freedom (n):-

γ = C_p/C_v = [1+(2/n)]

(a) For mono-atomic gas (n=3):- γ = [1+(2/n)] = 1+(2/3) = 5/3=1.67

(b) For diatomic gas (at medium temperatures (n=5)):- γ = [1+(2/5)] = 1+(2/5) = 7/5=1.4

(c) For diatomic gas (at high temperatures (n=7)):- γ = [1+(2/7)] = 9/7 = 1.29

Link-1 Chapter 12 Thermodynamics Class 11 Hand written notes Physics

Link-2 Chapter 12 Thermodynamics Class 11 Hand written notes Physics

Thermodynamics Class 11 notes Physics Chapter 12

Introduction

The foundation of thermodynamics is the conservation of energy and the fact that heat flows spontaneously from hot to cold body and not the other way around. The study of heat and its transformation to mechanical energy is called thermodynamics. It comes from a Greek word meaning “Movement of Heat”.

In this chapter, we shall study the laws of thermodynamics, various process, basic theory of heat engines, refrigerators and Carnot engine.

Thermal Equilibrium

When the temperature of the mixture becomes almost stable with the surrounding there is no further exchange of energy. This state in thermodynamics is called thermal equilibrium. So we may say in thermal equilibrium, the temperatures of the two systems are equal.

Zeroth Law of Thermodynamics

Zeroth law of thermodynamics states that “If two systems are in thermal equilibrium with a third system separately are in thermal equilibrium with each other.” Physical quantity whose value is equal for two systems in thermal equilibrium is called temperature (T).

Heat and Internal Energy

(i). Heat

Heat is that form of energy which gets transferred between a system and its surrounding because of temperature difference between them. Heat flows from the body at a higher temperature to the body at lower temperature. The flow stops when the temperature equalises. i.e., the two bodies are then in thermal equilibrium.

(ii). Internal Energy

It is sum of the kinetic energies and potential energies of all the constituent molecules of the system. It is denoted by ‘U’. U depends only on the state of the system. It is a state variable which is independent of the path taken to arrive at that state.

Work Done by a Gas

A container of cross sectional area A is fitted with a movable piston. Let the pressure of gas is P. Due to force applied by gas on piston, piston is displaced by Δx

Work done by gas,

W=F.△rW=F.△r

W=F△xcos0W=F△xcos0

W=F△xW=F△x

W=PA△xW=PA△x

W=P△VW=P△V

First Law of Thermodynamics

The first law of thermodynamics is a particular form of the general law of conservation of energy. Suppose the amount of heat Q is supplied to a system. It is normally spent in two ways.

1. Partially, it is spent in increasing internal energy of system.
2. The remaining part of it is spent in expanding the body against the external pressure, i.e. in doing external work W.

If ΔU the change in internal energy “since energy can neither be created nor destroyed but only convert from one form to another“, we have then

Q = ΔU + W…………(1)

If dQ, dU and dW are infinitesimal changes in heat, internal energy and work respectively, then equation (1) becomes

dQ = dU + dW

This equation represents the differential form of first law of thermodynamics.

Limitations of First Law of Thermodynamics

The first law of thermodynamics plays an important role in thermodynamics as it can be applied to know how much work will be obtained by transferring a certain amount of heat energy in a given thermodynamic process. However, first law of thermodynamics suffers from the following limitations :

• First law of thermodynamics does not indicate the direction of heat transfer.
• First law of thermodynamics does not tell anything about the conditions under which heat can be transformed into work.
• The first law does not indicate as to why the whole of the heat energy cannot be continuously converted into mechanical work.

Specific Heat Capacity

Specific heat capacity of a substance is defined as the heat required to raise the temperature of unit mass through 1°C (or 1 K).

Heat capacity of a substance is given by

S=△Q△TS=△Q△T

If we divide S by mass of the substance m in kg, we get

C=Sm=1m△Q△TC=Sm=1m△Q△T

here s is known as the specific heat capacity of the substance. It depends on the nature of the substance and its temperature. The unit of s is J kg^–1 K^–1.

The specific heat at constant volume C_v

It is defined as the amount of heat required to raise the temperature of a 1 mole of a gas through 1°C when its volume is kept constant. It is denoted by (C_v) and given by

CV=(△Q△T)VCV=(△Q△T)V

The specific heat at constant pressure C_p

It is defined as the amount of heat required to raise the temperature of 1 mole of the gas through 1°C when its pressure is kept constant. It is denoted by (C_p) and given by

CP=(△Q△T)PCP=(△Q△T)P

Derivation of Mayer’s Formula

From 1st law,

ΔQ = ΔU + ΔW = ΔU + PΔV

At constant volume ΔV = 0 so ΔQ = ΔU

Cv=(ΔQΔT)v=(ΔUΔT)vCv=(ΔQΔT)v=(ΔUΔT)v

Cv=ΔUΔTCv=ΔUΔT

On the other hand, at constant pressure,

ΔQ = ΔU + PΔV

Cp=(ΔQΔT)p=(ΔUΔT)p+P(ΔVΔT)pCp=(ΔQΔT)p=(ΔUΔT)p+P(ΔVΔT)p

Now, for a mole of an ideal gas

PV = RT

ΔVΔT=RPΔVΔT=RP

Cp=(ΔUΔT)p+P×RPCp=(ΔUΔT)p+P×RP

Cp=Cv+RCp=Cv+R

Cp−Cv=RCp-Cv=R

This formula is known as Mayer’s Formula. All the three quantities (C_p), (C_v) and R in this equation should be expressed in the same units either in joule/mole°C or in cal/mole°C.

Thermodynamic state variables and equation of state

The parameters or variables which describe equilibrium states of the system are called state variables.

(i). Intensive Variable

These are the variables which are independent of the size. e.g., pressure, density and temperature.

(ii). Extensive Variable

These are the variables which depend on the size of the system. e.g., volume, mass, internal energy.

(iii). Equation of State

The relation between the state variables is called the equation of state.

Thermodynamic processes

Any change in the thermodynamic coordinates of a system is called a process. The following are familiar processes in the thermodynamics.

(i). Isothermal Process

When a thermodynamic system undergoes a process under the condition that its temperature remains constant, then the process is said to be isothermal process. The essential condition for an isothermal process is that the system must be contained in a perfectly conducting chamber.

For isothermal process,

ΔU = 0

from the first lawof thermodynamics,

ΔU = Q-W

0 = Q-W

Q = W

Hence, for an ideal gas all heat is converted into work in isothermal process.

(ii). Adiabatic Process

When a thermodynamic system undergoes a process under the condition that no heat comes into or goes out of the system, then the process is said to be adiabatic process. Such a process can occur when a system is perfectly insulated from the surroundings.

For adiabatic process,

Q = 0

from the first lawof thermodynamics,

ΔU = Q-W

ΔU = 0-W

ΔU = -W

(iii). Isobaric Process

If the working substance is taken in expanding chamber in which the pressure is kept constant, the process is called isobaric process. In this process the gas either expands or shrinks to maintain a constant pressure and hence a net amount of work is done by the system or on the system.

(iv). Isochoric Process

If a substance undergoes a process in which the volume remains unchanged, the process is called an isochoric process. The increase of pressure and temperature produced by the heat supplied to a working substance contained in a non-expanding chamber is an example of isochoric process.

For isochoric process,

ΔV = 0,     W = PΔV,       W = 0

from the first law of thermodynamics,

ΔU = Q-W

ΔU = Q-0

ΔU = Q

(v). Quasi Static Process

A quasi-static process is defined as the process in which the deviation from thermodynamics equilibrium is infinitesimal and all the states through which the system passes during quasi-static process may be treated as aquarium states. Thus it may be defined as a succession of equilibrium states.

Heat engines

Any “cyclic” device by which heat is converted into mechanical work is called a heat engine. For a heat engine there are three essential requirements :

• Source:- A hot body, at a fixed high temperature T₁ from which the heat can be drawn heat, is called source or hot reservoir.
• Sink:- A cold body at a fixed lower temperature T₂ to which any amount of heat can be rejected, is called sink or cold reservoir.
• Working Substance:- The material, which on being supplied with heat, performs mechanical work is called the working substance.

Heat Engine

In a heat engine, the working substance takes in heat from the source, converts a part of it into external work, gives out the rest to the sink and returns to its initial state. This series of operations constitute a cycle. The work can be continuously obtained by performing the same cycle over and over again.

Suppose the working substance takes in an amount of heat Q₁ from the source, and gives out an amount Q₂ to the sink. Let W be the amount of work obtained. The net amount of heat absorbed by the substance is Q₁ – Q₂, which has been actually converted into work. Applying the  of thermodynamics to one complete cycle. We get

Q₁ – Q₂ = W

Thermal Efficiency

The thermal efficiency (e) of an engine is defined as the ratio of the work obtained to the heat taken in from the source, that is,

e=WQ1=Q1−Q2Q1e=WQ1=Q1-Q2Q1

e=1−Q2Q1e=1-Q2Q1

This equation indicates that the efficiency of the heat engine will be unity (efficiency 100%) when Q₂ = 0. This is, however, not possible in practice, This means that the engine cannot convert all the heat taken in from the source into work.

Refrigerators and heat pumps

(i). Reversible Process

A reversible process is one which can be retraced in opposite order by slightly changing the external conditions. The working substance in the reverse process passes through all the stages as in the direct process in such a way that all changes occurring in the direct process are exactly repeated in the opposite order and inverse sense and no changes are left in any of the bodies participating in the process or in the surroundings.

For reversible process,

ΔU = 0

from the first law of thermodynamics,

ΔU = Q-W

0 = Q-W

Q = W

(ii). Irreversible Process

Those process which can not be retraced in the opposite order by reversing the controlling factors are known as irreversible processes.

Second law of thermodynamics

This has two statements. First is Kelvin-Planck statement which is based upon the performance of heat engine and second is Clausius statement which is based upon the performance of refrigerator.

Kelvin-Planck statement

This may be stated as, “It is impossible to construct a device which operating in a cycle, has a sole effect of extracting heat from a reservoir at performing an equivalent amount of work“. Thus, a single reservoir at a single temperature can not continuously transfer heat into work.

Clausius statement

This may be stated as, “It is impossible for a self-acting machines working in a cycle process, unaided by any external agency to transfer heat from a body at a lower temperature to a body at a higher temperature.” In other words it may be stated as “Heat cannot flow itself from a colder to a hotter body”.

Reversible and Irreversible Processes

Reversible Process: A thermodynamic process is said to be reversible if the process can be turned back such that both the system and the surroundings return to their original states, with no other change anywhere else in the universe. Ex- extension of springs, slow adiabatic compression or expansion of gases.

Irreversible Process: An irreversible process can be defined as a process in which the system and the surroundings do not return to their original condition once the process is initiated. Ex- Relative motion with friction, Heat transfer.

Carnot Engine

A reversible heat engine operating between two temperatures is called a Carnot engine and the sequences of steps constituting one cycle is called the Carnot cycle.

Carnot Theorem

Carnot gave the most important results which are:

• No engine can have efficiency more than that of the Carnot engine.
• The efficiency of the Carnot engine is independent of the nature of the working substance.

Link-1 Chapter 11 Thermal Properties of Matter Handwritten notes Class 11 Physics

Link-2 Chapter 11 Thermal Properties of Matter Handwritten notes Class 11 Physics

Thermal Properties of Matter Class 11 notes Physics Chapter 11

Introduction

In this chapter, we shall examine some of the thermal properties of matter. We will first consider thermal expansion which plays an important role in everyday life and then discuss changes of phase and latent heat. At the end, we will discuss the phenomenon of heat transfer.

When a body is heated, various changes take place. Temperature is a measure of ‘hotness’ of a body. A kettle with boiling water is hotter than a box containing ice. When water boils or freezes, its temperature does not change during these processes even though a great deal of heat is flowing into or out of it.

Temperature and Heat

(i). Temperature

Temperature is a relative measure, or indication of hotness or coldness. A hot cooker is said to have a high temperature, and ice cube to have a lower temperature. An object at a higher temperature is said to be hotter than the one at a lower temperature. The SI unit of temperature is kelvin (K), whereas degree celsius (°C) is a commonly used unit of temperature.

(ii). Heat

When you put a cold spoon into a cup of hot coffee, the spoon warms up and the coffee cools down as they were trying to equalise the temperature. Energy transfer that takes place solely because of a temperature difference is called heat flow or heat transfer and energy transferred in this way is called heat. The SI unit of heat energy transferred is expressed in joule (J).

Measurement of Temperature

A physical property that changes with temperature is called a thermometric property. When a thermometer is put in contact with a hot body, the mercury expands, increasing the length of the mercury column.

(i). Celsius Scale

It defines ice-point temperature as 0°C and the steam point temperature as 100°C. The space between 0°C and 100°C marks is equally divided into 100 intervals.

(ii). Fahrenheit Scale

It defines the ice-point temperature as 32°F and the steam point temperature as 212°F. The space between 32°F and 212°F is divided into 180 equal intervals.

(iii). Kelvin Scale

Kelvin Scale is a scale of measuring of temperature, the melting point of ice is taken as 273 K and the boiling point of water as 373 K the space between these two points is divided into 100 equal intervals.

(iv). Relation between Different Scales of Temperatures

To convert a temperature from one scale to the other, we must take into account the fact that zero temperatures of the two scales are not the same.

C100=F−32180=K−273100=R80C100=F-32180=K-273100=R80

Note: The normal temperature of the human body measured on the Celsius scale is 37°C which is 98.6°F.

Ideal Gas Equation and Absolute Temperature

(i). Ideal Gas Equation

An equation which follows the law of Boyal, law of Charls and llaw of Avogadro is called ideal gas equation.

At constant temperature,

V∝1PV∝1P …( From Boyal’s law)

At constant pressure,

V∝TV∝T …( From charl’s law)

At constant T and P,

V∝nV∝n …( From Avogadro’s law)

By combinig all above equation, we get

V∝TnPV∝TnP

V=nRTPV=nRTP

PV=nRTPV=nRT

where, n = Number of moles of gas

R = Universal gas constant (R = 8.31 J mol–1 K–1)

P = Pressure of gas

V = Volume of gas

(ii). Absolute Temperature

The absolute minimum temperature is equal to –273.15ºC. This is also known as absolute zero. Absolute zero is the foundation of the kelvin temperature scale or absolute scale temperature.

Thermal Expansion

Increase in size of any matter on heating is called thermal expansion. There are three types of thermal expansion.

(i). Linear Expansion

The expansion in length is called linear expansion and the fractional change in length, ΔL/L is given by ΔL/L = αΔT where α is called coefficient of linear expansion.

(ii). Area Expansion

The expansion in area is called area expansion or superficial expansion and the fractional change in area, ΔA/A is given by ΔA/A = βΔT where β is called coefficient of area expansion.

(iii). Volume Expansion

The expansion in volume is called volume expansion and the fractional change in area, ΔV/V is given by ΔV/V = γΔT where γ is called coefficient of volume expansion.

(iv). Relation Between

α : β : γ = 1 : 2 : 3

αβ=12αβ=12 …..[ β = 2α ]

αγ=13αγ=13 …..[ γ = 3α ]

Specific Heat Capacity

If an amount of heat Q, when given to a body of mass m, increases its temperature by an amount ΔT, then

Q = mcΔT

where c is a constant and is called the specific heat capacity or simply specific heat of the material of the body.

If m = 1 kg and ΔT = 1C° then c = Q

Specific heat of the material of a substance is the amount of heat required to raise the temperature of unit mass of the substance through 1C°.

In SI, the unit of c is J/kg K.

Calorimetry

Calorimetry deals with the measurement of heat. The vessel which is largely used in such a measurement is called a calorimeter.

When two bodies at different temperatures are allowed to share heat, they attain a common temperature. If it is assumed that no heat is received from or given to any body outside the system and if there is no chemical action involved in the process of sharing, then

Heat gained = Heat lost

This simple statement based on the law of conservation of energy is called the principle of calorimetry.

Change of State

Depending on temperature and pressure, all matter can exist in a solid, liquid or gaseous state. These states or forms of matter are also called the phases of matter.

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. That is, both the solid and the liquid states of the substance coexist in thermal equilibrium during the change of states from solid to liquid.

The temperature at which the solid and the liquid states of the substance is in thermal equilibrium with each other is called its melting point. The change of state from liquid to vapour (or gas) is called vaporisation. It is observed that the temperature remains constant until the entire amount of the liquid is converted into vapour.

The temperature at which the liquid and the vapour states of the substance coexist is called its boiling point. The change from solid state to vapour state without passing through the liquid state is called sublimation, and the substance is said to sublime.

Latent Heat

Latent heat is defined as the heat or energy that is absorbed or released during a phase change of a substance. It could either be from a gas to a liquid or liquid to solid and vice versa. Latent heat is related to a heat property called enthalpy. It is denoted by L and its SI unit is J/kg.

L=QmL=Qm

There are two types of latent heat.

(i). Latent Heat of melting

It is a amount of heat which is required to change of phase from solid to liquid for unit mass at constant temperature. Ex- Latent heat of melting of ice is 3.33 x 10⁵ J/kg.

(ii). Latent Heat of Vaporization

It is a amount of heat which is required to change of phase from liquid to vapor for unit mass at constant temperature. Ex- Latent heat of vaporization of water is 22.6 x 10⁵ J/kg.

Heat Transfer

There are three mechanisms of heat transfer which name is given as- conduction, convection and radiation. Conduction occurs within a body or between two bodies in contact. Convection depends on motion of mass from one region of space to another. Radiation is heat transfer by electromagnetic radiation, such as sunshine, with no need for matter to be present in the space between bodies.

(i). Conduction

Conduction is the mechanism of transfer of heat between two adjacent parts of a body because of their temperature difference. Suppose, one end of a metallic rod is put in a flame, the other end of the rod will soon be so hot that you cannot hold it by your bare hands.

Here, heat transfer takes place by conduction from the hot end of the rod through its different parts to the other end. Gases are poor thermal conductors, while liquids have conductivities intermediate between solids and gases.

(ii). Convection

Convection is a mode of heat transfer by actual motion of matter. It is possible only in fluids. Convection can be natural or forced. In natural convection, gravity plays an important part. When a fluid is heated from below, the hot part expands and, therefore, becomes less dense. Because of buoyancy, it rises and the upper colder part replaces it. This again gets heated, rises up and is replaced by the relatively colder part of the fluid. The process goes on.

In forced convection, material is forced to move by a pump or by some other physical means. The common examples of forced convection systems are forced-air heating systems in home.

(iii). Radiation

Radiation is the transfer of heat by electromagnetic waves such as visible light, infrared, and ultraviolet rays. Everyone has felt the warmth of the sun’s radiation and intense heat from a charcoal grill or the glowing coals in a fireplace. Most of the heat from these bodies reaches you not by conduction or convection in the intervening air but by radiation. This heat transfer would occur even if there were nothing but vacuum between you and the source of heat.

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’ (e_λ) of the given surface, given temperature and wavelength. Its S.I. unit is Js^-1 m-^-2.

(ii). Absorptive Power

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 (a_λ). 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 nor 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, move towards shorter wavelengths. Wein found that “The product of the peak wavelength (λ_m) and the Kelvin temperature (T) of the black body should remain constant.”

λm×T=bλm×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 radiations 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

E∝AT4E∝AT4

E=σAT4E=σAT4

Where σ is a Stefan’s constant and its value is 5.67 x 10^-8 W m^-2 K^-4.

Newton’s Law of Cooling

According to Newton’s law of cooling, “The rate of loss of heat of a body is directly proportional to the excess of the temperature (T–T₀) of the body with respect to the surroundings”.

−dTdt∝(T−T0)-dTdt∝(T-T0)

Summary

• Temperature : The relative measure of hotness or coldness of a body is called its temperature.
• Heat : The energy that flows between two bodies by virtue of temperature difference between them is called heat. It flows from a hot body to cold body.
• Specific Heat Capacity : The amount of heat per unit mass absorbed or rejected by a substance to change its temperature by one unit is called its specific heat capacity (C).
• Molar Specific Heat Capacity : The amount of heat per unit mole absorbed or rejected by a substance to change its temperature by one unit is called its molar specific heat capacity (C).
• Calorimeter : A device in which heat measurement can be made is called a calorimeter.
• Melting Point : The temperature at which the solid and the liquid states of a substance exist in thermal equilibrium with each other is called its melting point.
• Boiling Point : The temperature at which the liquid and the vapour states of the substance coexist is called its boiling point.
• Triple Point : The temperature and pressure at which all the three phases of a substance co exist is called its triple point.
• Latent Heat : The heat per unit mass required (absorbed or evolved) to change the state of a substance at the same temperature and pressure is called its latent heat.
• The temperature of an object is measured with a device called thermometer.
• Heat transfer can take place by three modes namely, conduction, convection and radiation. Radiation is fastest of them all and does not require a material medium.

Chapter 10 Mechanical Properties of Fluids Handwritten notes Class 11 Physics

Mechanical Properties of Fluids Class 11 notes Physics Chapter 10

Introduction

In this chapter, we shall study some common physical properties of liquids and gases. How are fluids different from solids? What is common in liquids and gases? Unlike a solid, a fluid has no definite shape of its own. Solids and liquids have a fixed volume, whereas a gas fills the entire volume of its container.

Liquids and gases can flow and are therefore, called fluids. Earth has an envelop of air and two-thirds of its surface is covered with water. All the processes occurring in living beings including plants are mediated by fluids.

Pressure

We can define pressure as the normal force acting per unit area of a surface. It is denoted by P and SI unit of it is pascal (Pa). It is a scalar quantity.

P=FAP=FA

We observe that the same force (weight) exerts different pressures for different areas in contact. Lesser the area, more is the pressure exerted for a given force.

Density

The density of any material is defined as its mass per unit its volume. If a fluid of mass m occupies a volume V, then its density is given as

Density=mVDensity=mV

Density is usually denoted by the symbol ρ. It is a positive scalar quantity. Its SI unit is kg m^–3 and its dimensions are [ML^–3].

Pascal’s Law

This law states that the pressure in a fluid at rest is same at all points which are at the same height.

Imagine a small element of fluid in the shape of a right angled prism. All its points lie at the same depth inside the liquid. Therefore the effect of gravity is same at all these points. The forces exerted by rest of the fluid on different surfaces of this fluid element are as follows:

F1=F2=F3F1=F2=F3

If A₁, A₂ and A₃ are the surface areas of faces respectively, then by dividing.

F1A1=F2A2=F3A3F1A1=F2A2=F3A3

p1=p2=p3p1=p2=p3

So, the pressure exerted is same in all directions in a fluid at rest, at the points at equal height.

Variation of Pressure with Depth

Let’s find the difference in pressures at two points, whose levels differ by a height h in a fluid at rest. Let P₁ and P₂ be the pressures at two points 1 and 2 inside a fluid. Point 1 is at a height h above the point 2.

Imagine a fluid element in the shape of a cylinder as shown. If A be the area of the top and the bottom of this cylinder, then

F1=P1AF1=P1A

F2=P2AF2=P2A

Since, the fluid remains at rest, therefore the force F₂, which acts upwards should balance the two downward forces. These are, the force F₁ exerted at the top of the cylinder, and the weight W of the fluid confined within the cylinder.

F2=F1+mgF2=F1+mg

If ρ is the density of the fluid, then

F2=F1F2=F1+ρ.(volume of the cylinder).g

P2A=P1A+ρ.(A.h).gP2A=P1A+ρ.(A.h).g

P2=P1+ρghP2=P1+ρgh

P2−P1=ρghP2-P1=ρgh

This result tells us that as we go deep down a liquid the pressure goes on increasing. This pressure depends only on the height of the liquid column above the point.

Atmospheric Pressure

The atmospheric pressure at a point is equal to the weight of a column of air of unit cross-sectional area extending from that point to the top of the atmosphere. Its value is 1.013 × 10⁵ Pa at sea level. Atmospheric pressure is measured using an instrument called barometer.

Units os Atmospheric Pressure

• SI unit of pressure is N m^–2 or Pascal (Pa)
• Atmosphere, 1 atm = 1.013 × 10⁵ Pa = 760 mm of Hg
• 1 torr = 133 Pa
• 1 mm of Hg = 1 torr
• 1 bar = 10⁵ Pa
• 1 millibar = 100 Pa

Gauge Pressure

When we remove atmospheric pressure from total pressure of any system then this remaining pressure is called Gauge Pressure. The excess pressure P–P_a, at depth h is called a gauge pressure at that point.

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 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

1. Upward thrust (T) acting at the centre of buoyancy and whose magnitude is equal to the weight of the liquid displaced
2. 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.

Streamline Flow

When a liquid flows such that each particle of the liquid passing a given point moves along the same path and has the same velocity as its predecessor had at that point, the flow is called streamlined or steady flow. The path followed by a fluid particle in steady flow is called streamline.

Equation of Continuity

According to this theorem, “For the streamline flow of an incompressible fluid through a pipe of varying cross-section, product of cross-section area and velocity of streamline flow (Av) remains constant throughout the flow”.

Av = constant

Bernoulli’s Principle

It may be stated as follows: As we move along a streamline, the sum of the pressure (P), the kinetic energy per unit volume (ρv22ρv22) and the potential energy per unit volume (ρgh) remains a constant.

P+ρgh+12ρv2=P+ρgh+12ρv2=constant

Limitations of Bernoulli’s Equation

1. The equation is valid only for incompressible fluids having streamline flow. It is because it does not take into account the elastic energy of the fluids.
2. It is assumed that no energy is dissipated due to frictional force exerted by different layers of fluid on each other.
3. It does not hold for non-steady flow. In such situation velocity and pressure constantly fluctuate with time.

Surface Tension

Surface tension is defined as the surface energy per unit area or the force per unit length acting in the plane of the interface between the plane of the liquid and any other substance. The surface tension of a liquid usually decreases with increase in temperature.

S=FlS=Fl

Capillary Rise

A tube of very fine bore is called a capillary. ‘Capilla’ is a Latin word which means hair. Thus, capillary is a very thin tube. When such a tube, open at both ends, is dipped in a beaker containing water, water rises in it against gravity.

Let us find an expression for the height h, upto which a liquid rises in a capillary tube. Let a capillary tube of radius r be dipped in a liquid of surface tension S and density ρ.

Thus, capillary rise

h=2Scosθrρgh=2Scosθrρg

If the angle of contact for this liquid and the capillary tube is acute, the liquid forms a concave meniscus.

Viscosity

When a fluid moves, it flows in the form of parallel layers. These layers exert a force on each other which tends to oppose their relative motion. This is similar to what a frictional force does when two solids in contact move or tend to move over each other. The property of fluid which gives rise to such frictional force in them, is called viscosity. It is denoted by ‘η’. Its SI unit is N s m^–2 or Pa s which is also called poiseuille (Pl). The dimensions of viscosity are [ML^–1T^–1].

η=FAdvdxη=FAdvdx

where, dv/dx = velocity gardient

F = frictional force between layer of water

A = area of layer

Angle of Contact

The angle of contact is defined as the angle that the tangent to the liquid surface at the point of contact makes with the solid surface inside the liquid. The angle of contact depends on the nature of the solid and the liquid in contact. At the point of contact, the surface forces between the three media must be in equilibrium.

Stokes’ Law

When a spherical ball is dropped in a liquid, he observed that the viscous force F experienced by the ball is proportional to the

– velocity of the object through the fluid, v

– viscosity of the fluid, η

– radius of the sphere, r

Thus    F ∝ ηrv

Here, the constant of proportionality is found to be 6π.

Thus    F = –6πηrv   ← Stokes’ law.

The negative sign in the above expression just indicates that the retarding force is opposite to the direction of motion of the object.

Terminal Velocity

The maximum constant velocity acquired by a body while falling through a viscous medium is called its terminal velocity. It is usually denoted by V_T.

When the body acquires terminal velocity,

the upward viscous force + the upward buoyant force = weight of the ball

Reynolds Number

Whether a flow will be turbulent or not, is decided by a dimensionless parameter called Reynolds number R_e. This parameter is given by the relation

Re=ρvdηRe=ρvdη

where, ρ = density of the fluid

v = velocity of the fluid

η = viscosity of the fluid

d = diameter of the pipe through which the fluid flows

• If R_e < 1000, the flow is streamline or laminar
• If Re > 2000, the flow becomes turbulent
• If 1000 < Re < 2000, flow is unsteady i.e., it may change from laminar to turbulent and vice versa.

Poiseuille’s Formula

The Poiseuille’s formula gives an expression for volume flow rate through a capillary tube of inner radius r and length l due to a pressure difference between its ends, P. The volume flow rate is represented by Q such that

Q=dVdtQ=dVdt

According to Poiseuille’s,

dVdt=(π8)×(Pl)×r4ηdVdt=(π8)×(Pl)×r4η

Here, η is the coefficient of viscosity and all symbols have standard meaning.

Chapter 9 Mechanical Properties of Solids Hand Written Notes Class 11 Physics

Chapter 9 Mechanical Properties of Solids Class 11 notes Physics

Introduction

Can we design an aeroplane which is very light but sufficiently strong? Can we design an artificial limb which is lighter but stronger? Why does a railway track have a particular shape like I? Why is glass brittle while brass is not?

In this chapter, we will introduce the concepts of stress, strain and elastic modulus and a simple principle called Hooke’s law that help us predict what deformation will occur when forces are applied to a real kind of body.

Elastic Behaviour of Solids

In a solid, each atom or molecule is surrounded by neighbouring atoms or molecules. These are bonded together by interatomic or intramolecular forces and stay in a stable equilibrium position. When a solid is deformed, the atoms or molecules are displaced from their equilibrium positions causing a change in the interatomic distance.

When the deforming force is removed, the interatomic forces tend to drive them back to their original position. Thus the body regains its original shape and size.

(i) Deforming Force

If a force applied on a body produces a change in the normal positions of the molecules of the body, it is called deforming force.

(ii) Elasticity

The property of the body due to which, it tries to regain its original configuration when the deforming forces are removed is called elasticity.

(iii) Perfectly Elastic body

A body which completely regains its original configuration after the removal of deforming force, is called perfectly elastic body. Quartz and phosphor bronze are closest to perfectly elastic body known.

(iv) Perfectly Plastic Body

A body which does not regain its original configuration at all on the removal of deforming force, how so ever small the deforming force may be is called perfectly plastic body. For example, clay behaves like a perfectly plastic body.

(v) Restoring Force

When a deforming force is applied to a body to change its shape, the body develops an opposing force due to its elasticity. This opposing force tries to restore the original shape of the body, it is called restoring force.

Stress and Strain

(A) Strain

The strain is the relative change in dimensions of a body resulting from the external forces.

Strain = change in length / original length

It is a fractional quantity so, it has no unit.

(i) Tensile Strain

The tensile strain of the object is equal to the fractional change in length, which is the ratio of the elongation Δl to the original length l.

Tensile Strain=ΔlL=ΔlL

(ii) Shear Strain

We define shear strain as the ratio of the displacement x to the transverse dimension L.

Shear Strain=xL=xL

(iii) Bulk Strain

The fractional change in volume that is, the ratio of the volume change ΔV to the original volume V is called Bulk Strain.

Bulk Strain=ΔVV=ΔVV

(B) Stress

The restoring force developed per unit area in a body is called stress.

Stress = Restoring Force / area

In SI system, stress is measured in N / m² (pascal) and in CGS system in dyne/cm². The dimensional formula for stress is [M L^–1T^–2]

(i) Tensile Stress

We define the tensile stress at the cross-section as the ratio of the force F_⊥ to the cross-sectional area A.

Tensile Stress=F⊥A=F⊥A

(ii) Shear Stress

We define the shear stress as the force F_|| acting tangent to the surface, divided by the area A on which it acts.

Shear Stress=F∣∣A=F∣∣A

(iii) Bulk Stress

If an object is immersed in a fluid (liquid or gas) at rest, the fluid exerts a force on any part of the surface of the object. This force is perpendicular to the surface. The force F_⊥ per unit area that the fluid exerts on the surface of an immersed object is called the pressure p in the fluid (Bulk Stress).

Hooke’s Law

For small deformations the stress and strain are proportional to each other. This is known as Hooke’s law.

Thus,

stress ∝ strain

stress = k × strain

where k is the proportionality constant and is known as modulus of elasticity.

Elastic Moduli

The ratio of stress and strain, called modulus of elasticity, is found to be a characteristic of the material.

(i) Young’s Modulus

For a sufficiently small tensile stress, stress and strain are proportional. The corresponding elastic modulus is called Young’s modulus, denoted by Y.

Y = Tensile-stress / Tensile-strain=FAΔlL=FAΔlL

Y=FLAΔlY=FLAΔl

(ii) Shear Modulus

If the forces are small enough that Hooke’s law is obeyed, the shear strain is proportional to the shear stress. The corresponding elastic modulus is called the shear modulus, denoted by G. It is also called the modulus of rigidity.

G = Shear-stress / Shear-strain=FAΔxL=FAΔxL

G=F×LA×ΔxG=F×LA×Δx

G=FAθG=FAθ

SI unit of shear modulus is Nm^–2 or Pa.

(iii) Bulk Modulus

When Hooke’s law is obeyed, an increase in Bulk stress produces a proportional Bulk strain. The corresponding elastic modulus (ratio of stress to strain) is called the Bulk modulus, denoted by B.

When the pressure on a body changes by a small amount Δp, from p to (p+Δp), and the resulting Bulk strain is ΔV/V, Hooke’s law takes the form

B = Normal-stress / Volume-strain=−ΔpΔVV=-ΔpΔVV

We include a minus sign in this equation because an increase of pressure always causes a decrease in volume. The Bulk modulus B itself is a positive quantity.

SI unit of bulk modulus is Nm^–2 or Pa.

Compressibility

The reciprocal of the Bulk modulus is called the compressibility and is denoted by K. From equation

K=1BK=1B

The units of compressibility are those of reciprocal pressure, Pa^–1 or atm^–1.

Elastic Potential Energy

The excess of the energy of interaction between all atoms/molecules of a deformed object is elastic energy. When we remove the external force the body becomes undeformed and the elastic energy, will be retrieved back and converted into vibrational energy followed by heat, light, sound etc.

The elastic potential energy

ΔU=12=12stress × strain × volume

Also, the elastic potential energy per unit volume, i.e.,

ΔUvolume=12ΔUvolume=12stress × strain

Poisson’s Ratio

When a body is linearly extended, it contracts in the direction at right angles. Poisson’s ratio, σ is the ratio of the lateral strain to the longitudinal strain.

Longitudinal strain = Δl/L

Lateral strain = – ΔR/R

The Poisson’s ratio is given as,

σ = lateral strain / longitudinal strain

σ=–ΔRRΔlLσ=–ΔRRΔlL

–ve sign shows that if the length increases, then the radius of wire decreases. Poisson’s ratio is a unit less and dimensionless quantity.

Relation between Y, K, η and σ

• Y = 3K (1 – 2σ)
• Y = 2η (1 + σ)
• σ = (3K – 2η) /(2η + 6K)
• 9/Y = 1/K + 3/η

Applications of Elastic Behaviour of Materials

In our daily life, most of the materials which we use, undergo some kind of stress. That is why, while designing a structure of the material we give due consideration to the possible stresses, the material might suffer at one stage or the other. The following examples illustrate this concept.

1. The metallic parts of the machinery are never subjected to a stress beyond elastic limit, otherwise they will get permanently deformed.
2. The crane which is used to lift and move the heavy load is provided with thick and strong metallic ropes to which the load to be lifted is attached. The rope is pulled by using pulleys and motor.
3. The bridges are designed in such a way that they do not bend much or break under the load of heavy traffic, force of strongly blowing wind and its own weight.
4. Maximum height of a mountain on earth can be estimated from the elastic behaviour of earth. At the base of mountain, the pressure is given by p = ρgh, where h is the height of mountain, ρ is the density of material of mountain and g is the acceleration due to gravity.

Summary

• Elasticity : Elasticity is that property of the material of a body due to which the body opposes any change in its shape and size when deforming forces are applied on it and recovers its original configuration partially or wholly as soon as the deforming forces are removed.
• Stress : It is defined as the internal restoring force per unit area of cross-section of object.
• Strain : The change in dimensions of an object per unit original dimensions is called strain.
• Hooke’s law : For small deformation, the stress is proportional to strain.
• Young’s modulus : The ratio of tensile (or compressive) stress to the corresponding longitudinal strain is called Young’s modulus.
• Bulk modulus : The ratio of volumetric stress to volumetric strain is called Bulk modulus.
• Shear modulus : It is the ratio of shear stress to shearing strain.
• Poisson’s ratio : The lateral strain is proportional to longitudinal strain within the elastic limit and the ratio of two strains is called Poisson’s ratio.
• Elastic after effect : The slow process of recovering the original state by an object after the removal of the deforming force is called elastic after effect.

Chapter 8 Gravitation Handwritten Notes Class 11 notes Physics

Chapter 8 Gravitation Class 11 notes Physics

Summary

• Gravitational force: It is a force of attraction between the two bodies by the virtue of their masses.
• Acceleration due to gravity: The acceleration produced in the motion of a body freely falling towards earth under the force of gravity is known as acceleration due to gravity.
• Gravitational potential energy: The amount of work done in displacing the particle from infinity to a point under consideration.
• Gravitational potential: The gravitational potential due to the gravitational force of the earth is defined as the potential energy of a particle of unit mass at that point.
• Escape speed: The minimum speed with which the body has to be projected vertically upwards from the surface of the earth is called escape speed.
• Orbital speed: The minimum speed required to put the satellite into the given orbit around earth is called orbital speed.
• Satellite: It is a body which revolves continuously in an orbit around a comparatively much larger body.
• Polar satellite: It is the satellite which revolves in polar orbit around the earth.
• Geostationary satellite: It is the satellite which appears at a fixed position and at a definite height to an observer on earth.
• Kepler’s Ist law: All planets move in elliptical orbits, with the sun at one of foci of the ellipse.
• Kepler’s IInd law: The line that joins any planet to the sun sweeps out equal areas in equal intervals of time.
• Kepler’s IIIrd law: 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.
• Newton’s universal law of gravitation: Every particle in the universe attracts every other particle with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
• Gravitational force is a conservative force.
• The value of acceleration due ot gravity is maximum at the surface of the earth while zero at the centre of earth.
• Henry Cavendish was the first person who found the value of G experimentally.
• Gravitational force on a particle inside a spherical shell is zero.
• Gravitational shielding is not possible.
• An astronaut experiences weightlessness in a space satellite. It is because both the astronaut and the satellite are in “free fall” towards the earth.
• The value of g increases from equator to poles.
• The escape speed from a point on the surface of the earth may depend on its location on the earth e.g., escape speed is more on poles and less on equator.
• The orbital speed of satellite is independent of mass of the satellites.
• Kepler’s laws hold equally well for satellites.

Chapter 7 Systems of Particles and Rotational Motion Handwritten Notes Class 11 Notes Physics

Chapter 7 Systems of Particles and Rotational Motion Class 11 Notes Physics

Chapter 6 Work, Energy and Power Handwritten Notes Class 11th Physics

Chapter 6 Work, Energy and Power Class 11 Notes Physics

Chapter 5 Law of Motion Handwritten Notes Class 11th Physics

Chapter 5 Law of Motion Class 11 Notes Physics

 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 inertia is more 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’. It defines inertia, force and inertial frame of reference.
I here is always a need of ‘frame of reference’ to describe and understand the motion of particle, lhc simplest ‘frame of reference’ used are known as the inertial frames.
A frame of referent, e is known as an inertial frame it, within it, all accelerations of any particle are caused by the action of ‘real forces’ on that particle.
When we talk about accelerations produced by ‘fictitious’ or ‘pseudo’ forces, the frame of reference is a non-inertial one.

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 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 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 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 mass of body and g is acceleration due to gravity, weight of the body is Mg in 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 as Normal force.

Example. Let us consider a book resting on the table. It is acted upon by its weight in vertically downward direction and is at rest. It means there is another force acting on the block in opposite direction, which balances its weight. This force is provided by the table and we call it as 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 in extensible string, which passes over a light and friction less 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 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 area of contact between the surfaces.

Coefficient of Friction

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 Friction

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. It is given by

where, v is the linear velocity, r is the radius of circular path and ω is the angular velocity of the body.

Centrifugal Force
Centrifugal force is a force that arises when a body is moving actually along a circular path, by virtue of tendency of the body to regain its natural straight line path.
The magnitude of centrifugal force is 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 vertical circle. In horizontal circle, the motion is not effected by the acceleration due to gravity (g) whereas in the motion of vertical circle, the motion is not effected 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 move 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.

Let us consider a particle moving in a circular vertical path of radius V and centre o tide with a string. L be the instantaneous position of the particle such that

Here the following forces act on the particle of mass ‘m’.
(i) Its weight = mg (verticaly downwards).
(ii) The tension in the string T along LO.

We can take the horizontal direction at the lowest point ‘p’ as the position of zero gravitational potential energy. Now as per the principle of conservation of energy,

From this relation, we can calculate the tension in the string at the lowest point P, mid-way point and at the highest position of the moving particle.
Case (i) : At the lowest point P, θ = 0°

When the particle completes its motion along the vertical circle, it is referred to as “Looping the Loop” for this the minimum speed at the lowest position must be √5gr

IMPORTANT TABLES