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By going through these CBSE Class 11 Physics Notes Chapter 15 Waves, students can recall all the concepts quickly.

Waves Notes Class 11 Physics Chapter 15

→ A wave is a form of disturbance that transmits energy from one place to another without the actual flow of matter as a whole.

→ Waves are of three types:

1. Mechanical waves,
2. e.m. waves,
3. matter waves.

→ Water waves or sound waves are called mechanical or elastic waves as they require a material medium for their propagation.

→ A material medium possesses both elasticities as well as inertia.

→ Light waves don’t require any material medium for their propagation.

→ Light waves are electromagnetic waves or non-mechanical waves which can propagate through a vacuum.

→ Matter waves are associated with moving electrons, protons, neutrons and other fundamental particles and even atoms and molecules.

→ The matter is constituted by electrons, protons, neutrons and other fundamental particles.

→ The waves associated with matter particles are called matter waves.

→ Matter waves arise in the quantum mechanical description of nature.

→ Wave motion is a form of disturbance that is due to the repeated periodic vibrations of the particles of the medium about their mean positions.

→ The motion is handed over from one medium particle to another without any net transport of the medium during wave motion.

→ Mechanical waves are of two types

1. transverse waves and
2. longitudinal waves.

→ A wave is said to be a progressive or travelling wave if it travels from one point of the medium to another.

→ The waves on the surface of the water are of two types: capillary waves and gravity waves.

→ The restoring force that produces capillary waves is the surface tension of water.

→ The restoring force that produces gravity waves is the pull of gravity which tends to keep the water surface at its lowest level.

→ The oscillations of the particles in gravity waves are not confined to the surface only but extend with diminishing amplitude to the very bottom.

→ The particle motion in water waves involves a complicated motion, they not only move up and down but also back and forth.

→ The waves in an ocean are a combination of both longitudinal and transverse waves.

→ Transverse and longitudinal waves travel at different speeds in the same medium.

→ k is called propagation constant or angular wavenumber.

→ S.I. unit of k is radian (rad) per metre of rad m^-1.

→ The speed of transverse waves in a string is determined by two factors:

1. Linear mass density i.e. mass per unit length (m),
2. Tension (T) in the string.

→ Positions of zero amplitude are called nodes.

→ Positions of maximum amplitude are called antinodes.

→ Nodes and antinodes are separated by \(\frac{λ}{4}\).

→ Two successive nodes or antinodes are separated by \(\frac{λ}{2}\).

→ Audible sound waves have a frequency between 20 Hz to 20,000 Hz.

→ The equation of a simple harmonic wave travelling in the positive X-direction is given by
y = A sin (ωt – kx)
where ω = \(\frac{2 \pi}{\mathrm{T}}\) = 2πv
k = \(\frac{2 \pi}{\lambda}\)

→ The particle velocity in a wave is given by v = \(\frac{\mathrm{dy}}{\mathrm{dt}}\)

→ Wave velocity is given by C = \(\frac{\mathrm{dx}}{\mathrm{dt}}\).

→ A wave travelling in negative x-direction is given by
y = A sin (ωt + kx)

→ The speed of sound does not depend on the frequency or wavelength.

→ Sound waves are mechanical waves that can’t propagate in a vacuum.

→ Sound waves can’t travel in sawdust or dry sand because the medium is not continuous.

→ The damping of sound in wood is much larger as compared to that in metals.

→ The higher the frequency of sound greater is the pitch of the sound.

→ The voice of ladies and children is of higher pitch than that of men.

→ Unit of loudness is bell (B) = log\(\frac{\mathrm{I}}{\mathrm{I}_{0}}\).

→ The sound is reflected and refracted according to the same laws as the light does.

→ The wavelength for ultrasonics is very small, therefore they are not diffracted by the ordinary objects or holes etc.

→ The speed of mechanical waves is determined by the properties of the medium i.e. elasticity and inertia and not by the nature, intensity, amplitude or shape of the wave.

→ The velocity of sound is the largest in hydrogen among the gases.

→ Monosyllabic sound is produced in about 0.2 s.

→ The vibrations of the prongs of a tuning fork are transverse and that of the stem are longitudinal.

→ The point where the stem of the tuning fork is connected to the prongs is an antinode.

→ The ends of the prongs are also antinodes.

→ There is a node between them that is nearer to the stem than the ends of the prongs.

→ The speed of sound in the air is not affected by the changes in pressure.

→ For every 1°C rise in temperature, the speed of sound increases by 0. 61 ms^-1.

→ Due to a change in temperature, the wavelength of sound waves is affected.

→ Beats are not audible if the beat frequency is more than 10 Hz.

→ If the prong of a tuning fork is loaded near the stem its frequency increases and when it is filled near the stem, the frequency decreases.

→ The number of beats produced per second is equal to the difference in the frequencies of the superposing notes.

→ In the progressive wave, the crest and troughs or compressions and rarefactions move with the speed of the wave.

→ When there is no relative motion between the source and listener, the Doppler’s effect is not observed.

→ When a source of sound moves, it causes a change in the wavelength of k the sound received by the listener.

→ When the listener moves, it causes a change in the number of waves ( received by the listener.

→ If source and listener move in mutually perpendicular .directions, no Doppler’s effect is observed.

→ Not Doppler’s effect is produced when only the medium moves.

→ A musical sound consists of a quick, regular and periodic succession of compressions and rarefactions without a sudden change in amplitude.

→ Pitch, loudness and quality are the characteristics of musical sound.

→ Pitch depends on frequency, loudness depends on intensity and quality depends on the number and intensity of overtones.

→ Pitch increases with an increase in frequency.

→ The ratio of the frequencies of the two nodes is called the interval between them. e.g. interval between 256 and 512 Hertz is 1: 2.

→ Two nodes are said to be in unison if their frequencies are equal i. e. if the interval between them is 1: 1.

→ Some other common intervals found useful in producing musical founds are as follows: octave (1: 2), major tone (8: 9), minor tone (9: 10), semitone (15: 16).

→ The fundamental note is called the first harmonic.

→ If n, be the fundamental frequency, then 2n₁, 3n₁, 4n₁, …. are respectively called second, third, fourth,…. harmonics respectively.

→ Harmonics are the integral multiples of the fundamental frequency.

→ Overtones are the notes of frequency higher than the fundamental frequency actually produced by the instrument.

→ In the strings, all harmonics are produced.

→ In the open organ pipe, all the harmonics are produced.

→ In the closed organ pipe, only the odd harmonics are produced.

→ In an open organ pipe as well as the string the second harmonics is the first overtone.

→ In the closed organ pipe, the third harmonic is the first overtone.

→ The ratio of the frequencies of the overtones in an open organ pipe is 2: 3: 4: 5:…

→ The ratio of the frequency of the overtones in the closed organ pipe is 3: 5: 7: …..

→ The frequency of the notes produced by the organ pipe varies:

1. directly as \(\sqrt{λ}\) , where λ, is a constant.
2. ∝ \(\sqrt{T}\) where T is the absolute temperature of the gas.
3. inversely as \(\sqrt{ρ}\) where ρ is the density of the gas.
4. inversely as length (l) of the tube.

→ The sound produced by the open organ pipe is comparatively pleasant as compared to that produced by the closed organ pipe.

→ The rarefactions are the regions of decrease in density or pressure and compressions are the regions of increase in density or pressure in air, gas when wave propagates through it.

→ Two waves travelling along the same path in the same or opposite direction superpose.

→ Superposition of waves gives rise to the phenomenon of interference, stationary waves and beats.

→ Interference of waves: Superposition of two waves of the same frequency and same wavelength travelling in the same direction with the same speed results in interference of waves.

→ Constructive interference: Interference is said to be constructive if two waves of the same frequency travelling in the same direction with the same speed superpose on each other such that the resultant displacement is more than the individual displacements.

→ Destructive interference: If the resultant displacement due to the superposition of two waves is less than the individual displacements then it is called destructive interference.

→ The wavelength of a wave: It is defined as the distance between two consecutive points (i.e. two consecutive troughs or crests) in the same phase of wave motion.

→ The fundamental mode of the first harmonic: It is defined as the oscillation mode with the lowest frequency.

→ Infrasonics: Sound waves of frequency less than 20Hz are called infrasonics. They can’t be heard by the human ear.

→ Beats: They are defined as the periodic variations in the intensity of sound due to the superposition of two sound waves of slightly different frequencies.

→ Mechanical or elastic waves: The waves set up and propagated due to the presence of a material medium and its properties of elasticity and inertial are called mechanical waves.

→ Electromagnetic waves: They are defined as the waves set up by the variation in electric and magnetic fields of an oscillating charge.

→ Transverse wave: It is defined as the wave motion set up due to vibrations of medium particles perpendicular to the direction of propagation of the wave.

→ Longitudinal wave: It is defined as the wave motion set up due to the vibrations of the medium particles along the direction of wave propagation.

→ Phase (Φ): It is defined as the argument of sine or cosine function representing a wave. It is the angular periodic position of a wave.

→ Time period (T): It is defined as the time taken by the medium particles to complete one oscillation.

→ Velocity of wave motion (v): It is defined as the ratio of wavelength to the time period i.e. v = \(\frac{λ}{T}\) = vλ, (∵ v = \(\frac{1}{T}\))

→ Stationary wave: It is defined as the wave due to the superposition of two progressive waves of the same frequency and amplitude but travelling in the opposite directions along the same line. It is also called a standing wave.

→ Harmonics: The wave of frequencies having integral multiples of a fundamental frequency are called harmonics of the fundamental wave including itself.

→ Overtones: They are defined as the waves of frequencies having integral multiples of a fundamental frequency but excluding it.

→ The 2nd harmonics is the first overtone, the third harmonics is 2nd overtone and so on.

→ Taut string: It is defined as a string vibrating in any mode/modes fixed at one end and loaded at the other end.

→ Musical sound: It is defined as a sound having series of harmonic waves following each other rapidly at regular intervals of time without a Sudden change in their amplitude. It produces a pleasant effect on the ear of the listener.

→ Noise: It is defined as a sound having series of harmonic waves following each other at irregular intervals of time with a sudden change in their amplitude. It produces a displeasing effect on the ear of the listener.

→ The intensity of sound at a point (I): It is defined as the amount of energy passing per unit time per unit area held perpendicular to the incident sound waves at that point.

→ Temperature coefficient of the velocity of sound (α): It is defined as the change in velocity of sound per Kelvin change in temperature.

→ Capillary waves: They are defined as the ripples of a fairly short wavelength not more than a few centimetres.

→ Gravity waves: They are defined as waves that have wavelengths typically ranging from several metres to several hundred metres.

→ Superposition Principle: It states that when two or more waves of the same nature travel in a medium, then the resultant displacement at a point is the vector sum of the displacement due to the individual waves.

→ Threshold of hearing or zero levels (I₀): It is defined as the lowest intensity of sound that can be heard by the human ear. It is about 10^-12 Wm^-2 for a sound of frequency I KHz.

→ The loudness of a sound: It ¡s defined as the degree of sensation of sound produced ¡n the car. It distinguishes between a loud and a faint sound.

→ Weber Fechner Law : It states that the loudness of sound is proportional to the logarithm of its intensity i.e. L = log₁₀ \(\left(\frac{\mathrm{I}}{\mathrm{I}_{0}}\right)\)

→ Bel (B): Loudness is said to be one bel if the intensity of sound is 10 times the threshold of hearing.

→ Pitch: It ¡s defined as that characteristic of musical sound which helps one to classify a note as a high note or low note.

→ Quality or Timber: Ills defined as that characteristic of musical sound which helps us to distinguish between.two sounds of the same intensity and pitch.

→ Musical scale: It consists of a series of flotes (frequencies) separated by definite and simple intervals so as to produce a musical effect when played in Succession.

→ Decibel (dB): \(\frac{1}{10}\)th of bel is called decible i.e. 1 dB = \(\frac{1}{10}\)B.

→ Keynote: The first note of the lowest frequency is called keynote. Octave: Two notes are said to be octave if the ratio of their frequencies is 2. It is also a musical scale called the diatomic scale which has 8 intervals (octave + 7 other intervals).

→ Shock wave: It is defined as the wave produced by a body moving with a speed greater than the speed of sound. Shock waves carry a large amount of energy and when strike a building rattling sound due to the vibration of the building is produced.

→ Mach number: It is defined as the ratio of the velocity of the body producing shock waves to the velocity of sound.
∴ Mach number = \(\frac{\mathrm{V}_{\mathrm{s}}}{\mathrm{v}}\)

→ Echo: It is defined as the repetition of the sound of short duration. It (echo) is heard if the minimum distance between the obstacle reflecting sound waves and the source of sound is 17 m.

→ Reverberation: It is defined as the persistence or prolongation of audible sound after the source has stopped emitting sound. It is due to multiple reflections of sound waves.

→ Reverberation time: It is defined as the time during which the intensity of sound falls to one million of its original value after the source has stopped producing it.

→ The acoustics of Building: It is that branch of science which deals with the design of big halls and auditoriums so that a speech delivered or music produced in them is distinctly and clearly heard at all places in the building.

Important Formulae:
→ Velocity of wave: v = vλ
v = frequency of oscillator generating the wave
λ = wavelength of the wave
v = velocity of wave

→ Velocity of transverse wave in a string:
v = \(\sqrt{\frac{T}{m}}=\sqrt{\frac{T}{\pi r^{2} \rho}}\), where

ρ = density of the material of string
r = radius of string
T = tension Applied on the string
m = mass per unit length of the string

→ Newton’s form ula for velocity of sound in air:
v = \(\sqrt{\frac{P}{\rho}}\)
P = air pressure
ρ = density of air

→ Velocity of elastic waves or longitudinal waves in a medium is:
v = \(\sqrt{\frac{E}{\rho}}\)
E = coefficient of elasticity of the medium
ρ = density of the medium

→ Leplace’s formula for velocity of sound is air/gases:
v = \(\sqrt{\frac{\gamma \mathrm{P}}{\rho}}\) where

E = γP = adiabatic elasticity of air/gas
ρ = density of air/gas
γ = C_P/C_V.

→ Velocity of wave in gas/liquid medium (Longitudinal wave):
V = \(\sqrt{\frac{Y}{\rho}}\), where

Y = Young’s modulus
ρ = coefficient of rigidity

→ Velocity as a function of:
1. temperature, \(\frac{v_{1}}{v_{2}}=\sqrt{\frac{T_{1}}{T_{2}}}\)

2. density, \(\frac{v_{1}}{v_{2}}=\sqrt{\frac{\rho_{1}}{\rho_{2}}}\)

→ The equation of a plane simple harmonic wave (progressive wave) travelling from left to right is:
y = A sin 2π(\(\frac{\mathrm{t}}{\mathrm{T}}-\frac{\mathrm{x}}{\lambda}\))
= A sin \(\frac{2 \pi}{\lambda}\)(vt – x)
= A sin (ωt – kx)
and from right to left i.e. along – X axis is obtained by replacing
x = -x, i.e. y = A sin \(\frac{2 \pi}{\lambda}\)(vt – x)

→ Phase difference = \(\frac{2 \pi}{\lambda}\) × path difference
or
ΔΦ = \(\frac{2 \pi}{\lambda}\) × Δx

→ Total energy transmitted per Unit volume in waves is given by
E = 2π² ρ v² A²
= \(\frac{2 \pi^{2} \rho v^{2} A^{2}}{\lambda^{2}}\)

→ Intensity of wave = \(\frac{2 \pi^{2} \rho v^{2} A^{2}}{\text { area } \times \text { time }}\)

→ I_max = (A₁ + A₂)².

→ I_min = (A₁ – A₂)².

→ Apparent frequency of sound when:
1. Source moves towards listener at rest is
ν’ = \(\frac{v}{v-v_{s}}\)ν

2. When source moves away from listener at rest is
ν’ = \(\frac{v}{v+v_{s}}\)ν

3. When listener moves towards source at rest is
ν’ = \(\frac{\mathbf{v}+\mathbf{v}_{0}}{\mathbf{v}}\)ν

4. When listener moves away from source at rest is
ν’ = \(\frac{\mathbf{v}-\mathbf{v}_{0}}{\mathbf{v}}\)ν

5. When both source and listener move towards each other
ν” = \(\frac{v-v_{0}}{v+v_{s}}\)ν

6. If both move away from each other, then
ν” = \(\frac{v-v_{0}}{v+v_{s}}\)ν

→ Sabine’s formula for reverberation time is
t = \(\frac{0.166 \mathrm{~V}}{\sum \alpha \mathrm{s}}\), Where
k = constant
V = volume of the hall
α = coefficient of absorption
s = area exposed to sound

→ Particle velocity at any instant in a progressive wave is
v = v_o cos 2π (\(\frac{t}{T}-\frac{x}{\lambda}\))
Where v_o = \(\frac{2 \pi}{\lambda}\) A = 2πAv
= velocity amplitude.

→ Particle acceleration at any instant of time in a progressive wave is
where a_o = a_o sin 2π (\(\frac{t}{T}-\frac{x}{\lambda}\))
where a_o = 4π² v² = ω²
= acceleration amplitude.

By going through these CBSE Class 11 Physics Notes Chapter 14 Oscillations, students can recall all the concepts quickly.

Oscillations Notes Class 11 Physics Chapter 14

→ All oscillatory motions are periodic motions but all periodic motions may not be oscillatory.

→ Oscillatory or Vibratory motions are harmonic motions of the simplest type, so they are called simple harmonic motions (S.H.M.).

→ Simple Harmonic Motion is defined as the projection of a uniform circular motion on any diameter of a cycle of reference.

→ The periodic motions are described by fundamental concepts of period$, frequency, amplitude, and displacement.

→ v is the number of oscillations per second.

→ A measurable physical quantity that changes with time is called displacement.

→ The phase difference between displacement and velocity is \(\frac{π}{2}\)

→ The phase difference between displacement and acceleration is π.

→ The S.H.M. is characterized by displacement, amplitude, period, frequency, velocity, acceleration, vibration, and phase.

→ Angular frequency (ω) is related to the period and frequency of the motion by: ω = 2πv = \(\frac{2π}{T}\) .

→ One full oscillation back and forth is known as a cycle or a vibration.

→ A liquid in a U-tube set in oscillations executes S.H.M. with a period T = \(\sqrt{\frac{h}{g}}\), where h is the rise or depression of liquid from the mean position in one limb,

→ The velocity amplitude (v_o) of S.H.M. and the acceleration amplitude (a_o) are related as follows:
a_o = ω v_o

→ The necessary and sufficient condition for a particle to execute S.H.M. is that the acceleration is directly proportional to the displacement and is always directed towards the mean position i.e. opposite to the displacement.

→ The work done by a simple pendulum in one complete oscillation is zero.

→ The total energy of S.H.M. is directly proportional to the square of the amplitude.

→ The total energy of S.H.M. is directly proportional to the square of the frequency.

→ The simple pendulum cannot oscillate in weightlessness but the spring can do so.

→ The driving force is a time-dependent force and can be represented by F(t) = f_o cos ωt = f_o cos 2πvt, v = frequency of driving force.

→ Restoring force must act on the particle executing S.H.M.

→ S.H.M. is represented by y = r sin ωt, where y = displacement of the particle, r = amplitude of oscillation of the particle.

→ Velocity of a particle executing S.H.M. is v = rω cosωt = ω\(\sqrt{r^{2}-y^{2}}\)

→ The maximum velocity of the particle is called velocity amplitude (v_o) which is equal to rω.

→ Acceleration of a particle executing a = – ω²y.

→ Acceleration amplitude (i.e. maximum acceleration), a_o = ω²r.

→ The velocity of a particle executing S.H.M. is zero at the extreme position and maximum at the mean position.

→ Acceleration is maximum at the extreme position and zeroes at the mean position.

→ The phase difference between velocity and acceleration is \(\frac{π}{2}\)

→ The time period of a simple pendulum is independent of its mass.

→ The graph between l and T² is a straight line in the case of a simple pendulum.

→ When length of the spring is made n times, its spring constant becomes \(\frac{1}{n}\) times and hence time period will increase \(\sqrt{n}\) times

→ When spring is cut into n equal pieces, the spring constant of each piece will become n times and hence time period will become \(\frac{1}{\sqrt{n}}\) times.

→ The time period of a simple pendulum is oo at the center of the earth because g = 0 at the center of the earth.

→ The time period of a simple pendulum decreases if it accelerates upward with an acceleration a.

→ The time period of a simple pendulum increase if it accelerates downward with an acceleration ‘a’.

→ The time period of the pendulum increases with an increase in length. If its length is increased n times, its time period becomes \(\sqrt{n}\) times.

→ The time period of a simple pendulum increase when it is immersed in a liquid of density σ.

→ The time period of a simple pendulum increase when the temperature of the wire of the bob is increased.

→ The length of a second’s pendulum is 99.3 cm ≈ 1 m.

→ The time period of a simple pendulum of infinite length is 84.6 minutes. In a medium, all oscillations are damped oscillations as their, amplitude decreases with time.

→ Oscillations of a simple pendulum in a room are damped ones.

→ For resonance, frequency of an applied periodic force = natural frequency of the body.

→ The energy-time graph of damped oscillations is shown in the figure here.

→ Periodic Motion: A motion that repeats itself after regular intervals of time is called periodic motion.

→ Oscillatory or Vibratory Motion: A periodic motion in which a body moves to and fro about a central fixed point (called mean position) is called the oscillatory or vibratory motion of the body of the particle.

→ Driving Force: A time-dependent force applied on an oscillator to increase its vibrations is called a driving force.

→ Second, ’s Pendulum: A pendulum whose time period is 2 seconds is called a second’s pendulum.

→ Undamped Oscillations: The oscillations whose amplitude does not change with time are called undamped oscillations. Such oscillations exist only in a vacuum.

→ Restoring Force: It is defined as the periodic force which comes into play when an object moves away from its equilibrium position while executing S.H.M.

→ Phase: The phase of a vibrating particle at any instant is its state regarding its displacement and direction of vibration at that particular instant. It is denoted by Φ. It is a function of time and is expressed as
Φ = ωt + Φ_o = \(\frac{2π}{T}\)t + Φ_o ………..(1)

→ Epoch: It is defined as the initial phase of the vibrating particle i. e. phase at f = 0. It is denoted by Φ_o. From (1), at t = 0, Φ = Φ_o.

→ Free Vibrations: When a body vibrates with its own natural frequency, it is said to execute free vibrations.

→ Forced Vibrations: When a body is maintained in a state of vibration by a strong periodic force of frequency other than the natural frequency of the body, the vibrations are said to be forced vibrations.

→ Resonant Vibrations: When a body vibrates with a frequency equal to its natural frequency of vibration, then the vibrations are called resonant vibrations.

→ Resonant vibrations are merely a special case of forced vibrations.

→ Coupled system: A system of two or more oscillators linked together in such a way that there is a mutual exchange of energy between them is called a coupled system.

→ Coupled Oscillations: The oscillations of a coupled system are called coupled oscillations.

→ Force Constant of Spring Constant (k): It is defined as the restoring force per unit displacement, i.e. k = \(\frac{F}{x}\), when F = force, x = displacement of particle executing S.H.M.

→ Phase Difference: The difference in phase angles of two positions of a body or oscillator in periodic motion is called phase difference.

→ Amplitude: The maximum displacement on either side of the mean position of the particle executing S.H.M. is known as the amplitude (A) of the particle.

→ Displacement: Displacement is the change in the position with time from the mean position of oscillatory motion.

→ Period of periodic motion: The smallest time interval after which the process repeats itself is called the period of the periodic motion (T) i.e. It is the time required for one complete cycle or oscillation.

→ Frequency: The reciprocal of the Time Period of motion is known as the frequency. It is the number of oscillations per second, v = \(\frac{1}{T}\).

Important Formulae:
→ Angular frequency/Angular velocity, ω = 2πv = \(\frac{2π}{T}\)

→ Displacement in S.H.M., y(t) = r sin (ωt + Φ_o).

→ Velocity of particle in S.H.M., v(t) = rωcos (ωt + Φ_o) = ω\(\sqrt{r^{2}-y^{2}}\)

→ Acceleration in S.H.M., a(c) = – rω² sin (ωt – Φ_o) = – ω²y.

→ Time period of a particle in S.H.M. is
T = 2π\(\sqrt{\frac{y}{a}}\) = 2π\(\sqrt{\frac{\mathrm{m}}{\mathrm{k}}}\)

= 2π\(\sqrt{\begin{array}{c}
\text { inertia factor } \\
\hline \text { spring constant }
\end{array}}\)

→ Time period of a mass m suspended by two springs connected in parallel.
T = 2π\(\sqrt{\frac{\mathrm{m}}{\mathrm{k}_{\mathrm{I}}+\mathrm{k}_{2}}}\)

→ Time period of a mass m suspended by two springs connected in . series.
T = 2π\(\sqrt{\left(\frac{1}{k_{1}}+\frac{1}{k_{2}}\right) m}\)

= 2π\(\sqrt{\frac{m\left(k_{1}+k_{2}\right)}{k_{1} k_{2}}}\)

→ Time period of the simple pendulum, T = 2π\(\sqrt{\frac{l}{\mathrm{~g}}}\)

→ Time period of pendulum decreases if it accelerates upward with an acceleration ‘a’ s.t.
T = 2π\(\sqrt{\frac{l}{g+a}}\)

→ Time period of a simple pendulum increases if it accelerates downward with an acceleration ‘a’ s.t.
T = 2π\(\sqrt{\frac{l}{g-a}}\)

→ Time period of a simple pendulum immersed in a liquid of density σ is T’ = T\(\sqrt{\frac{\rho}{\rho-\sigma}}\) = 2π\(\sqrt{\frac{l}{g} \frac{\rho}{(\rho-\sigma)}}\) , ρ = density of the material of the bob.

→ Increase in time period of a simple pendulum with increase in temperature = \(\frac{\alpha \mathrm{d} \theta \mathrm{T}}{2}\), where T = 2π\(\sqrt{\frac{l}{g}}\)

→ Time period of a cylinder floating in a liquid of density ρ_l, is T = 2π\(\sqrt{\frac{h \rho}{\rho_{l} g}}\) , where h = height of the cylinder, ρ = density of the material of the cylinder, ρ_l = density of the liquid.

→ Time period of a liquid in a U-tube is
T’ = 2π\(\sqrt{\frac{h}{g}}\) where h = height of liquid column.

→ Energy of a particle in S.H.M. = 2π² mv² r² = \(\frac{1}{2}\) m ω²r² = \(\frac{1}{2}\) kr², where ω = angular frequency, r = amplitude, m = mass of particle.

→ Time period of oscillation of a ball in the neck of an air chamber under isothermal conditions is:
T = 2π\(\sqrt{\frac{\mathrm{mV}}{\mathrm{EA}^{2}}}=\frac{2 \pi}{\mathrm{A}} \sqrt{\frac{\mathrm{mV}}{\mathrm{P}}}\)

Where E = coefficient of elasticity
P = atmospheric pressure
m = mass of ball
V = volume of air in the chamber
A = area of cross-section of the neck of the air chamber
Under isothermal conditions E = P.

→ The mechanical energy of damped oscillations for small damping is given by
E(t) = \(\frac{1}{2}\)kx_m² e^bt/m
where x_m = r = amplitude. It is obtained by replacing r by x_me^bt/2m in the equation of energy of a particle in S.H.M.

→ The frequency of damped oscillations is given by
W’ = \(\sqrt{\frac{\mathrm{k}}{\mathrm{m}}-\frac{\mathrm{b}^{2}}{4 \mathrm{~m}^{2}}}\)

By going through these CBSE Class 11 Physics Notes Chapter 13 Kinetic Theory, students can recall all the concepts quickly.

Kinetic Theory Notes Class 11 Physics Chapter 13

→ The molecules of the ideal gas are point masses with zero volume.

→ P.E. for the molecules of an ideal gas is zero and they possess K.E. only.

→ There is no. intermolecular force for the molecules of an ideal gas.

→ An ideal gas cannot be converted into solids or liquids which is a consequence of the absence of intermolecular force.

→ No gas in the universe is ideal. Gases such as H₂, N₂, O₂, etc. behave very similarly to ideal gases.

→ The behavior of real gases at high temperatures and low pressure is very similar to ideal gases.

→ NTP stands for normal temperature and pressure.

→ STP stands for standard temperature and pressure.

→ STP and NTP both carry the same meaning and they refer to a temperature of 273 K or 0°C and 1 atm pressure.

→ The kinetic theory of an ideal gas makes use of a few simplifying assumptions for obtaining the relation:
P = \(\frac{1}{3}\)ρC² = \(\frac{1}{3} \frac{\mathrm{M}}{\mathrm{V}}\)C² = \(\frac{1}{3} \frac{\mathrm{mn}}{\mathrm{V}}\)C²

where m = mass of each molecule,
n = no. of molecules in the gas.

→ Combined with the ideal gas equation, it yields a kinetic interpretation of temperature
\(\frac{1}{2}\) mC² = \(\frac{3}{2}\) k_BT.

→ Using the law of equipartition of energy, the molar specific heats of gases can be predicted as:
For Monoatomic gases: C_V = \(\frac{3}{2}\) R, C_P = \(\frac{5}{2}\) R, γ = \(\frac{5}{2}\)

For Diatomic gases: C_V = \(\frac{5}{2}\) R, C_P = \(\frac{7}{2}\) R, γ = \(\frac{7}{5}\)

For Polyatomic gases: C_V = 3R, C_P = 4R, γ = \(\frac{4}{3}\)

→ These predictions are in agreement with the experimental values of the specific heat of several gases.

→ The agreement can be improved by including vibrational modes of motion.

→ The mean free path λ is the average distance covered by a molecule between two successive collisions.

→ Brownian motion is a striking confirmation of the kinetic molecular picture of matter.

→ Any layer of gas inside the volume of a container is in equilibrium because the pressure is the same on both sides of the layer.

→ The intermolecular force is minimum for the real gases and zero for ideal gases.

→ Real gases can be liquified as well as solidified.

→ The internal energy of real gases depends on volume, pressure as well as temperature.

→ Real gases don’t obey the gas equation PV = nRT.

→ The volume and pressure of ideal gas become zero at the absolute zero.

→ The molecules of a gas are rigid and perfectly elastic spheres.

→ The molecules of each gas are identical but different from that of the other gases.

→ The molecules of the gases move randomly in all directions with all possible velocities.

→ The molecules of the gas continuously collide with one another as well as with the walls of the containing vessels.

→ The molecular collisions are perfectly elastic.

→ The total energy of the molecules remains constant during collisions.

→ The molecules move with constant velocity along a straight line between the two successive collisions.

→ The density of the gas does not change due to collisions.

→ 1 atm pressure =1.01 × 10⁵ Pa.

→ Maxwell’s law proved that the molecules of a gas move with all possible speeds from 0 to ∞.

→ The no. of molecules having speeds tending to zero or infinity is very very small (almost tending to zero).

→ There is a most probable speed (C_mp) which is possessed by a large number of molecules.

→ C_mp increases with the increase in temperature.

→ C_mp varies directly as the square root of the temperature i.e.
C_mp ∝ \(\sqrt{T}\)

→ Absolute temperature can never be negative.

→ The peak of the no. of molecules (n) versus speed (C) curve corresponds to the most probable speed (C_mp).

→ The number of molecules with higher speeds increases with the rise in temperature.

→ At the constant temperature of the gas, λ decreases with the increase in pressure because the volume of the gas decreases.

→ At constant pressure, the λ increases with an increase in temperature due to the increase in volume.

→ The numerical value of the molar mass in grams is called molecular weight.

→ Law of Gaseous Volumes: It states that when gases react together, they do so in volumes which will be a simple ratio to one another and also to the volumes of product.

→ Law of equipartition of energy: It states that the energy for each degree of freedom in thermal equilibrium is \(\frac{1}{2}\)K_BT.

→ Monoatomic gases: The molecule of a monoatomic gas has three translational degrees of freedom and no other modes of motion.
Thus the average energy of a molecule at temperature T is \(\frac{3}{2}\) K_BT.

→ The total internal energy of a mole of such a gas is
U = \(\frac{3}{2}\)K_B T × N_A = \(\frac{3}{2}\)RT

→ Diatomic Gases: The molecule of a diatomic gas has five translational and two rotational degrees of freedom. Using the law of equipartition of energy, the total internal energy of a mole of such a diatomic gas is
U = \(\frac{5}{2}\) K_BT × N_A = \(\frac{5}{2}\) RT

→ Polyatomic Gases: In general, a polyatomic molecule has three translational and three rotational degrees of freedom and a certain number (0 of vibrational modes. According to the law of equipartition of energy, one mole of such gas has
U = [\(\frac{3}{2}\)K_BT + \(\frac{3}{2}\)KBT + fK_BT]N_A

→ Mean Free Path: Mean free path is the average distance covered between two successive collisions by the gas molecule moving along a straight line.

→ Degree of freedom: It is defined as the number of ways in which a gas molecule can absorb energy.
Or
It is the number of independent quantities that must be known to specify the position and configuration of the system completely.

→ Molar mass: It is defined as the mass of 1 mole of a substance. Molar mass = Avogadro’s no. × mass of one molecule.

→ The law of equilibrium of energy states that if a system is in equilibrium at absolute temperature T, the total energy is distributed equally in different energy modes of absorption, the energy of each mode being equal to \(\frac{1}{2}\)K_BT. Each translational and rotational degree of freedom corresponds to one energy model of absorption and has energy \(\frac{1}{2}\)K_BT. Each vibrational frequency has two modes of energy (Kinetic and Potential) with corresponding energy equal to 2 × \(\frac{1}{2}\)K_BT = K_BT.

Important Formulae:
→ K.E./mole of a gas = \(\frac{1}{2}\)MC² = \(\frac{3}{2}\)RT
K.E./molecule = \(\frac{1}{2}\)mC² = \(\frac{3}{2}\)k_B T

C_rms = \(\sqrt{\frac{3 P}{\rho}}=\sqrt{\frac{3 P V}{M}}\)
γ = \(\frac{\mathrm{C}_{\mathrm{P}}}{\mathrm{C}_{\mathrm{v}}}\)

→ PV = nRT is ideal gas equation.

→ PV = rT is gas equation for one gram of gas.
where r = \(\frac{\mathrm{R}}{\mathrm{M}}\),
M = molecular weight of the gas.

→ The gases actually found in nature are called real gases.

→ Real gases don’t obey Boyle’s law at all temperature.

→ The mean free path is given by:
γ = \(\frac{1}{\sqrt{2 \pi n d^{2}}}\)
= \(\frac{\mathrm{m}}{\sqrt{2} \pi \mathrm{d}^{2} \mathrm{mn}}=\frac{\mathrm{m}}{\sqrt{2} \pi \mathrm{d}^{2} \rho}\)
Where ρ = mn = mass/volume of the gas
= density of gas.

d = diameter of molecule.
n = number densisty = \(\frac{\mathrm{N}}{\mathrm{V}}\)

Also P = \(\frac{\mathrm{RT}}{\mathrm{V}}=\frac{\mathrm{N}}{\mathrm{V}} \frac{\mathrm{R}}{\mathrm{N}}\)T = nkT

∴ n = \(\frac{\mathrm{P}}{\mathrm{kT}}\)

∴ λ = \(\frac{\mathrm{kT}}{\sqrt{2} \pi \mathrm{d}^{2} \mathrm{P}}\)

→ Graham’s law of diffusion:
\(\frac{\mathrm{R}_{1}}{\mathrm{R}_{2}}=\sqrt{\frac{\mathrm{M}_{2}}{\mathrm{M}_{1}}}\)
where R₁ and R₂ are diffusion rates of gases 1 and 2 having molecular masses M₁ and M₂.

By going through these CBSE Class 11 Physics Notes Chapter 12 Thermodynamics, students can recall all the concepts quickly.

Thermodynamics Notes Class 11 Physics Chapter 12

→ A thermodynamic system is a collection of a large no. of atoms or molecules confined within the boundaries of a closed surface so that it has definite values of P, V, and T.

→ Work is done during expansion or contraction of the system and is given by dW = PdV where dV = change in volume at constant pressure P.

→ The temperature of the system decreases during expansion and increases during contraction.

→ The slope of the adiabatic curve is steeper than that of the isothermal curve.

→ W_iso > W_adia during expansion if the initial (V_t) and final (V_f) volumes are the same in both the cases.

→ Work done during isothermal compression is less than that during adiabatic compression if V_t and V_f are the same in both cases.

→ Δp = 0 in isobaric process and ΔV = 0 for an isochoric process.

→ Heat engines are devices that convert heat into work.

→ The refrigerator is regarded as a heat engine in the reverse direction.

→ 1 litre = 10^-3 m³.

→ SI and G.G.S. unit of heat capacity is JK^-1 and cal/°C respectively.

→ η of Carrot heat engine is independent of the nature of the working substance.

→ C_P – C_V is constant for all gases.

→ C_P/C_V is not constant for all gases.

→ C_P/C_V has different values for mono, di, and triatomic gases.

→ U for a system is the unique function of the state of the system i.e. U is a unique function of P, V, T.

→ The refrigerator absorbs heat from the cold reservoir and rejects the heat to the hot reservoir..

→ The liquid used as a working substance in the refrigerator is called refrigerant.

→ The most commonly used refrigerants are pheon (dichlorodifluoromethane), SO₂ and ammonia.

→ Freon or SO₂ are used in household refrigerators.

→ NH₃ is used for large-scale refrigeration.

→ U for real gas depends on T and V i.e. U = f (T, V).

→ U for ideal gas depends only on T i.e. U = f (T).

→ For isothermal process, dU = 0 and dQ = dW.

→ For an adiabatic process, dQ = 0 and dU = – dW.

→ PV^γ = constant for an adiabatic process.

→ Open system: The system which can exchange energy with the surroundings is called an open system.

→ Closed system: The system which cannot exchange energy with its surroundings is called a closed system.

→ The first law of thermodynamics: According to this law, the total energy of an isolated system remains the same. However, it can change the form, Mathematically,
ΔQ = ΔU + ΔW

where ΔQ = amount of heat supplied,
ΔU = change in the internal energy and
ΔW = the amount of work done by the system
ΔW = ΔQ – ΔU.

→ Zeroth law of thermodynamics: If two given bodies are in thermal equilibrium with a third body individually, then the given bodies will also be in thermal equilibrium with each other.

→ The second law of thermodynamics:

1. It is impossible to get a continuous supply of work from a body by cooling it to a temperature lower than that of its surroundings.
2. In other words, a perpetual motion of the second kind is impossible without doing anything else.
3. It is impossible to make heat flow from a body at a lower temperature to a body at a higher temperature without doing any work.
4. It is impossible to construct a device that can without other effect lift one object by extracting internal energy from another.

→ Isothermal process: The variation of P with V at T remaining constant is called the isothermal process.

→ Isobaric process: A process in which volume (V) and temperature (T) vary but the pressure (P) remains constant is known as the isobaric process.

→ Isochoric process: A process in which volume remains constant but P and T can change is known as the isochoric process.

→ Adiabatic process: A process in which the total heat content of the system (Q) remains conserved when it undergoes various changes is called an adiabatic process.

→ Indicator diagram: The graph between (P) and volume (V) of a thermodynamic system undergoing certain changes is called a P-V diagram or an indicator diagram as it is drawn with the help of a device called an indicator.

→ Non-cyclic process: A process in which a system after undergoing certain changes does not return to its initial state is called a non-cyclic process.

→ Cyclic process: A process in which a system after undergoing certain changes returns to its initial state is called a cyclic process.

→ External combustion engine: An engine in which fuel is burnt in a separate unit than the main engine is called an external combustion engine.

→ Internal combustion engine: An engine in which the fuel is burnt within the working cylinder of the engine is called an internal combustion engine.

→ Heat engine: A device that uses thermal energy to deliver mechanical energy is called a heat engine.

→ Heat reservoir: A source of heat at constant temperature is called a heat reservoir.

→ Heat sink: A sub-system of the engine in or out of it in which unspent heat is rejected at constant temperature for use is called a heat sink.

→ Working substance: A substance that receives some heat from a source and after converting a part of it into work rejects the remaining heat into the sink. Gas, steam are usual working substances in an engine.

→ Critical pressure: It is the pressure that is necessary to produce liquefaction at the critical temperature.

→ Critical volume: It is the volume of 1 mole of a gas at the critical temperature and critical pressure.

→ Real gas: The real gases are those in which the molecular energy is both kinetic and potential due to attraction between the molecules.

→ Ideal or perfect gas: A gas in which intermolecular attractive force is zero and energy of molecules is only kinetic are called ideal or perfect gases.

→ The internal energy of a perfect gas depends only on its temperature and not on its volume.

→ Phases: The existence of a substance in liquid, vapor, or solid-state are known as three phases of a substance on a given pressure-temperature graph.

→ Phase diagram: The way of showing different phases of substance on a pressure-temperature graph is known as a phase diagram.

→ Reversible process: A process is said to be reversible when the various stages of an operation to which it is subjected can be traversed back in the opposite direction in such a way that the substance passes through exactly the same conditions at every step in the reverse process as in the direct process.

→ Irreversible process: A process in which any one of the conditions stated for the reversible process is not fulfilled is called an irreversible process.

Important Formulae:
→ Equation of state for an ideal gas of μ moles is
PV = μRT

→ Equation of state for a real gas is
(P + \(\frac{\mathrm{a}}{\mathrm{V}^{2}}\))(V – b) = RT

→ Internal energy of the gas molecules is given by
U = KE. + P.E.

→ First law of thermodynamics is the law of conservation of energy and is mathematically expressed as
dQ = dU + dW
= dU + PdV.

→ Work done during isothermal and adiabatic processes are given by

1. W_iso = 2.303 RT log10(\(\frac{\mathrm{V}_{2}}{\mathrm{~V}_{1}}\))
2. W_adia = \(\frac{R}{\gamma-1}\)(T₁ – T₂)

→ Efficiency of heat engine is given by
η = \(\frac{\mathrm{W}}{\mathrm{Q}_{1}}=\frac{\mathrm{Q}_{1}-\mathrm{Q}_{2}}{\mathrm{Q}_{1}}\)

= 1 – \(\frac{\mathrm{Q}_{2}}{\mathrm{Q}_{1}}\) = 1 – \(\frac{\mathrm{T}_{2}}{\mathrm{T}_{1}}\)

where Q₁ = heat absorbed from the source at temperature T₁
Q₂ = heat rejected to the sink at temperature T₂.

→ P₁V₁ = P₂V₂ for an isothermal process.

→ P₁V₁^r = P₂V₂^r for an adiabatic process.

→ Coefficient of performance of refrigerator is given by
β = \(\frac{\text { Heat extracted from cold body }}{\text { Work doneon the refrigerant }}\)
= \(\frac{\mathrm{Q}_{2}}{\mathrm{~W}}=\frac{\mathrm{Q}_{2}}{\mathrm{Q}_{1}-\mathrm{Q}_{2}}\)

→ In a true camot cycle,
\(\frac{\mathrm{Q}_{2}}{\mathrm{Q}_{1}}=\frac{\mathrm{T}_{2}}{\mathrm{~T}_{1}}\)

∴ β = \(\frac{\mathrm{T}_{2}}{\mathrm{~T}_{1}-\mathrm{T}_{2}}\)

→ C_P – C_V = \(\frac{\mathrm{r}}{\mathrm{J}}\)

→ Work done is given by
dW = PdV(J) = \(\frac{\mathrm{PdV}}{4.2}\)cal.

→ Internal energy gained or lost by a perfect gas is
ΔU = n C_VΔT.

→ For isochoric process,
ΔQ = n C_u ΔT.

→ For isobaric process,
ΔQ = n C_P ΔT.

By going through these CBSE Class 11 Physics Notes Chapter 11 Thermal Properties of Matter, students can recall all the concepts quickly.

Thermal Properties of Matter Notes Class 11 Physics Chapter 11

→ Heat is the thermal energy that transfers from a body at a higher temperature to the other body at a lower temperature.

→ Temperature is the property of a body that determines whether or not it is in thermal equilibrium with its surroundings.

→ Thermometry is the branch of heat that deals with the measurement of temperature.

→ S.I. Unit of coefficient of thermal expansion in K^-1.

→ The volume of water decreases with the increase in temperature from 0°C to 4°C. It is called the anomalous expansion of water.

→ The density of water is maximum at 4°C and. its maximum value is 1 g cm^-3 or 10³ kg m^-3.

→ Water (0° to 4°C) and silver iodide (80°C to 141°C) contract on heating.

→ Quartz, pyrex glass, fused silica and invar neither expand nor contract on heating.

→ On a freezing, the volume of ice becomes more than that of water in cold countries when the temperature goes below 0°C and thus the pipe expands and may burst.

→ The principle of Calorimetry is:
Heat gained = Heat lost.

→ A sensitive thermometer is one that shows a large change in the position of mercury meniscus for a small change in temperature.

→ The critical temperature is that temperature up to which gas can be liquified by applying pressure alone.

→ Vapour is a gas above the critical temperature and gas is a vapour below the critical temperature.

→ ΔC = ΔK.

→ In order to convert the temperature from one scale to another, the following relation is used :
\(\frac{\mathrm{C}-0}{100}=\frac{\mathrm{F}-32}{180}=\frac{\mathrm{R}-0}{80}=\frac{\mathrm{K}-273.15}{100}\)

→ Ideal gas equation is PV = nRT.

→ Heat Capacity = mC = W = water equivalent.

→ There are three modes of transfer of heat i.e. conduction, convection and radiation.

→ Radiation mode is the fastest mode of heat transfer.

→ A body that neither reflects nor transmits any heat radiation but absorbs all the radiation is called a perfectly black body.

→ Q = mL
where Q = quantity of heat required for a change from one state to another.
L = Latent heat, m = mass of substance.

→ Melting point is a characteristic of the substance and it also depends 7 on the pressure.

→ Skating is possible on snow due to the formation of water below the skates. It is formed due to the increase of pressure and it acts as a lubricant.

→ The change from solid-state to vapour state without passing through the liquid state is called sublimation and the substance is said to be sublime.

→ Solid CO₂ is called dry ice and it sublimes.

→ During the sublimation process, both the solid and the vapour states of a substance coexist in thermal equilibrium.

→ Melting: The change of state from solid to liquid is called melting.

→ Fusion: The change of state from liquid to solid is called fusion.

→ Melting point: The melting point is the temperature at which the solid and the liquid states of the substance co-exist in thermal equilibrium with each other.

→ Regelation: Regelation is the process of refreezing.

→ Vaporisation: Change of state from liquid to vapour is called vaporisation.

→ Boiling point: Boiling point is the temperature at which the liquid and the vapour states of the substance co-exist in thermal equilibrium with each other.

→ Normal melting point: The melting point of a substance at standard atmospheric pressure is called its normal melting point.

→ Normal boiling point: The boiling point of a substance at standard atmospheric pressure is called its normal boiling point.

Important Formulae:
→ \(\frac{T}{T_{t r}}=\frac{P}{P_{t r}}\)

→ Change in length is given by
Δ l = l_o α Δθ

→ Change in area is given by
Δ S = S_o β Δθ.

→ Change in volume is given by
ΔV = V_o Y Δθ.
l_t = l_o(1 + α Δθ).
S_t = S_o (1 + β Δθ).
V_t = V_o (1 + γ Δθ).
where α, β & γ are called coefficient of linear, superficial and volume expansion respectively.

→ Thermal conductivity of a composite rod made of two conductors. of equal lengths and joined in series is given by
K = \(\frac{2 \mathrm{~K}_{1} \mathrm{~K}_{2}}{\mathrm{~K}_{1}+\mathrm{K}_{2}}\)

→ Temperature of the interface connecting two rods of different lengths d₁ and d₂ is given by
T_o = \(\frac{\mathbf{K}_{1} \mathrm{~d}_{2} \theta_{1}+\mathrm{K}_{2} \mathrm{~d}_{1} \theta_{2}}{\mathbf{K}_{2} \mathrm{~d}_{1}+\mathrm{K}_{1} \mathrm{~d}_{2}}\)
and
T_o = \(\frac{\mathrm{K}_{1} \theta_{1}+\mathrm{K}_{2} \theta_{2}}{\mathrm{~K}_{1}+\mathrm{K}_{2}}\) if their lengths are equal i.e. if d₁ = d₂.

→ If areas of the cross-section are equal, Then
K = \(\frac{\mathrm{K}_{1}+\mathrm{K}_{2}}{2}\)

→ \(\frac{\text { Change in temperature }}{\text { Time }}\) = KΔθ
where Δθ = difference of average temperature and room temperature.

→ Specific heat capacity of a substance is given by
C = \(\frac{\Delta \mathrm{Q}}{\mathrm{M} \Delta \theta}\)
or
ΔQ = MCΔθ.

→ The relation between Kelvin temperature (T) and the celcius temperature t_c is
T = t_c + 273.15.

→ Resistance varies with temperature as:
R_t = R_o(l + α Δθ)
where R_o = resistance at 0°C
R_t = resistance at t°C
α = temperature coefficient of resistance
Δθ = change in temperature.

→ Q = mL, where L = latent heat.

→ Temperature difference Δ°F equivalent to Δ°C is
ΔF = \(\frac{9}{5}\) × ΔC

→ Temperature difference ΔK equivalent to ΔF is
ΔF = \(\frac{9}{5}\)ΔK.

→ T_K = T_c + 273.15

By going through these CBSE Class 11 Physics Notes Chapter 10 Mechanical Properties of Fluids, students can recall all the concepts quickly.

Mechanical Properties of Fluids Notes Class 11 Physics Chapter 10

→ Fluids are substances that can flow e.g. liquids and gases. Fluids don’t possess a definite shape.

→ When a liquid is in equilibrium, the force acting on its surface is perpendicular everywhere.

→ In a liquid, 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, the density of liquid and acceleration due to gravity.

→ Pressure is the same in all directions.

→ If two drops of the same volume but different densities are mixed together, then the density of the mixture is the arithmetic mean of their densities i.e. ρ = \(\frac{\rho_{1}+\rho_{2}}{2}\)

→ The upthrust on a body immersed in a liquid depends only on the volume of the body and is independent of the mass, density or shape of the body.

→ The weight of the plastic bag full of air is the same as that of the empty bag because the upthrust is equal to the weight of the air enclosed.

→ The wooden rod can’t float vertically in a pond of water because the centre of gravity lies above the metacentre.

→ The cross-section of the water stream from a tap decreases as it goes down in accordance with the equation of continuity.

→ The loss in weight of a body = Weight of the fluid displaced by the body.

→ Upthrust = Weight of the liquid displaced.

→ The floating body is in stable equilibrium when the metacentre is above the C.G. (C.G. is below the centre of buoyancy).

→ The floating body is in unstable equilibrium when the metacentre lies below the C.G. (i.e. C.G. is above the centre of buoyancy).

→ The floating body is in the neutral equilibrium when the C.G. coincides with the metacentre {i.e. C.G. coincides with the C.B.).

→ When a gale blows over a roof, the force on the roof is upwards.

→ If a beaker is filled with a liquid of density ρ up to height h, then the mean pressure on the walls of the beaker is \(\frac{\mathrm{h} \rho \mathrm{g}}{2}\)

→ The viscosity of liquids decreases with the rise in temperature i.e.
η ∝ \(\frac{1}{\sqrt{\mathrm{T}}}\)

→ The viscosity of gases increases with the rise in temperature i.e.
η ∝ \(\sqrt{T}\)

→ The streamlined or turbulent nature of flow depends on the velocity of flow of the liquid.

→ Streamline flow is also called laminar flow.

→ Reynolds number is low for liquids of higher viscosity.

→ N_R < 2000 for streamline flow.

→ N_R > 3000 for turbulent flow.

→ N_R lies between 2000 and 3000 for unstable flow.

→ Viscosity is due to the transport of momentum.

→ Bernoulli’s theorem is based on the conservation of energy.

→ Bernoulli’s theorem is strictly applicable to non-viscous fluids.

→ Viscosity arises out of tangential dragging force acting on the fluid layer.

→ Grease is more viscous than honey.

→ The coefficient of viscosity is measured in Nm^-2.

→ Stake’s law can be used to find the size of tiny spherical objects.

→ The flow of fluid under pressure may be zig-zag or in parallel layers of slow velocity in which the velocity vector is parallel at each point of the fluid.

→ The flow of fluid whose velocity varies from point to point is called turbulent flow.

→ The flow of fluid whose velocity at every point remains constant is called streamline flow.

→ For streamline flow, conservation of energy law-holds good and this law is known as Bernoulli’s Theorem.

→ A large number of phenomena like the flow of fluids through constructed pipes, the flight of planes, birds, burners, filter pumps and many other devices work on the principle of Bernoulli’s theorem.

→ The flow of fluids through pipes and capillaries is described by Poiseuille’s formula.

→ Pascal’s law accounts for the Principle of transmission of pressure in fluids.

→ The equation of continuity always holds good which is A₁v₁ = A₂v₂. . The force acting per unit length of the imaginary line drawn on the liquid surface parallel to the surface is called the force of surface tension.

→ Due to surface tension, free surfaces of fluids tend to have minimum surface and so, the liquid drops tend to be spherical and also bubbles are formed in such a film.

→ The free surface has surface energy per unit area equal to surface tension.

→ Free surfaces in tubes, pipes of negligible bore tend to be concave sides which forces the liquid to rise in the capillary.

→ There is the force of pressure inside a soap bubble equal to \(\frac{4 \mathrm{~T}}{\mathrm{R}}\) due to two surfaces in the bubble.

→ Practical use of surface tension made in the capillary rise of liquids f (rise of ink in fountain pen) and cleaning of other stains by detergents.

→ Molecular forces don’t obey the inverse square law.

→ Molecular forces are of electrical origin.

→ Work done in forming a soap bubble of radius R is 8πR²T, where T = surface tension.

→ The angle of contact increases with the rise in temperature and it decreases with the addition of soluble impurities.

→ The angle of contact is independent of the angle of inclination of the walls.

→ The materials used for waterproofing increase’s the angle of contact e as well as the surface tension.

→ Detergents decrease both the angle of contact as well as surface tension.

→ Surface tension does not depend on the area of the surface.

→ When there is no external force, the shape of a liquid is determined by the surface tension of the liquid.

→ Soap helps in better cleaning of clothes because it reduces the surface tension of the liquid.

→ A liquid having an obtuse angle of contact does not wet the walls of containing vessel.

→When force of adhesion is less than \(\frac{1}{\sqrt{2}}\) times the force of cohesion (F_A < \(\frac{\mathrm{F}_{\mathrm{c}}}{\sqrt{2}}\)) the liquid does not wet the walls of vessel and meniscus is convex.

→ The height of a liquid column in a capillary tube is inversely proportional to acceleration due to gravity.

→ Energy is released when the liquid drops merge into each other to form a larger drop.

→ The liquid rises in a capillary tube when angle of contact is acute and F_A > \(\frac{\mathrm{F}_{\mathrm{c}}}{\sqrt{2}}\)

→ The surface tension of molten cadmium increases with the increase in temperature.

→ Surface tension is numerically equal to surface energy.

→ Surface energy is the potential energy of the surface molecules per unit area.

→ The surface tension of lubricants, paints, antiseptics should below so that they may spread easily.

→ C.G.S. and S.L units of rare poise (dyne s cm^-2 or g cm^-1 s^-1 ) and decompose (Nsm^-2 or kg m^-1 s^-1) respectively.

→ 1 decapoise= 10 poise..

→ Thrust: It ¡s defined as the total force exerted by the fluid on any surface in contact.

→ Atmospheric Pressure: It is defined as the weight of a column of air of unit cross-sectional area extending from that point to the top of the atmosphere.
= 1.013 × 10⁵ Pa = 76cm of Hg column.

→ Gauge pressure: It is the difference between absolute pressure and atmospheric pressure.

→Archimede’s Principle: It states that when a body is dipped wholly or partially in a fluid, it loses its weight.

→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 ft behaves like a stretched elastic membrane.

→ Poiseuille’s Formula: According to it, the volume of the fluids flowing through ¡h pipe-isdireçly pLoportona1 to the pressure difference across the ends of the pipe and fourth power of the radius, it is inversely proportional to the coefficient of viscosity and length of the pipe.
i.e. mathematically. V = \(\frac{\pi}{8} \frac{\mathrm{pr}^{4}}{\eta l}\)

→ 1 Torr: It is the pressure exerted by a mercury column of 1 mm in height.

→ Law of Floatation: It states that a body floats in a fluid if the weight of the fluid displaced by the immersed portion of the body is equal to the weight of the body.
i.e V₁ ρ₁ g =V₂ ρ₂ g
or
\(\frac{\rho_{1}}{\rho_{2}}=\frac{V_{2}}{V_{1}}\)
or
\(\frac{\text { density of solid }}{\text { density of liquid }}=\frac{\text { Volume of immersed part of solid }}{\text { Total volume of solid }}\)

→ Force of Cohesion: It is the force of attraction between the molecules of the same substance or the same kind.

→ Force of adhesion: It is the force of attraction between the molecules of different substances.

→ The angle of Contact: It is defined as the angle at which the tangent to the liquid surface at the point of contact makes with the solid surface inside the liquid.

→ Capillarity: It is the phenomenon of rising or fall of a liquid in a capillary tube.

→ Jurin’s Law: It states that the liquid rises more in a narrow tube and lesser in a wider tube.

→ Viscosity: It is the property of fluid layers to oppose the relative motion among them.

→ Coefficient of Viscosity: It is defined as the tangential force required per unit area of the fluid surface to maintain a unit velocity gradient between two adjacent layers.

→ Stoke’s Law: It states that the viscous drag on a spherical body of radius r moving with terminal velocity v_T in a fluid of viscosity r| is given by F – 6πηrv_T.

→ Central line: The line joining the C.G. and centre of buoyancy is called the Central line.

→ Metacentre: It is defined as the point where the vertical line through the centre of buoyancy intersects the central line.

→ Terminal Velocity: It is defined as the constant velocity attained by a spherical body falling through a viscous medium when the net force on it is zero.

→ Pascal’s Law: It states that in an enclosed fluid, the increased pressure is transmitted equally in all possible directions if the effect of gravity is neglected.

→ Streamline: It is defined as the path straight or curved, the tangent to which at any point gives the direction of flow of the liquid at ‘ that point.

→ Tube of flow: It is a bundle of streamlines having the same velocity of liquid elements over any cross-section perpendicular to the direction of flow.

→ Streamline flow: The flow of a liquid is said to streamline flow or steady flow if all its particles pass through a given point with the same velocity.

→ Turbulent flow: The flow of a liquid in which the velocity of all particles crossing a given point is not the same and the motion of fluid becomes disorderly is called turbulent flow,

→ Laminar flow: The flow is said to be laminar if the liquid flows over a horizontal surface in the form of layers of different velocities.

→ Critical velocity: It is defined as the maximum velocity of a liquid or fluid up to which the flow is streamlined and above which it is turbulent.

→ Reynolds’ number: It is a pure number that tells about the type of flow. It is the ratio of inertial force and the viscous force for a fluid in motion.

→ Equation of Continuity: It expresses the law of conservation of ‘ mass in fluid dynamics.
i. e. a₁v₁ =a₂v₂ .

→ Bernoulli’s Theorem: It states that the total energy (sum of pressure energy, K..E. and P.E.) per unit mass is always constant for an ideal fluid.
i.e. \(\frac{\mathrm{P}}{\mathrm{\rho}}\) + gh + \(\frac{1}{2}\) v² = constant

→ Surface film: It is the topmost layer of the liquid at rest with a thickness equal to the molecular range.

Important Formulae:
→ Pressure is given by P = \(\frac{F}{A}\).

→ Pressure exerted by a liquid column.
P = hρg

→ Downward acceleration of a body falling down in a fluid
(i.e. effective value of g) is
a = (\(\frac{\text { density of body }-\text { density of fluid }}{\text { density of body }}\))g

→ Pascal’s law, \(\frac{\mathrm{F}_{1}}{\mathrm{a}_{1}}=\frac{\mathrm{F}_{2}}{\mathrm{a}_{2}}\) = Constant.

→ Surface tension, T = \(\frac{F}{l}=\frac{\text { Force }}{\text { Length }}\).

→ Excess of pressure inside an air bubble is
p_i – p_o = \(\frac{2 \mathrm{~T}}{\mathrm{R}}\)

→ Excess of pressure inside a soap bubble is
p_i – p_o = \(\frac{4 \mathrm{~T}}{\mathrm{R}}\)
And inside a liquid drop,
p_i – p_o = \(\frac{2 \mathrm{~T}}{\mathrm{R}}\)

→ Ascent formula is h = \(\frac{2T cosθ}{rρg}\)

→ Shape of drops is decided by using
cos θ = \(\frac{\mathrm{T}_{\mathrm{SA}}-\mathrm{T}_{\mathrm{SL}}}{\mathrm{T}_{\mathrm{LA}}}\)

→ Viscous force is given by
F = – η A \(\frac{\mathrm{d} \mathrm{v}}{\mathrm{dx}}\)

→ Volume of liquid flowing per second is given by
V = \(\frac{\pi \mathrm{pr}^{4}}{8 \eta l}\)

→ Terminal velocity is given by
VT = \(\frac{2}{9} \frac{r^{2}}{\eta}\)(ρ – σ)g
where ρ = density of body
σ = density of liquid (fluid).

→ P + ρgh + \(\) ρv² = constant.
If h = constant, Then
P₁ + \(\) ρv₁² = P₂ + \(\) ρv₂².

→ The weight of the aircraft is balanced by the upward lifting force due to pressure difference
Let mg = Δp × A
or
mg = \(\frac{1}{2}\) ρ(v₁² – v₂²) × A.

→ Inertial force = (avρ)v = av²ρ

→ Viscous force = \(\frac{ηav}{D}\).