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Grade XIPhysics

(Important Concepts)

Units and Measurement Parallax method: Used for measuring large distances.

where b is the basis and 𝛳 is the parallax angle. Applications of Dimensional Analysis:

● Checking the dimensional consistency of equations ● Deducing the relations among physical quantities ● Converting one system of units to another

Motion in A Straight Line

Velocity Acceleration

Velocity = Displacement / Time taken

Its unit is m/s.

Its dimensional formula is [MoT -1].

A vector quantity

Can be positive, zero and negative

Acceleration (a) = Change in velocity (Δv) / Time interval (Δt) Its unit is m/s2.

Its dimensional formula is [MoLT-2 ]. A vector quantity Can be positive, zero or negative

Graphical Representation of Accelerating Bodies:

Distance travelled in nth second of uniformly accelerated motion is given by the relation:

Relative Velocity: The relative velocity of a body A with respect to another body B

is the time rate at which A changes its position with respect to B.

Case 1: Both bodies moving in same direction:

Case 2: The bodies moving in opposite directions:

Motion in A Plane Projectile Motion:

Path of Projectile:

Time of Flight: It is the total time for which the object is in flight.

Maximum Height: It is the maximum height reached by the projectile.

Horizontal Range: It is the distance covered by the object between its point of projection and the point of hitting the ground.

Centripetal acceleration:

Laws of Motion Newton’s First Law: Every body continues to be in its state of rest or uniform motion in a straight line, unless compelled by some external force acting on it.

Newton’s Second Law: The rate of change of linear momentum of a body is directly proportional to the external force applied on the body, and this change takes place always in the direction of force applied. Mathematically,

where, k is a constant of proportionality. In S.I., k = 1

Thus, the second law of motion gives us the measure of force.

Newton’s Third Law: For every action there is an equal and opposite reaction and both acts on two different bodies. Mathematically, F12 = – F21.

Impulse: Impulse of a force is a measure of the total effect of the force

Impulse = Force × Time

Law of Conservation of Momentum: In an isolated system, the vector sum of the linear momenta of all the bodies of the system is conserved and is not affected due to their mutual action and reaction.

Final momentum = Initial momentum

Apparent Weight in a Lift:

(i) When a lift is at rest or moving with a constant speed, then

R = mg

The weighing machine will read the actual weight.

(ii) When a lift is accelerating upward, then apparent weight

R1 = m(g + a)

The weighing machine will read the apparent weight, which is more than the actual weight.

(iii) When a lift is accelerating downward, then apparent weight

R2 = m (g – a)

The weighing machine will read the apparent weight, which is less than the actual weight.

(iv) When lift is falling freely under gravity, then

R2 = m(g – g) = 0

The apparent weight of the body becomes zero.

(v) If lift is accelerating downward with an acceleration greater than g, then body will lift from floor to the ceiling of the lift.

Friction: Friction is the property due to which a force is set up at the surface of contact of two bodies and which prevents any relative motion between them.

Sliding friction is of two types: Static friction and Dynamic friction.

Laws of static friction: ● First law: The magnitude of the limiting force of static friction (Fs) between any two

bodies in contact is directly proportional to the normal reaction (N) between them.

Fs ∝∝ N

Fs = μsN

where μs is proportionality constant and is called the coefficient of static friction.

● Second law: The limiting force of static friction is independent of the area of contact, as long as the normal reaction between two surfaces in contact remains the same.

● Third law: The limiting force of static friction depends on the nature of material of the surfaces in contact.

● Fourth law: The direction of limiting friction force is always opposite to the direction in which one body is at verge of moving over the other body.

Laws of kinetic friction: ● First law: The magnitude of force of kinetic friction (Fk) between any two bodies in

contact is directly proportional to the normal reaction (N) between them.

Fk ∝∝ N

Fk = μkN

where μk is the proportionality constant and is called the coefficient of kinetic friction.

● Second law: The force of kinetic friction is independent of the area of contact, as long as the normal reaction between two surfaces in contact remains the same.

● Third law: The force of kinetic friction depends on the nature of material of the surface in contact.

● Fourth law: The force of kinetic friction is approximately independent of relative velocity, provided relative velocity is neither too high nor too low.

Centripetal Force:

Motion of a Car on a Banked Road:

Work, Energy and Power

Work Done (W) by a Variable Force:

W = Area of ABCDA

Work-Energy Theorem: Work done by a force in displacing a body is equal to change in its kinetic energy.

W = Kf - Ki = KEΔ Regarding the work-energy theorem it is worth noting that (i) If Wnet is positive, then Kf – Ki = positive, i.e., Kf > Ki or kinetic energy will increase and vice-versa. (ii) This theorem can be applied to non-inertial frames also. In a non-inertial frame it can be written as: Work done by all the forces (including the Pseudo force) = change in kinetic energy in non-inertial frame Potential Energy of A Spring:

Potential energy of spring is given as:

Collisions:

● Collision between two or more particles is the interaction for a short interval of time in which they apply relatively strong forces on each other.

● In a collision physical contact of two bodies is not necessary. 1. Elastic collision: The collision in which both the momentum and the kinetic energy of the system remains conserved are called elastic collisions.

● In an elastic collision all the involved forces are conservative forces. ● Total energy remains conserved.

● Elastic Collision in One Dimension:

The ratio of relative velocity of separation after the collision to the relative velocity of the approach before the collision is known as coefficient of restitution or coefficient of resilience.

For perfectly elastic collision, e = 1 Calculation of velocities after collision:

2. Inelastic collision: The collision in which only the momentum remains conserved but kinetic energy does not remain conserved are called inelastic collisions.

● In an inelastic collision some or all the involved forces are non-conservative forces. ● Total energy of the system remains conserved. ● If after the collision two bodies stick to each other, then the collision is said to be

perfectly inelastic.

System of Particles and Rotational Motion Centre of Mass: Centre of mass of a system is the point that behaves as whole mass of the system is concentrated at it and all external forces are acting on it. For rigid bodies, centre of mass is independent of the state of the body i.e., whether it is in rest or in accelerated motion centre of mass will remain same.

Centre of Mass of System of two Particles:

Let us consider a system of two particles having mass m1 and m2. Let the distances of two particles be x1 and x2 respectively from some origin O.

The centre of mass of the system is that point C, which is at a distance X from O, where X is given by,

Centre of Mass of a System of n Particles:

Centre of Mass of a Rigid body Having Continuous Distribution of Mass:

If is the position vector of an elementary portion of mass dm of the body, then the position

vector of the centre of mass is given by,

Moment of Inertia: ● The inertia of rotational motion is called moment of inertia. ● It is denoted by L.

● Moment of inertia is the property of an object by virtue of which it opposes any change in its state of rotation about an axis.

● The moment of inertia of a body about a given axis is equal to the sum of the products of the masses of its constituent particles and the square of their respective distances from the axis of rotation

K.E. of rotation

If ω =1, then

K.E of rotation

I = 2 × K.E. of rotation

Thus, moment of inertia of a body about a given axis is equal to twice the K.E. of rotation of the body rotating with unit angular velocity about the given axis.

Theorem of Perpendicular Axis:

Statement: The moment of inertia of a planar body about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with perpendicular axis and lying in the plane of the body.

Theorem of Parallel Axes:

Statement: The moment of inertia of a body about any axis is equal to the sum of the moments of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of the distance between the two parallel axes.

Torque: ● Torque or moment of a force about the axis of rotation

τ = r x F = rF sinθ n ● It is a vector quantity. ● If the nature of the force is to rotate the object clockwise, then torque is called negative

and if rotate the object anti-clockwise, then it is called positive. ● Its SI unit is ‘newton-metre’ and its dimension is [ML2T -2 ]. ● In rotational motion, torque, τ = Iα

Gravitation Newton’s Law of Gravitation:

● Gravitational force is a attractive force between two masses m1 and m2 separated by a distance r.

● The gravitational force acting between two point objects is proportional to the product of their masses and inversely proportional to the square of the distance between them.

∴

where, G is constant of proportionality known as gravitational constant

● Its S.I value is 6.67 × 10−11 Nm2 kg−2.

● The universal gravitational constant (G) is numerically equal to the force of attraction between two bodies for unit masses, separated by unit distance.

Acceleration Due to Gravity:

● The uniform acceleration produced in a freely falling object due to the gravitational pull of the earth is known as acceleration due to gravity.

● It is denoted by g and its unit is m/s2 .

● Relation between g and a is given by

g = GM / R2

where M = mass of the earth = 6.0 * 1024 kg and R = radius of the Earth = 6.38 * 106 m Factors Affecting Acceleration Due to Gravity: (i) Effect of Altitude: The value of g at height h from earth’s surface

g’ = g / (1 + h / R)2 == g (1 – 2h / R) Therefore, g decreases with altitude. (ii) Effect of Depth: The value of g at depth h A from earth’s surface

g’ = g * (1 – h / R)

Therefore g decreases with depth from earth’s surface. The value of g becomes zero at earth’s centre. Gravitational Potential: Gravitational potential at a point in a gravitational field is defined as the amount of work done in bringing a body of unit mass from infinity to that point without acceleration. ∴ Gravitational potential at any point

Where, W is the amount of work done in bringing a body of mass m0 from infinity to that point. Expression for gravitational potential at a point:

Gravitational Potential Energy: Gravitational potential energy of a body at a point in a gravitational field of another body is defined as the amount of work done in bringing the given body from infinity to that point without acceleration. Expression for gravitational potential energy:

Orbital Velocity of A Satellite: Orbital velocity of a satellite is the minimum velocity required to the satellite into a given orbit around earth.

vo = √GM / r = R √g / (R + h )

where, M = mass of the planet, R = radius of the planet and h = height of the satellite from planet’s surface. If v is the speed of a satellite in its orbit and vo is the required orbital velocity to move in the orbit, then

(i) If v < vo, then satellite will move on a parabolic path and satellite falls back to earth. (ii) If V = vo then satellite revolves in circular path/orbit around earth. (iii) If vo < V < ve then satellite shall revolve around earth in elliptical orbit.

Escape Velocity: Escape velocity on the earth is the minimum velocity with which a body has to be projected vertically upwards from the earth’s surface so that it just crosses the earth’s gravitational field and never returns. Escape velocity of any object:

ve = √2GM / R = √2gR = √8πp GR2 / 3 Escape velocity does not depend upon the mass or shape or size of the body as well as the direction of projection of the body.

Mechanical Properties of Solids Elasticity: It is the property of a body by virtue of which it tends to regain its original size and shape after the applied force is removed.

Plasticity: It is the inability of a body in regaining its original status on the removal of the deforming forces. Examples of plastic materials − bakelite, plastic

Stress: The restoring force or deforming force experienced by a unit area is called stress. Its S.I

unit is Nm−2.

(a) Tensile Stress: When there is an increase in the length or the extension of the body in the direction of the force applied, the stress set up is called tensile stress

Here,

l = Original length, Δl = Increase in length

(b) Compressive Stress: When there is a decrease in the length or the compression of the body due to the force applied, the stress set up is called compressive stress.

Here,

l = Original length, Δl = Increase in length

(c) Tangential or Shearing Stress: When the elastic restoring force or deforming force acts parallel to the surface area, the stress is called tangential stress.

Strain: Ratio of change in configuration to the original configuration

Strain =

It is a dimensionless quantity.

(a) Longitudinal Strain:

Longitudinal Strain =

(b) Volumetric Strain:

Volumetric Strain =

(c) Shearing Strain: An angle (in radian) through which a plane perpendicular to the fixed surface of the cubical body gets turned under the effect of a tangential force.

Shearing Strain

Hooke’s Law: For small deformations, stress and strain are proportional to each other

Stress α strain

Stress = k × strain

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

Stress-strain curve for a Wire:

Modulus of Elasticity: We know,

= k = constant Here, k is known as the modulus of elasticity. Types of Modulus of Elasticity: (a) Young’s modulus of elasticity (Y):

Y =

∴ Y = Here, F = force applied, r = radius of the wire, l = original length, Δl = change in length Its unit is Nm−2 or Pascal (denoted by Pa). (b) Bulk modulus of elasticity (B):

B =

B = If P is the increase in pressure applied on the spherical body, then P = F/a

∴ B = Here, F = force applied, a = area of the object, V = original volume, ΔV = change in volume Its unit is Nm−2 or Pascal. Compressibility (k): It is the reciprocal of bulk modulus of elasticity (B). i.e., k = 1/B (c)Modulus of rigidity or shear modulus of elasticity (G)

G =

Shearing strain =θ = Tangential stress = F/a

∴ G = Here, F = force applied, a = area, L = original length, ΔL = change in length Its unit is Nm−2 or Pascal. Poisson's ratio(σ):

Poisson's ratio(σ) = lateral strain/ longitudinal strain ● Poisson's ratio (σ) is a unitless and dimensionless quantity. ● The value of Poisson's ratio for isotropic materials lies in the range [−1,0.5]. ● For practical purposes, the value of Poisson's ratio is always positive and lies in range

[0.2,0.4].

Mechanical Properties of Fluids Pascal’s Law:

● Pressure inside a fluid at rest is same at all points if they are at the same height.

● Pressure exerted in all directions in a fluid at rest is same.

Hydraulic Lift:

Force supported by the large piston, F2 = PA2

F2 Since A2 is greater than A1, force F2 on the larger piston will also be much larger than the force F1

applied on the smaller piston. Mechanical advantage of the device is . Equation of Continuity: A1v1 = A2v2 Bernoulli’s Principle:

The sum of pressure energy, kinetic energy, and potential energy per unit mass is always constant for the streamline flow of a non-viscous and incompressible fluid.

General expression for Bernoulli’s equation,

Torricelli’s Law:

Speed of efflux (fluid outflow) is same as the speed of a free falling body.

Velocity of efflux, v = √2gh where, h = depth of orifice below the free surface of liquid

Horizontal range, S = √4h(H — h) where, H = height of liquid column Horizontal range is maximum, equal to height of the liquid column H, when orifice is at half of the height of liquid column Venturimeter:

Viscosity: It is the resistance of the fluid motion. This force exists when there is relative motion between the layers of liquid. Coefficient of viscosity is given as:

Stoke’s Law: When a small spherical body falls in a long liquid column, then after sometime it falls with a constant velocity, called terminal velocity. When a small spherical body falls in a liquid column with terminal velocity then viscous force acting on it is

F = 6πηrv where, r = radius of the body, V = terminal velocity and η = coefficient of viscosity This is called Stoke’s law. Terminal Velocity (vt): When a spherical body falls through a viscous fluid, it experiences a viscous force. The magnitude of viscous force increases with the increase in velocity of the falling body under the action of its weight. As a result, the viscous force soon balances the driving force (weight of the body) and the body starts moving with a constant velocity known as its terminal velocity.

Reynolds Number: Reynolds (Re) number implies if the flow would be turbulent or not.

Surface Tension: Surface tension is the force acting per unit length on either side of the imaginary line.

Relation between Surface Energy and Surface Tension:

Work done = σ × Increase in area

Excess Pressure Inside a Liquid Drop:

Excess Pressure Inside a Soap Bubble:

Capillary Rise:

● Pressure difference between two sides of the top surface: Pa − P 0 = 2S /r

● Consider points A and B. They should be at the same pressure.

● Height of water rise,

Thermal Properties of Matter Ideal Gas Equation:

Where, P − Pressure, V − Volume, T − Temperature, μ − Number of moles, R− Universal gas constant (8.31 J mol−1K−1)

Heat capacity (S): The change in temperature (Δt) of a substance when heat is absorbed or rejected (ΔQ) by it is characterised by a quantity called the heat capacity.

Specific heat capacity (s) of a substance determines the change in temperature when a given amount of heat is absorbed or rejected per unit mass of the substance.

SI unit is J kg−1K−1

Molar specific heat (C): When the amount of substance is specified in moles (μ) instead of mass

SI unit is J mol−1K−1

● Heat transfer at constant pressure is called molar specific heat capacity at constant pressure (Cp).

● Heat transfer at constant volume is called molar specific heat capacity at constant volume (Cv).

Thermal expansion: Increase in dimension of a body due to heat (a) Linear expansion (Δl): Expansion of length

Linear expansion ∝ Original length × Temperature change

→ Coefficient of linear expansion (characteristic property of material)

(b) Area expansion (ΔA) − Expansion in area

Δ A ∝ Original area × Temperature change

α A → Coefficient of area expansion

● Volume expansion (ΔV) − Expansion in complete volume of body

Δ V ∝ Original volume × Temperature change

α V → Coefficient of volume expansion Relation between αl, αA, and αV:

Pressure- Temperature Diagram for Co2:

Latent Heat: It is the amount of heat energy required to change the state of a unit mass of a substance from solid to liquid or from liquid to vapour, without a change in temperature. Quantity of heat required,

Where, L → Latent heat A plot of temperature versus heat energy for a quantity of water is shown in the figure.

Thermal Conductivity: Conduction is the mechanism of heat transfer due to temperature difference between two adjacent parts of a body.

Heat flows by conduction in the bar. In this steady state, the rate of heat (H) flow,

Where K → Thermal conductivity ● Thermal conductivity − Metals have high thermal conductivity whereas non-metals and

gases have low thermal conductivity. Newton's Law of Cooling: Rate of cooling of a body is directly proportional to the temperature difference between the body and surroundings (T2 − T1) [provided the temperature difference is small]

Where, k → a positive constant. k depends on area and nature of the surface of the body.

Calorimetry: Calorimetry means measurement of heat. When a body at higher temperature is brought in contact with a body at lower temperature, the heat lost by the former is equal to the heat gained by the latter (no heat should escape to the surroundings). Calorimeter A device used for heat measurement is called a calorimeter.

Heat gained by the water and the calorimeter = (m1 + w) (t − t1) Heat lost by the substance = s. m2 (t2 − t)

According to calorimetry principle, (m1 + w) (t − t1) = s.m2 (t2 − t)

Thermodynamics

Zeroth Law of Thermodynamics: When two systems are separately in thermal equilibrium with a third system, they are also in thermal equilibrium with each other.

First Law of Thermodynamics: According to first law of thermodynamics, when an amount of heat ΔQ is added to a system, a part of it increases its internal energy by ΔU and the remaining part is used up as the external ΔW done by the system.

ΔQ = ΔU + ΔW

Specific Capacity of Gas: For gases, we can define two specific heats:

Types of Thermodynamic Processes:

Isothermal Process: Occurs at a constant temperature; PV = Constant Work done for the entire process

Adiabatic Process: During this process, no heat enters or leaves the thermodynamic system during the change. System is insulated from its surroundings. PVγ = Constant γ = Ratio of specific heats

The following graph shows the P-V curves of an ideal gas for two adiabatic processes connecting two isotherms.

Work done for this process can be given as

Isochoric Process:

● Occurs at constant volume. ● No work is done on or by the gas. ● Heat absorbed by the gas is used for changing the internal energy and temperature of the

gas. Isobaric Process

● Occurs at constant pressure. ● Work done, W = P (V2 − V1) = μR(T2 − T1)

● Change in temperature also changes the internal energy. ● Heat absorbed is used partly for increasing internal energy and partly for doing work.

Second Law of Thermodynamics: Kelvin−Planck Statement

It is not possible to design a heat engine which works in cyclic process and whose only result is to take heat from a body at a single temperature and convert it completely into mechanical work.

Clausius Statement

It is impossible for a self-acting machine, unaided by any external agency, to transfer heat from a body at lower temperature to another at higher temperature.

Carnot Engine: A reversible heat engine operating between two temperatures is called a Carnot engine.

The sequence of steps constituting one cycle of a Carnot engine is called a Carnot cycle.

Heat Engines: Heat engine is a device which converts heat energy into mechanical energy.

Efficiency (η) of a Heat Engine:

W = Q1 − Q2

Refrigerators or Heat Pumps: Reverse of heat engine.

Coefficient of performance (α),

α can be greater than 1.

Kinetic Theory Boyle’s Law: At constant temperature the volume (V) of given mass of a gas is inversely proportional to its pressure (p), i.e.,

V ∝∝ 1/p ⇒⇒ pV = constant For a given geas, p1V1 = p2V2

Charles’ Law: At constant pressure the volume (V) of a given mass of gas is directly proportional to its absolute temperature (T), i.e.,

V ∝∝ T ⇒⇒ V / T = constant For a given gas, V1/T1 = V2/T2 Expression For Pressure Due to an Ideal Gas:

Kinetic Interpretation of Temperature:

Gay Lussac's’ or Regnault’s Law: At constant volume the pressure p of a given mass of gas is directly proportional to its absolute temperature T, i.e. ,

p ∝∝ T ⇒⇒ V/T = constant For a given gas, p1/T1 = p2/T2 At constant volume (V) the pressure p of a given mass of a gas increases or decreases by 1/273.15 of its pressure at 0°C for each l°C rise or fall in temperature. Volume of the gas at t°C,

pt = p0 (1 + t/273.15) where P0 is the pressure of gas at 0°C. Avogadro’s Law: Avogadro stated that equal volume of all the gases under similar conditions of temperature and pressure contain equal number molecules. This statement is called Avogadro’s hypothesis.

Law of Equipartition of Energy: A dynamic system in thermal equilibrium has the energy system equally distributed amongst the various degrees of freedom and the energy

associated with each degree of freedom per molecule is . Where, kB − Boltzmann constant, T − Temperature, Monoatomic gas: It has three degrees of freedom for translational motion. Mean kinetic energy

of translational motion of gas is Components along 3 axes:

Average kinetic energy for each degree of freedom is

● Translational and rotational degree of freedom of molecule contributes to the energy.

● Vibrational mode has both kinetic and potential energy. Therefore, each vibrational

frequency contributes Mean Free Path: The average distance travelled by a molecule between two successive collisions is called mean free path (γ).

γ = kT / √2 π σ2 p (derivation is important) where σ = diameter of the molecule, p = pressure of the gas, T= temperature and k = Boltzmann's constant Mean free path λ ∝ T and λ ∝ 1/p.

Oscillations Energy in Simple Harmonic Motion:

● Kinetic energy (Ek) of a particle is given as

● Potential energy (Ep) of a particle is given as

● Total energy of the system is given as

E = Ek + Ep

Time period of Simple Pendulum:

Seconds Pendulum: Period of a simple pendulum is given by . For a seconds pendulum, the time period is given by T = 2 seconds.

Length of the seconds pendulum,

Characteristic of Simple Harmonic Motion (SHM): (a) Displacement:

In Δ OPM,

Consider the particle has some phase (Φ0) initially.

Here, θ = ωt + Φ0

● Amplitude − Maximum displacement on either side of the mean position Maximum value of y is a. (b) Velocity: Velocity is obtained by directly differentiating displacement with respect to time (t). y(t) = A sin (ωt + Φ0) …(i) A = Amplitude

(c) Acceleration: It is obtained by differentiating velocity [v(t)] with respect to time (t).

Force Law for SHM:

F(t) = −mω2y(t) = −k y (t)

∴∴ k = mω2

Here, k is the spring constant. ● Force acting in simple harmonic motion is proportional to displacement and is always

directed towards the centre of motion. Period of SHM:

Frequency of SHM:

Oscillations Due to a Spring:

● The angular speed of the spring:

● Time period (T) of the oscillator is,

Forced Oscillations: The motion of a particle under the combined action of a linear restoring force, a damping force and a time-dependent driving force is given by

The displacement after the natural oscillation dies out, x(t) = A cos(ωdt + Φ) Amplitude, A, is the function of the forced frequency (ωd) and the natural frequency, ω. Analysis shows that it is given by

Here, m = Mass of the particle, v0 = Velocity of the particle (at t = 0), x0 = Displacement of the particle (at t = 0), ω = Natural frequency

Damped Simple Harmonic Motion: A simple harmonic system that oscillates with decreasing amplitude with time is called damped simple harmonic oscillation.

When a mass (m) attached to a spring is released, it settles to a height. When the mass is pulled up/down, the restoring force (Fs) on the spring is proportional to the displacement (x) from its equilibrium position.

According to Newton’s law of motion,

Where, A = Amplitude ω' = Angular frequency of the damped oscillator

● x(t) is not periodic because e−bt/2m decreases continuously with time. ● Mechanical energy is represented as,

● ● For small damping,

Oscillations

Displacement Relation in a Progressive Wave: ● Representation of a sinusoidal wave travelling along the positive x-axis is ●

y(x, t) = a sin(kx − ωt + Φ)

● The equation can also be represented as linear combination of sine and cosine function.

y(x, t) = A sin(kx − ωt) + B cos(kx − ωt), where a = and Φ Where, y(x, t) → Displacement function of position x and time t, k → Propagation constant or angular wave number, ω → Angular frequency, Φ → Initial phase angle, A, B, a → Amplitude Frequency (ν): Number of oscillations per second (Unit → hertz)

ω is angular frequency; unit is rad/s. Speed of a Travelling Wave:

Speed of a Transverse Wave on a Stretched String:

Speed of a Longitudinal Wave in Fluid:

Speed of longitudinal waves in a solid bar:

Newton’s Formula For Velocity of Sound in Gases:

Factors affecting velocity of sound: (i) Effect of density:

Hence, velocity of sound is inversely proportional to the square root of density of gas. (ii) Effect of pressure:

Boyle's Law is obeyed when temperature T is constant, i.e. PV = constant. As M and γ are also constant, v = constant Thus, velocity of sound is independent of pressure, provided the temperature is constant. (iii) Effect of temperature:

So, velocity of sound in a gas is directly proportional to the square root of absolute temperature. (iv) Effect of humidity: If ρ m and vm represent the density of moist air and velocity of sound in moist air, respectively. And ρd and vd represent the density of dry air and velocity of sound in dry air, respectively. Then

Since, water vapour reduces the density of air,

So, vm > vd Hence, velocity of sound in moist air is greater than velocity of sound in dry air. The Principle of Superposition of Waves: If y1, y2, y3,… are the displacements of waves, then the resultant displacement (y) is

y = y1 + y2 + y3 + …

Normal Modes of Oscillation of an Air Column (One end closed and other end open):

● Closed end → Pressure is highest → Displacement of air particles is minimum (zero) → Node is formed (x = 0)

● Open end → Pressure is least → Displacement is maximum → Anti-node is formed (x = L)

● Natural frequencies: normal modes

● Fundamental frequency → for n = 0 → ● Odd harmonics → Higher frequencies → Odd multiples of fundamental frequencies

Beat:

● Two sound waves of nearly the same frequency and amplitude produce beats. ● The resultant sound has alternate maxima and minima.

Beat frequency,

Doppler Effect: Relative motion between a source of sound and a listener causes the apparent frequency of the sound heard by the listener to be different from the frequency of the sound emitted by the source. Apparent frequency of sound heard,

Medium at rest, vm = 0 .

Special Cases

● Source moving towards observer − Observer stationary: vs = + ve and v0 = 0

● Source moving away from observer − Observer stationary:vs = − ve and v0 = 0

● Source stationary − Observer moving away from source: vs = 0 and v0 = + ve

● Source stationary − Observer moving towards source: vs = 0 and v0 = − ve

● Source and observer approaching each other: vs = + ve and v0 = − ve

● Source and observer moving away from each other: vs = − ve and v0 = + ve

Standing Wave and Normal Modes in a Stretched String: (a) Boundary condition of a string:

L = Length of the string

Corresponding Frequency,

Vibrating String

Mode of vibration Harmonic Tone Nodes Anti- nodes Frequency

First or fundamental First Fundamental 2 1

Second Second First 3 2

nth nth (n − 1) tones n + 1 n

Laws of Vibrating Strings: The fundamental frequency of the vibrations in a stretched string,

Here, L is the length of the vibrating string, T is the tension in the vibrating string and μ is the linear mass density of the vibrating string. (1) Law of length: The fundamental frequency of transverse vibration of a stretched string is inversely proportional to the length of the vibrating string. This is subject to the condition that the tension (T) in the string and the linear density (μ) of the string remain constant.

(2) Law of tension: The fundamental frequency of transverse vibration is directly proportional to the square root of the tension (T) in the string if the length of the vibrating string (L) and linear density (μ) are kept constant.

(3) Law of linear density: The fundamental frequency of transverse vibration of a stretched string is inversely proportional to the square root of the linear density of the string if the length of the vibrating string (L) and tension (T) are constant.

But . From equation (1),

If the strings of same material are used ( ), then

If the strings of different materials are used ( ), then

(r1 = r2)

Ray optics and Optical Instruments

Mirror Formula:

Magnification

or, m = -v/u Laws of refraction:

● The incident ray, the normal to the refracting surface at the point of incidence and the refracted ray lie in the same plane.

● The ratio of the sine of the angle of incidence to the sine of the angle of refraction is

constant for two given media. This constant is denoted by and is called the relative refractive index of medium b with respect to medium a.

This law is also called Snell’s Law of Refraction.

Real and Apparent Depths:

Refraction by a Lens:

Assumptions made in the derivation:

● The lens is thin so that distances measured from the poles of its surfaces can be taken as equal to the distances from the optical centre of the lens.

● The aperture of the lens is small. ● The object consists only of a point lying on the principle axis of the lens. ● The incident ray and refracted ray make small angles with the principle axis of the lens.

A convex lens is made up of two convex spherical refracting surfaces. The first refracting surface forms image I of the object O [figure (b)]. Image I1 acts as virtual object for the second surface that forms the image at I [figure (c)]. Applying the equation for spherical refracting surface to the first interface ABC, we obtain

A similar procedure applied to the second interface ADC gives

For a thin lens, BI1= DI1

Adding equations (i) and (ii), we obtain

Suppose the object is at infinity i.e., OB → ∞and DI → f Equation (iii) gives

The point where image of an object placed at infinity is formed is called the focus (F) of the lens

and the distance f gives its focal length. A lens has two foci, Fand , on either side of it by the sign convention. BC1= R1

CD2= −R2

Therefore,equation (iv) can be written as

Equation (v) is known as the lens maker’s formula. From equations (iii) and (iv), we obtain

As B and D both are close to the optical centre of the lens, BO = − u, DI = + v, we obtain

Power of Lens: The ability of a lens to converge or diverge the rays of light incident on it is called the power of the lens.

or,

Combination of Thin Lenses in Contact

Total Internal Reflection: Two essential conditions for total internal reflection:

● Incident ray should travel in the denser medium and refracted ray should travel in the rarer medium.

● Angle of incidence (i) should be greater than the critical angle for the pair of media in contact.

Relation between refractive index and critical angle (C): When, i = C and r = 90°: Applying Snell’s law at A2, μb sin C = μa sin 90° = μa × 1

Refraction through prism:

δ = ( i + e) − (r1+ r2) r1+ r2 = A δ = (i + e) − A When δ = δ m [prism in minimum deviation position], e = i and r2 = r1 = r

Factors affecting angle of deviation

● Angle of incidence (i) ● Angle of prism (A) ● Refractive index ( ) of the material of the prism ● Colour or wavelength ( ) of light

Dispersion: Dispersion is the phenomenon of splitting of light into its component colours.

● Cause of dispersion: Different colours of white light have different wavelengths. The

wavelength of violet light is smaller than that of red light. The refractive index of a material in terms of the wavelength of the light is given by Cauchy’s expression.

Here, a, b and c are constants for the material. ∴ Refractive index for violet light > Refractive index for red light i.e., μv > μ r

For a small-angled prism, we have: δ = A(μ − 1) Since μv> μ r, violet light will have greater deviation than red light. δv> δ r

Angular Dispersion Angular dispersion is the difference in the angle of deviation of two extreme colours in the dispersed beam of light. δv δ r = A (μ v− 1) A (μ r− 1) δv δ r = A (μ v μ r) The unit for angular dispersion is degree or radian. Dispersive Power Dispersive power of a prism is the ratio of angular dispersion between the extreme colours to the deviation of the mean colour produced by the prism. For a thin prism,

Scattering of Light

● Scattering of light takes place when the size of the scattering object is smaller compared to the wavelength of the light.

● Rayleigh’s Law of Scattering: The intensity of light corresponding to the wavelength in the scattered light varies inversely as the fourth power of the wavelength.

Amount of scattering:

● The sky appears blue on a clear day because of the scattering of blue light by the air molecules in the Earth's atmosphere.

● At sunrise or sunset, the Sun looks almost reddish. This is because all the components of light except red are scattered by the air molecules in Earth’s atmosphere.

Defects of Vision

● Nearsightedness or Myopia − A person suffering from myopia can see only nearby objects clearly, but cannot see the objects beyond a certain distance clearly.

Correction − In order to correct the eye for this defect, a concave lens of suitable focal length is placed close to the eye so that the parallel ray of light from an object at infinity after refraction

through the lens appears to come from the far point of the myopic eye.

If x is the distance of the far point from the eye, then for the concave lens placed before the eye, u = ∞ v = − x Let ‘f’ be the focal length of the required concave lens. Then

Thus, myopic eye is cured against the defect by using a concave lens of focal length equal to the distance of its far point from the eye.

● Farsightedness or Hypermetropia − A person suffering from hypermetropia can see distant objects clearly, but cannot see nearby objects.

Correction − To correct this defect, a convex lens of suitable focal length is placed close to the eye so that the rays of light from an object placed at the point N after refraction through the lens

appear to come from the near point of the hypermetropic eye.

Let

x → Distance of the near point from the eye D → Least distance of distinct vision u = − D ( distance is measured against the incident rays) v = − x ( distance is measured against the incident rays) If f is the focal length of the required convex lens, then

Simple Microscope When image is formed at the near point

The angular magnification of a simple microscope is the ratio of the angle β subtended at the eye by the image at the near point and the angle α subtended at the unaided eye by the object at the near point.

In case the eye is placed behind the lens at a distance ‘a’, then

When the image is formed at infinity

In this case, and

Compound Microscope

We know , therefore

Magnifying power, when final image is at infinity:

If the object is very close to the principal focus of the objective and the image formed by the objective is very close to the eyepiece, then

Where, L = Length of the microscope In this case, the microscope is said to be in normal adjustment.

Astronomical Refracting Telescope When the final image is formed at infinity

When the final image is formed at the least distance of distinct vision