1. Every gas consists of extremely small particles known as molecules. The molecules of a given gas are all identical but are different from those of another gas.
2. The molecules of a gas are identical spherical, rigid and perfectly elastic point masses.
3. Their molecular size is negligible in comparison to intermolecular distance (10^-9 m).
4. The speed of gas molecules lies between zero and infinity (very high speed).
5. The distance covered by the molecules between two successive collisions is known as free path and mean of all free path is known as mean free path.
6. The number of collision per unit volume in a gas remains constant.
7. No attractive or repulsive force acts between gas molecules.
8. Gravitational to extremely attraction among the molecules is ineffective due small masses and very high speed of molecules.

Gas laws

Assuming permanent gases to be ideal, through experiments, it was established that gases irrespective of their nature obey the following laws.

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

At constant pressure the volume (V) of a given mass of a gas increases or decreases by 1/273.15 of its volume at 0°C for each 1°C rise or fall in temperature.

Volume of the gas at t°Ce

V_t = V₀ (1 + t/273.15)

where V₀ is the volume of gas at 0°C.

Gay Lussacs’ 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, p_t = 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. According Avogadro’s law

(i) Avogadro’s number The number of molecules present in 1g mole of a gas is defined as Avogadro’s number.

N_A = 6.023 X 10²³ per gram mole

(ii) At STP or NTP (T = 273 K and p = 1 atm 22.4 L of each gas has 6.023 x 10²³ molecules.

(iii) One mole of any gas at STP occupies 22.4 L of volume.

Standard or Perfect Gas Equation

Gases which obey all gas laws in all conditions of pressure and temperature are called perfect gases.

Equation of perfect gas pV=nRT

where p = pressure, V = volume, T = absolute temperature, R = universal gas constant and n = number of moles of a gas.

Universal gas constant R = 8.31 J mol^-1K^-1.

Real Gases

Real gases deviate slightly from ideal gas laws because

• Real gas molecules attract one another.
• Real gas molecules occupy a finite volume.

Real or Van der Waal’s Gas Equation

(p + a/V²) (V – b) = RT

where a and b are called van der Waals’ constants.

Pressure due to an ideal gas is given by
p = (1/3).(mn/V). c² = 1/3 ρ c²

For one mole of an ideal gas
P = (1/3).(M/V).c²

where, m = mass of one molecule, n = number of molecules, V = volume of gas, c = (c₁² + c₂² + … + c_n²) / n allde root mean square (rrns) velocity of the gas molecules and M = molecular weight of the gas. If p is the pressure of the gas and E is the kinetic energy per unit volume is E, then

p = (2/3).E

Kinetic Energy of a Gas

(i) Average kinetic energy of translation per molecule of a gas is given by

E = (3/2) kt

where k = Boltzmann’s constant.

(ii) Average kinetic energy of translation per mole of a gas is given by

E = (3/2) Rt

where R = universal gas constant.

(iii) For a given gas kinetic energy

E ∝ T ⇒ E₁/E₂ = T₁/T₂

(iv) Root mean square (rms) velocity of the gas molecules is given by

(v) For a given gas c ∝ √T

(vi) For different gases c ∝1/√M

(vii) Boltzmann’s constant k = R/N

where R is ideal gas constant and N = Avogadro number.

Value of Boltzmann’s constant is 1.38 x 10^-28 J/K.

(viii) The average speed of molecules of a gas is given by

(ix) The most probable speed of molecules of a gas is given by

Degree of Freedom

The degree of freedom for a dynamic system is the number of directions in which it can move freely or the number of coordinates required to describe completely the position and configuration of the system.

It is denoted by for N.

Degree of freedom of a system is given by

f or N = 3A – R

where A = number of particles in the system and R = number of independent relations.

Degree of Freedom

1. For monoatomic gas = 3
2. For diatomic gas = 5
3. For non-linear triatomic gas = 6
4. For linear triatomic gas = 7

Specific heat of a gas

(a) At constant volume, C_V = f/2 R

(b) At constant pressure, c_p = (f/2 + 1)R

(c) Ratio of specific heats of a gas at constant pressure and at constant volume is given by
γ = 1 + 2/f

Mean Free Path

The average distance travelled by a molecule between two successive collisions is called mean free path (γ).

Mean free path is given by

γ = kT / √2 π σ²p

where σ = diameter of the molecule, p = pressure of the gas,
T = temperature and k = Botlzmann’s constant.

Mean free path

λ ∝ T and λ ∝ 1/p

Brownian Motion

The continuous random motion of the particles of microscopic size suspended in air or any liquid, is called Brownian of microscopic motion.

Brownian suspended motion in both is observed with many liquids and gases.

Brownian motion is due to the unequal bombardment of the suspended Particles by the molecules of the surrounding medium.

Heat

Heat is a form of energy called thermal energy which flows from a higher temperature body to a lower temperature body when they are placed in contact.

Heat or thermal energy of a body is the sum of kinetic energies of all its constituent particles, on account of translational, vibrational and rotational motion.

The SI unit of heat energy is joule (J).

The practical unit of heat energy is calorie.

1 cal = 4.18 J

1 calorie is the quantity of heat required to raise the temperature of 1 g of water by 1°C.

Mechanical energy or work (W) can be converted into heat (Q) by 1 W = JQ

where J = Joule’s mechanical equivalent of heat.

J is a conversion factor (not a physical quantity) and its value is 4.186 J/cal.

Temperature

Temperature of a body is the degree of hotness or coldness of the body. A device which is used to measure the temperature, is called a thermometer.

Highest possible temperature achieved in laboratory is about 108 while lowest possible temperature attained is 10-8 K.

Branch of Physics dealing with production and measurement temperature close to 0 K is known as cryagenics, while that deaf with the measurement of very high temperature is called pyromet Temperature of the core of the sun is 107 K while that of its surface 6000 K.

NTP or STP implies 273.15 K (0°C = 32°F).

Different Scale of Temperature

1. Celsius Scale In this scale of temperature, the melting point ice is taken as 0°C and the boiling point of water as 100°C and space between these two points is divided into 100 equal parts
2. Fahrenheit Scale In this scale of temperature, the melt point of ice is taken as 32°F and the boiling point of water as 211 and the space between these two points is divided into 180 equal parts.
3.  Kelvin Scale In this scale of temperature, the melting pouxl ice is taken as 273 K and the boiling point of water as 373 K the space between these two points is divided into 100 equal pss

Relation between Different Scales of Temperatures

Thermometric Property

The property of an object which changes with temperature, is call thermometric property. Different thermometric properties thermometers have been given below

(i) Pressure of a Gas at Constant Volume

where p, p₁₀₀. and p_t, are pressure of a gas at constant volume 0°C, 100°C and t°C.

A constant volume gas thermometer can measure tempera from – 200°C to 500°C.

(ii) Electrical Resistance of Metals

R_t = R₀(1 + αt + βt²)

where α and β are constants for a metal.

As β is too small therefore we can take

R_t = R₀(1 + αt)

where, α = temperature coefficient of resistance and
R₀ and R_t, are electrical resistances at 0°C and t°C.

where R₁ and R₂ are electrical resistances at temperatures t₁ and t₂.

where R₁₀₀ is the resistance at 100°C.

Platinum resistance thermometer can measure temperature from —200°C to 1200°C.

(iii) Length of Mercury Column in a Capillary Tube

l_t = l₀(1 + αt)

where α = coefficient of linear expansion and l₀, l_t are lengths of mercury column at 0°C and t°C.

Thermo Electro Motive Force

When two junctions of a thermocouple are kept at different temperatures, then a thermo-emf is produced between the junctions, which changes with temperature difference between the junctions. Thermo-emf

E = at + bt²

where a and b are constants for the pair of metals.

Unknown temperature of hot junction when cold junction is at 0°C.

Where E₁₀₀ is the thermo-emf when hot junction is at 100°C.

A thermo-couple thermometer can measure temperature from —200°C to 1600°C.

Thermal Equilibrium

When there is no transfer of heat between two bodies in contact, the the bodies are called in thermal equilibrium.

Zeroth Law of Thermodynamics

If two bodies A and B are separately in thermal equilibrium with thirtli body C, then bodies A and B will be in thermal equilibrium with each other.

Triple Point of Water

The values of pressure and temperature at which water coexists inequilibrium in all three states of matter, i.e., ice, water and vapour
called triple point of water.

Triple point of water is 273 K temperature and 0.46 cm of mere pressure.

Specific Heat

The amount of heat required to raise the temperature of unit mass the substance through 1°C is called its specific heat.

It is denoted by c or s.

Its SI unit is joule/kilogram-°C'(J/kg-°C). Its dimensions is [L²T^-2θ^-1].

The specific heat of water is 4200 J kg^-1°C^-1 or 1 cal g^-1 C^-1, which high compared with most other substances.

Gases have two types of specific heat

1. The specific heat capacity at constant volume (C_v).
2. The specific heat capacity at constant pressure (C_r).

Specific heat at constant pressure (C_p) is greater than specific heat constant volume (C_V), i.e., C_p > C_V .

For molar specific heats C_p – C_V = R
where R = gas constant and this relation is called Mayer’s formula.

The ratio of two principal sepecific heats of a gas is represented by γ.

The value of y depends on atomicity of the gas.

Amount of heat energy required to change the temperature of any substance is given by

Q = mcΔt

• where, m = mass of the substance,
• c = specific heat of the substance and
• Δt = change in temperature.

Thermal (Heat) Capacity

Heat capacity of any body is equal to the amount of heat energy required to increase its temperature through 1°C.

Heat capacity = me

where c = specific heat of the substance of the body and m = mass of the body.

Its SI unit is joule/kelvin (J/K).

Water Equivalent

It is the quantity of water whose thermal capacity is same as the heat capacity of the body. It is denoted by W.

W = ms = heat capacity of the body.

Latent Heat

The heat energy absorbed or released at constant temperature per unit mass for change of state is called latent heat.

Heat energy absorbed or released during change of state is given by

Q = mL

where m = mass of the substance and L = latent heat.

Its unit is cal/g or J/kg and its dimension is [L²T^-2].

For water at its normal boiling point or condensation temperature (100°C), the latent heat of vaporisation is

L = 540 cal/g
= 40.8 kJ/ mol
= 2260 kJ/kg

For water at its normal freezing temperature or melting point (0°C), the latent heat of fusion is

L = 80 cal/ g = 60 kJ/mol
= 336 kJ/kg

It is more painful to get burnt by steam rather than by boiling was 100°C gets converted to water at 100°C, then it gives out 536 heat. So, it is clear that steam at 100°C has more heat than wat 100°C (i.e., boiling of water).

After snow falls, the temperature of the atmosphere becomes very This is because the snow absorbs the heat from the atmosphere to down. So, in the mountains, when snow falls, one does not feel too but when ice melts, he feels too cold.

There is more shivering effect of ice cream on teeth as compare that of water (obtained from ice). This is because when ice cream down, it absorbs large amount of heat from teeth.

Melting

Conversion of solid into liquid state at constant temperature is melting.

Evaporation

Conversion of liquid into vapour at all temperatures (even below boiling point) is called evaporation.

Boiling

When a liquid is heated gradually, at a particular temperature saturated vapour pressure of the liquid becomes equal to atmospheric pressure, now bubbles of vapour rise to the surface d liquid. This process is called boiling of the liquid.

The temperature at which a liquid boils, is called boiling point The boiling point of water increases with increase in pre sure decreases with decrease in pressure.

Sublimation

The conversion of a solid into vapour state is called sublimation.

Hoar Frost

The conversion of vapours into solid state is called hoar fr..

Calorimetry

This is the branch of heat transfer that deals with the measorette heat. The heat is usually measured in calories or kilo calories.

Principle of Calorimetry

When a hot body is mixed with a cold body, then heat lost by ha is equal to the heat gained by cold body.

Heat lost = Heat gain

Thermal Expansion

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

1. Expansion of solids
2. Expansion of liquids
3. Expansion of gases

Expansion of Solids

Three types of expansion -takes place in solid.

Linear Expansion Expansion in length on heating is called linear expansion.

Increase in length

l₂ = l₁(1 + α Δt)

where, l_l and l₂ are initial and final lengths,Δt = change in temperature and α = coefficient of linear expansion.

Coefficient of linear expansion

α = (Δl/l * Δt)

where 1= real length and Δl = change in length and

Δt= change in temperature.

Superficial Expansion Expansion in area on heating is called superficial expansion.

Increase in area A₂ = A₁(1 + β Δt)

where, A₁ and A₂ are initial and final areas and β is a coefficient of superficial expansion.

Coefficient of superficial expansion

β = (ΔA/A * Δt)

where. A = area, AA = change in area and At = change in temperature.

Cubical Expansion Expansion in volume on heating is called cubical expansion.

Increase in volume V₂ = V₁(1 + γΔt)

where V₁ and V₂ are initial and final volumes and γ is a coefficient of cubical expansion.

Coefficient of cubical expansion

where V = real volume, AV =change in volume and Δt = change in temperature.
Relation between coefficients of linear, superficial and cubical expansions
β = 2α and γ = 3α
Or α:β:γ = 1:2:3

2. Expansion of Liquids

In liquids only expansion in volume takes place on heating.

(i) Apparent Expansion of Liquids When expansion of th container containing liquid, on heating is not taken into accoun then observed expansion is called apparent expansion of liquids.

Coefficient of apparent expansion of a liquid

(ii) Real Expansion of Liquids When expansion of the container, containing liquid, on heating is also taken into account, then observed expansion is called real expansion of liquids.

Coefficient of real expansion of a liquid

Both, y_r, and y_a are measured in °C^-1.

We can show that y_r = y_a + y_g

where, y_r, and y_a are coefficient of real and apparent expansion of liquids and y_g is coefficient of cubical expansion of the container.

Anamalous Expansion of Water

When temperature of water is increased from 0°C, then its vol decreases upto 4°C, becomes minimum at 4°C and then increases. behaviour of water around 4°C is called, anamalous expansion water.

3. Expansion of Gases

There are two types of coefficient of expansion in gases

(i) Volume Coefficient (γv) At constant pressure, the change in volume per unit volume per degree celsius is called volume coefficient.

where V₀, V₁, and V₂ are volumes of the gas at 0°C, t₁°C and t₂°C.

(ii) Pressure Coefficient (γ_p) At constant volume, the change in pressure per unit pressure per degree celsius is called pressure coefficient.

where p₀, p₁ and p₂ are pressure of the gas at 0°C, t₁° C and t₂° C.

Practical Applications of Expansion

1. When rails are laid down on the ground, space is left between the end of two rails.
2. The transmission cables are not tightly fixed to the poles.
3. The iron rim to be put on a cart wheel is always of slightly smaller diameter than that of wheel.
4. A glass stopper jammed in the neck of a glass bottle can be taken out by warming the neck of the bottles.

Important Points

• Due to increment in its time period a pendulum clock becomes slow in summer and will lose time.
• Loss of time in a time period ΔT =(1/2)α ΔθT
  ∴ Loss of time in any given time interval t can be given by
  ΔT =(1/2)α Δθt
• At some higher temperature a scale will expand and scale reading will be lesser than true values, so that
  true value = scale reading (1 + α Δt)
  Here, Δt is the temperature difference.
• However, at lower temperature scale reading will be more or true value will be less.

Surface tension of a liquid is measured as the force acting per length on an imaginary line drawn tangentially on the free surface the liquid.

Surface tension S = Force/Length = F/l = Work done/Change in area

Its SI unit is Nm^-1 or Jm^-2 and its dimensional formula is [MT^-2].

It is a scalar quantity. Surface tension is a molecular phenomenon which is due to cohesive force and root cause of the force is electrical in nature.

Surface tension of a liquid depends only on the nature of liquid and independent of the surface area of film or length of the line .

Small liquid drops are spherical due to the property of surface tension.

Adhesive Force

The force of attraction acting between the molecules of different substances is called adhesive force, e.g., the force of attracts acting between the molecules of paper and ink, water and glass, etc.

Cohesive Force

The force of attraction acting between the molecules of same substan is called cohesive force. e.g., the force of attraction acting between molecules of water, glass, etc.

Cohesive forces and adhesive forces are van der Waals’ forces.

These forces varies inversely as the seventh power of distance between the molecules.

Molecular Range

The maximum distance upto which a molecule can exert a force of attraction on other molecules is called molecular range.

Molecular range is different for different substances. In solids and liquids it is of the order of 10^-9 m.

If the distance between the molecules is greater than 10-9 m, the force of attraction between them is negligible.

Surface Energy

If we increase the free surface area of a liquid then work has to be done against the force of surface tension. This work done is stored in liquid stu-face as potential energy,

This additional potential energy per unit area of free surface of liquid is called surface energy.
Surface energy (E) = S x &ΔM
where. S = surface tension and ΔA = increase in surface area.

(i) Work Done in Blowing a Liquid Drop If a liquid drop is blown up from a radius r₁ to r₂ then work done for that is
W = S . 4π (r₂² – r₁²)

(ii) Work Done in Blowing a Soap Bubble As a soap bubble has two free surfaces, hence work done in blowing a soap bubble so as to increase its radius from r¹ to r² is given by
W = S.8π(r₂² – r₁²)

(iii) Work Done in Splitting a Bigger Drop into n Smaller Droplets
If a liquid drop of radius R is split up into n smaller droplets, all of same size. then radius of each droplet

r = R. (n)^-1/3
Work done, W = 4π(nr² – R²)
= 4πSR² (n^1/3 – 1)

(iv) Coalescance of Drops If n small liquid drops of radius reach combine together so as to form a single bigger drop of radius R=n^1/3.r, then in the process energy is released. Release of energy is given by

ΔU = S.4π(nr² – R²)

= 4πSπn(1 – n^1/3)

Angle of Contact

The angle subtended between the tangents drawn at liquid surface at solid surface inside the liquid at the point of contact, is called of contact (9).

Angle of contact depends upon the nature of the liquid and solid contact and the medium which exists above the free surface of liquid.

When wax is coated on a glass capillary tube, it becomes water-proof.
The angle of contact increases and becomes obtuse. Water does not in it. Rather it falls in the tube by virtue of obtuse angle of contact.

If θ is acute angle, i.e; θ <90°, then liquid meniscus will be concave upwards.

• If θ is 90°, then liquid meniscus will be plane.
• If θ is obtuse, i.e; θ >90°, then liquid meniscus will be convex upwards.
• If angle of contact is acute angle, i.e; θ <90°, then liquid will wet the surface.
• If angle of contact is obtuse angle, ie; θ > 90°, then liquid will not wet the surface.

Angle of contact increases with increase in temperature of Angle of contact decreases on adding soluble impurity to a liquid.

Angle of contact for pure water and glass is zero. For ordinary water and glass is 8°. For mercury and glass is 140°. For pure water silver is 90°. For alcohol and clean glass θ = 0°.

Angle of contact, meniscus, shape of liquid surface

Capillarity

The phenomenon called capillarity. of rise or fall of liquid column in a capillary tube is Ascent of a liquid column in a capillary tube is given by

h = (2S cos θ / rρg) – (r / 3)

If capillary is very narrow, then

h=2S cos θ / rρg

where, r = radius of capillary tube, p = density of the liquid, and
θ = angle of contact and S = surface tension of liquid.

• If θ < 90°, cos e is positive, so h is positive, i.e., liquid rises in a capillary tube.
• If θ > 90°, cos 9 is negative, so h is negative, i.e., liquid falls in a capillary tube.
• Rise of liquid in a capillary tube does not violate law of conservation of energy.

Some Practical Examples of Capillarity

1. The kerosene oil in a lantern and the melted wax in a candle, rise in the capillaries formed in the cotton wick and burns.
2. Coffee powder is easily soluble in water because water immediately wets the fine granules of coffee by the action of capillarity.
3. The water given to the fields rises in the innumerable capillaries formed in the stems of plants and trees and reaches the leaves.

Zurin’s Law

If a capillary tube of insufficient length is placed vertically in a then liquid never come out from the tube its own, as

Rh = constant ⇒ R₁h₁ = R₂h₂

where, R = radius of curvature of liquid meniscus and
h = height of liquid column.

When a tube is kept in inclined position in a liquid the vertical height remains unchanged then length of liquid column.

Liquid rises (water in glass capillary) or falls (mercury in capillary) due to property of surface tension

T = Rρgh / 2 cos θ

where, R = radius of capillary tube, h = height of liquid, p = density of liquid, e = angle of contact,

T = surface tension of liquid and 9 = acceleration due to gravity.

Excess Pressure due to Surface Tension

(i) Excess pressure inside a liquid drop = 2S / R

(ii) Excess pressure inside an air bubble in a liquid = 2S / R

(iii) Excess pressure inside a soap bubble = 4S / R

where, S = surface tension and R = radius of drop/bubble.

(iv) Work done in spraying a liquid drop of radius R into n droplets of radius r = T x increase in surface area

= 4πTR³ (1/r – 1/R)

Fall in temperature

Δθ = 3T/J (1/r – 1/R)

where. J = 4.2 J/cal.

(v) When n small drops are combined into a bigger drop, then work done is given by

W = 4πR²T (n ^1/3 – 1)

Temperature increase

Δθ = 3T/J (1/r – 1/R)

(vi) When two bubbles of radii r₁ and r₂ coalesce into a bubble of radius r isothermally, then
r² = r₁2 + r₂²

(vii) When two soap bubbles of radii ‘1 and ‘2 are in contact with each other, then radius (r) of common interface.

Factors Affecting Surface Tension

1. Surface tension of a liquid decreases with increase temperature and becomes zero at critical temperature.
2. At boiling point, surface tension of a liquid becomes zero becomes maximum at freezing point.
3. Surface tension decreases when partially soluble impurities such as soap, detergent, dettol, phenol, etc are added in water.
4. Surface tension increases when highly soluble impurities such as salt is added in water.
5. When dust particles or oil spreads over the surface of water, its surface tension decreases.

When charge is given to a soap bubble, its size increases surface tension of the liquid decreases due to electrification.

In weightlessness condition liquid does not rise in a capillary tube.

Some Phenomena Based on Surface Tension

1. Medicines used for washing wounds, as detol, have a surface tension lower than water.
2. Hot soup is more tasteful than the cold one because the surface tension of the hot soup is less than that of the cold and so spreads over a larger area of the tongue.
3. Insects and mosquitoes swim on the surface of water in ponds and lakes due to surface tension. If kerosence oil is sprayed on the water surface, the surface tension of water is lowered and the insects and mosquitoes sink in water and are dead.
4. If we deform a liquid drop by pushing it slightly, then due to surface tension it again becomes spherical.

The property of a fluid by virtue of which an internal frictional force acts between its different layers which opposes their relative motion is called viscosity.

These internal frictional force is called viscous force.

Viscous forces are intermolecular forces acting between the molecules of different layers of liquid moving with different velocities.

where, (dv/dx) = rate of change of velocity with distance called velocity gradient, A = area of cross-section and = coefficient of viscosity.

SI unit of η is Nsm^-2 or pascal-second or decapoise. Its dimensional formula is [ML^-1T^-1].

The knowledge of the coefficient of viscosity of different oils and its variation with temperature helps us to select a suitable lubricant for a given machine.

Viscosity is due to transport of momentum. The value of viscosity (and compressibility) for ideal liquid is zero.

The viscosity of air and of some liquids is utilised for damping the n.ving parts of some instruments.

The knowledge of viscosity of some organic liquids is used in determining the molecular weight and shape of large organic moleculars like proteins and cellulose.

Variation of Viscosity

The viscosity of liquids decreases with increase in temperature

where, η₀ and η_t is are coefficient of viscosities at 0°C t°C, α and β are constants.

The viscosity of gases increases with increase in temperatures as

η ∝ √T

The viscosity of liquids increases with increase in pressure but the viscosity of water decreases with increase in pressure.

The viscosity of gases do not changes with pressure.

Poiseuille’s Formula

The rate of flow (v) of liquid through a horizontal pipe for steady flow is given by

where, p = pressure difference across the two ends of the tube. r = radius of the tube, n = coefficient of viscosity and 1 = length of th tube.

The Rate of Flow of Liquid

Rate of flow of liquid through a tube is given by

v = (P/R)

where, R = (8 ηl/πr⁴), called liquid resistance and p = liquid pressure.

(i) When two tubes are connected in series

• Resultant pressure difference p = p₁ + p₂
• Rate of flow of liquid (v) is same through both tubes.
• Equivalent liquid resistance, R = R₁ + R₂

(ii) When two tubes are connected in parallel

1. Pressure difference (p) is same across both tubes.
2. Rate of flow of liquid v = v₁ + v₂
3. Equivalent liquid resistance (1/R) = (1/R₁) + (1/R₂)

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.

where,

• ρ = density of body,
• σ = density of liquid,
• η = coefficient of viscosity of liquid and,
• g = acceleration due to gravity

1. If ρ > ρ₀, the body falls downwards.
2. If ρ < ρ₀, the body moves upwards with the constant velocity.
3. If po << ρ, v = (2r²ρg/9η)

Importance of Stoke’s Law

1. This law is used in the determination of electronic charge by Millikan in his oil drop experiment.
2. This law helps a man coming down with the help of parachute.
3. This law account for the formation of clouds.

Flow of Liquid

1. Streamline Flow The flow of liquid in which each of its particle follows the same path as followed by the proceeding particles, is called streamline flow.
2. Laminar Flow The steady flow of liquid over a horizontal surface in the form of layers of different velocities, is called laminar flow.
3. Turbulent Flow The flow of liquid with a velocity greater than its critical velocity is disordered and called turbulent flow.

Critical Velocity

The critical velocity is that velocity of liquid flow, below which its fl is streamlined and above which it becomes turbulent.

Critical velocity v_c = (k_η/rρ)

where,

• K = Reynold’s number,
• η = coefficient of viscosity of liquid
• r = radius of capillary tube and ρ = density of the liquid.

Reynold’s Number

Reynold’s number is a pure number and it is equal to the ratio of inertial force per unit area to the viscous force per unit area for flowing fluid.

where, p = density of the liquid and v_c = critical velocity.

For pure water flowing in a cylindrical pipe, K is about 1000.

When 0< K< 2000, the flow of liquid is streamlined.

When 2000 < K < 3000, the flow of liquid is variable betw streamlined and turbulent.

When K > 3000, the flow of liquid is turbulent.

It has no unit and dimension.

Equation of Continuity

If a liquid is flowing in streamline flow in a pipe of non-unif cross-section area, then rate of flow of liquid across any cross-sec remains constant.

a₁v₁ = a₂v₂ av = constant

The velocity of liquid is slower where area of cross-section is larger faster where area of cross-section is smaller.

The falling stream of water becomes narrower, as the velocity of f stream of water increases and therefore its area of cross-s decreases.

Energy of a Liquid

A liquid in motion possess three types of energy

(i) Pressure Energy Pressure energy per unit mass = p/ρ

where,

p= pressure of the liquid and p = density of the liquid.
Pressure energy per unit volume = p

(ii) Kinetic Energy

• Kinetic energy per unit mass = (1/2v²)
• Kinetic energy per unit volume = 1/2ρv²

(iii) Potential Energy

• Potential energy per unit mass = gh
• Potential energy per unit volume = ρgh

Bernoulli’s Theorem

If an ideal liquid is flowing in streamlined flow then total energy, i.e., sum of pressure energy, kinetic energy and potential energy per unit
volume of the liquid remains constant at every cross-section of the tube.

Mathematically

It can be expressed as

where, (p/ρg) = pressure head, (v²/2g) = velocity head and h = gravitational head.

For horizontal flow of liquid,

• where, pis called static pressure and (1/2 ρv²) is called dynamic pressure.
• The erefore in horizontal flow of liquid, if p increases, v decreases and Nce-versa.
• The theorem is applicable to ideal liquid, i.e., a liquid which is non-viscous incompressible and irrotational.

Applications of Bernoulli’s Theorem

1. The action of carburetor, paintgun, scent sprayer atomiser insect sprayer is based on Bernoulli’s theorem.
2. The action of Bunsen’s burner, gas burner, oil stove exhaust pump is also based on Bernoulli’s theorem.
3. Motion of a spinning ball (Magnus effect) is based on Bernoulli theorem.
4. Blowing of roofs by wind storms, attraction between two close parallel moving boats, fluttering of a flag etc are also based Bernoulli’s theorem.

Venturimeter

It is a device used for measuring the rate of flow of liquid t pipes. Its working is based on Bernoulli’s theorem.

Rate of flow of liquid,

where, a₁ and a₂ are area of cross-sections of tube at bra and narrower part and h is difference of liquid columns in ver tubes.

Torricelli’s Theorem

Velocity of efflux (the velocity with which the liquid flows out orifice or narrow hole) is equal to the velocity acquired by a falling body through the same vertical distance equal to the dep orifice below the free surface of liquid.

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.

Important Points

• In a pipe the inner layer .central layer) have maximum velocity and the layer in contact with pipe have least velocity.
• Velocity profile of liquid flow in a pipe is parabolic.
• Solid friction is independent of area of surfaces in contact while viscous force depends on area of liquid layers.
• A lubricant is chosen ac:ording to the nature of machinary. In heavy machines lubricating oils of high viscosity are used and in light machines low viscosity oils are used.
• The cause of viscosity 1 liquids is the cohesive forces among their molecules while cause of viscosity in gases is diffusion.

Fluids are those substances which can flow when an external force is applied on it.

Liquids and gases are fluids.

Fluids do not have finite shape but takes the shape of the containing vessel,

The total normal force exerted by liquid at rest on a given surface is called thrust of liquid.

The SI unit of thrust is newton.

In fluid mechanics the following properties of fluid would be considered

(i) When the fluid is at rest – hydrostatics

(ii) When the fluid is in motion – hydrodynamics

Pressure Exerted by the Liquid

The normal force exerted by a liquid per unit area of the surface in contact is called pressure of liquid or hydrostatic pressure.

Pressure exerted by a liquid column

p = hρg

Where, h = height of liquid column, ρ = density of liquid

and g = acceleration due to gravity

Mean pressure on the walls of a vessel containing liquid upto height h is (hρg / 2).

Pascal’s Law

The increase in pressure at a point in the enclosed liquid in equilibrium is transmitted equally in all directions in liquid and to the Walls of the container.

The working of hydraulic lift, hydraulic press and hydraulic brakes are based on Pascal’s law.

Atmospheric Pressure

The pressure exerted by the atmosphere on earth is atmospheric pressure.

It is about 100000 N/m².

It is equivalent to a weight of 10 tones on 1 m².

At sea level, atmospheric pressure is equal to 76 cm of mercury column. Then, atmospheric pressure

= hdg = 76 x 13.6 x 980 dyne/cm²

[The atmospheric pressure does not crush our body because the pressure of the blood flowing through our circulatory system] balanced this pressure.]

Atmospheric pressure is also measured in torr and bar.

1 torr = 1 mm of mercury column

1 bar = l0⁵ Pa

Aneroid barometer is used to measure atmospheric pressure.

Buoyancy

When a body is partially or fully immersed in a fluid an upward force acts on it, which is called buoyant force or simply buoyancy.

The buoyant force acts at the centre of gravity of the liquid displaced] by the immersed part of the body and this point is called the centre buoyancy.

Archimedes’ Principle

When a body is partially or fully immersed in a liquid, it loses some of its weight. and it is equal to the weight of the liquid displaced by the immersed part of the body.

If T is the observed weight of a body of density σ when it is fully immersed in a liquid of density p, then real weight of the body

w = T / ( 1 – p / σ)

Laws of Floatation

A body will float in a liquid, if the weight of the body is equal to the weight of the liquid displaced by the immersed part of the body.

If W is the weight of the body and w is the buoyant force, then

(a) If W > w, then body will sink to the bottom of the liquid.

(b) IfW < w, then body will float partially submerged in the liquid.
(c) If W = w, then body will float in liquid if its whole volume is just immersed in the liquid,

The floating body will be in stable equilibrium if meta-centre (centre of buoyancy) lies vertically above the centre of gravity of the body.

The floating body will be in unstable equilibrium if meta-centre (centre of buoyancy) lies vertically below the centre of gravity of the body.

The floating body will be in neutral equilibrium if meta-centre (centre of buoyancy) coincides with the centre of gravity of the body.

Density and Relative Density

Density of a substance is defined as the ratio of its mass to its volume.

Density of a liquid = Mass / Volume

Density of water = 1 g/cm³ or l0³ kg/m³

It is scalar quantity and its dimensional formula is [ML^-3].

Relative density of a substance is defined as the ratio of its density to the density of water at 4°C,

Relative density = Density of substance / Density of water at 4°C

= Weight of substance in air / Loss of weight in water

Relative density also known as specific gravity has no unit, no dimensions.

For a solid body, density of body = density of substance

While for a hollow body, density of body is lesser than that of Substance.

When immiscible liquids of different densities are poured in a container, the liquid of highest density will be at the bottom while, that of lowest density at the top and interfaces will be plane.

Density of a Mixture of Substances

When two liquids of mass m₁ and m₂ having density p₁ and p₂ are mixed together then density of mixture is

p = m₁ + m₂ / (m₁ /p₁ ) + (m₂ + p₂)

= p₁p₂ (m₁ + m₂) / (m₁p₂ + m₂p₁)

When two liquids of same mass m but of different densities p₁ and p₂ are mixed together then density of mixture is

p = 2p₁p₂ / p₁ + p₂

When two liquids of same volume V but of different densities p₁ and p₂ are mixed together then density of mixture is

p = p₁ + p₂ / 2

Density of a liquid varies with pressure

p = p_o [ 1 + Δp / K]

where, p_o = initial density of the liquid, K = bulk modulus of elasticity of the liquid and Δp = change in pressure.

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 rermain same.

Centre of Mass of System of n Particles

If a system consists of n particles of masses m₁, m₂, m₃ ,… m_n having position vectors r_l, r₂, r₃,… r_n. then position vector of centre of mass of

Centre of Mass of Two Particle System

Choosing O as origin of the coordinate axis.

(ii) Position of centre of mass from m₂ = (m₁d) / m₁ + m₂

iii) If position vectors of particles of masses m₁ and m₂ are r₁ and r₂respectively, then

(iv) If in a two particle system, particles of masses m₁ and m₂ moving with velocities v₁ and v₂ respectively, then velocity the centre of mass

(v) If accelerations of the particles are a₁, and a₁respectively, then acceleration of the centre of mass

(vi) Centre of mass of an isolated system has a constant velocity.

(vii) It means isolated system will remain at rest if it is initially rest or will move with a same velocity if it is in motion initially.

(viii) The position of centre of mass depends upon the shape, size and distribution of the mass of the body.

(ix) The centre of mass of an object need not to lie with in the object.

(x) In symmetrical bodies having homogeneous distribution mass the centre of mass coincides with the geometrical centre the body.

(xi) The position of centre of mass of an object changes translatory motion but remains unchanged in rotatory motion,

Translational Motion

A rigid body performs a pure translational motion, if each particle the body undergoes the same displacement in the same direction in given interval of time.

Rotational Motion

A rigid body performs a pure rotational motion, if each particle of the body moves in a circle, and the centre of all the circles lie on a straight line called the axes of rotation.

Rigid Body

If the relative distance between the particles of a system do not changes on applying force, then it called a rigtd body. General motion of a rigid body consists of both the translational motion and the rotational motion.

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.

Its unit is kg.m² and its dimensional formula is [ML²].

The moment of inertia of a body depends upon

• position of the axis of rotation
• orientation of the axis of rotation
• shape and size of the body
• distribution of mass of the body about the axis of rotation.

The physical significance of the moment of inertia is same in rotational motion as the mass in linear motion.

The Radius of Gyration

The root mean square distance of its constituent particles from the axis of rotation is called the radius of gyration of a body.

It is denoted by K.

Radius of gyration

The product of the mass of the body (M) and square of its radius gyration (K) gives the same moment of inertia of the body about rotational axis.

Therefore, moment of inertia I = MK² ⇒ K = √1/M

Parallel Axes Theorem

The moment of inertia of any object about any arbitrary axes is equal to the sum of moment of inertia about a parallel axis passing through the centre of mass and the product of mass of the body and the square of the perpendicular distance between
the two axes.

Mathematically I = I_CM + Mr²

where I is the moment of inertia about the arbitrary axis, I_cM is moment of inertia about the parallel axis through the centre of mass, M is the total mass of the object and r is the perpendicular distance between the axis.

Perpendicular Axes Theorem

The moment of inertia of any two dimensional body about an axis perpendicular to its plane (I_z) is equal to the sum of moments of inertia of the body about two mutually perpendicular axes lying in its own plane and intersecting
each other at a point, where the perpendicular axis passes through it.

Mathematically I_z = I_x + I_y

where I_x and I_y are the moments of inertia of plane lamina about perpendicular axes X and Y respectively which lie in the plane lamina an intersect each other.

Theorem of parallel axes is applicable for any type of rigid body whether it is a two dimensional or three dimensional, while the theorem of perpendicular is applicable for laminar type or two I dimensional bodies only.

Moment of Inertia of Homogeneous Rigid Bodies

For a Thin Circular Ring

For a Circular Disc

For a Thin Rod

For a Solid Cylinder

For a Rectangular Plate

For a Thin Spherical Shell

For a Solid Sphere

Equations of Rotational Motion
(i) ω = ω₀ + αt
(ii) θ = ω₀t + 1/2 αt²
(iii) ω² = ω₀² + 2αθ
where θ is displacement in rotational motion, ω₀ is initial velocity, omega; is final velocity and a is acceleration.

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 anticlockwise, then it is called positive.

Its SI unit is ‘newton-metre’ and its dimension is [ML²T^-2].

In rotational motion, torque, τ = Iα

where a is angular acceleration and 1is moment of inertia.

Angular Momentum
The moment of linear momentum is called angular momentum.

It is denoted by L.
Angular momentum, L = I ω = mvr
In vector form, L = I ω = r x mv
Its unit is ‘joule-second’ and its dimensional formula is [ML²T^-1].

Torque, τ = dL/dt

Conservation of Angular Momentum

If the external torque acting on a system is zero, then its angular momentum remains conserved.
If τ_ext 0, then L = I(ω) = constant ⇒ I₁ω₁== I₂ω₂

Angular Impulse

Total effect of a torque applied on a rotating body in a given time is called angular impulse. Angular impulse is equal to total change in angular momentum of the system in given time. Thus, angular impulse

When a force acts on an object and the object actually moves in the direction of force, then the work is said to be done by the force.

Work done by the force is equal to the product of the force and the displacement of the object in the direction of force.

If under a constant force F the object displaced through a distance s, then work done by the force

W = F * s = F s cos θ

where a is the smaller angle between F and s.

Work is a scalar quantity, Its S1 unit is joule and CGS unit is erg.

∴ 1 joule = 10⁷ erg

Its dimensional formula is [ML²T^-2].

Work done by a force is zero, if

(a) body is not displaced actually, i.e., s = 0

(b) body is displaced perpendicular to the direction of force, i.e.,

θ = 90°

Work done by a force is positive if angle between F and s is acute angle.

Work done by a force is negative if angle between F and s is obtuse angle.

Work done by a constant force depends only on the initial and final Positions and not on the actual path followed between initial and final positions.

Work done in different conditions

(i) Work done by a variable force is given by

W = ∫ F * ds

It is equal to the area under the force-displacement graph along with proper sign.

Work done = Area ABCDA

(ii) Work done in displacing any body under the action of a number of forces is equal to the work done by the resultant force.

(iii) In equilibrium (static or dynamic), the resultant force is zero therefore resultant work done is zero.

(iv) If work done by a force during a rough trip of a system is zero, then the force is conservative, otherwise it is called non-conservative force.

• Gravitational force, electrostatic force, magnetic force, etc are conservative forces. All the central forces are conservative forces.
• Frictional force, viscous force, etc are non-conservative forces.

(v) Work done by the force of gravity on a particle of mass m is given by W = mgh

where g is acceleration due to gravity and h is height through particle one displaced.

(vi) Work done in compressing or stretching a spring is given by

W = 1 / 2 kx²

where k is spring constant and x is displacement from mean position.

(vii) When on end of a spring is attached to a fixed vertical support and a block attached to the free end moves on a horizontal

table from x = x₁ to x = x₂ then W = 1 / 2 k (x²x₂ – x²x₁)

(viii) Work done by the couple for an angular displacement θ is given by W = i * θ

where i is the torque of the couple.

power

The time rate of work done by a body is called its power.

Power = Rate of doing work = W­ork done / Time taken

If under a constant force F a body is displaced through a distance s in time t, the power

p = W / t = F * s / t

But s / t = v ; uniform velocity with which body is displaced.

∴ P = F * v = F v cos θ

where θ is the smaller angle between F and v.

power is a scalar quantity. Its S1 unit is watt and its dimensional formula is [ML²T^-3].

Its other units are kilowatt and horse power,

1 kilowatt = 1000 watt

1 horse power = 746 watt

Energy

Energy of a body is its capacity of doing work.

It is a scalar quantity.

Its S1 unit is joule and CGS unit is erg. Its dimensional formula is [ML³T^-3].

There are several types of energies, such as mechanical energy (kinetic energy and potential energy), chemical energy, light energy, heat energy, sound energy, nuclear energy, electric energy etc.

Mechanical Energy

The sum of kinetic and potential energies at any point remains constant throughout the motion. It does not depend upon time. This is known as law of conservation of mechanical energy.

Mechanical energy is of two types:

1. Kinetic Energy

The energy possessed by any object by virtue of its motion is called its kinetic energy.

Kinetic energy of an object is given by

k = 1 / 2 mv² = p² / 2m

where m = mass of the object, U = velocity of the object and p = mv = momentum of the object.

2. Potential Energy

The energy possessed by any object by virtue of its position or configuration is called its potential energy.

There are three important types of potential energies:

(i) Gravitational Potential Energy If a body of mass m is raised through a height h against gravity, then its gravitational potential energy = mgh,

(ii) Elastic Potential Energy If a spring of spring constant k is stretched through a distance x. then elastic potential energy of the spring = 1 . 2 kx²

The variation of potential energy with distance is shown in figure.

Potential energy is defined only for conservative forces. It does not exist for non-conservative forces.

Potential energy depends upon frame of reference.

(iii) Electric Potential Energy The electric potential energy of two point charges ql and q’l. separated by a distance r in vacuum is given by

U = 1 / 4πΣ₀ * q₁q₂ / r

Here 1 / 4πΣ₀ = 9.0 * 10¹⁰ N-m² / C² constant.

Work-Energy Theorem

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

where, K_i = initial kinetic energy

and K_f = final kinetic energy.

Regarding the work-energy theorem it is worth noting that

(i) If W_net is positive, then K_f – K_i = positive, i.e., K_f > K_i 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.

Mass-Energy Equivalence

According to Einstein, the mass can be transformed into energy and vice – versa.

When Δm. mass disappears, then produced energy

E = Δmc²

where c is the speed of light in vacuum.

Principle of Conservation of Energy

The sum of all kinds of energies in an isolated system remains constant at all times.

Principle of Conservation of Mechanical Energy

For conservative forces the sum of kinetic and potential energies of any object remains constant throughout the motion.

According to the quantum physics, mass and energy are not conserved separately but are conserved as a single entity called ‘mass-energy’.

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. rrhere are two types of collisions:

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.

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.

Coefficient of Restitution or Resilience

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

It is represented by e and it depends upon the material of the collidingI bodies.

For a perfectly elastic collision, e = 1

For a perfectly inelastic collision, e = 0

For all other collisions, 0 < e < 1

One Dimensional or Head-on Collision

If the initial and final velocities of colliding bodies lie along the same line, then the collision is called one dimensional or head-on collision.

Inelastic One Dimensional Collision

Applying Newton’s experimental law, we have

Velocities after collision

v₁ = (m₁ – m₂) u₁ + 2m₂u₂ / (m₁ + m₂)

and v₂ = (m₂ – m₁) u₂ + 2m₁u₁ / (m₁ + m₂)

When masses of two colliding bodies are equal, then after the collision, the bodies exchange their velocities.

v₁ = u₂ and v₂ = u₁

If second body of same mass (m₁ = m₂) is at rest, then after collision first body comes to rest and second body starts moving with the
initial velocity of first body.

v₁ = 0 and v₂ = u₁

If a light body of mass m₁ collides with a very heavy body of mass m₂ at rest, then after collision.

v₁ = – u₁ and v₂ = 0

It means light body will rebound with its own velocity and heavy body will continue to be at rest.

If a very heavy body of mass m₁ collides with a light body of mass m₂(m₁ > > m₂₁) at rest, then after collision

v₁ = u₁ and v₂ = 2u₁

In Inelastic One Dimensional Collision

Loss of kinetic energy

ΔE = m₁m₂ / 2(m₁ + m₂) (u₁ – u₂)² (1 – e²)

In Perfectly Inelastic One Dimensional Collision

Velocity of separation after collision = 0.

Loss of kinetic energy = m₁m₂ (u₁ – u₂)² / 2(m₁ + m₂)

If a body is dropped from a height h_o and it strikes the ground with velocity v_o and after inelastic collision it rebounds with velocity v₁ and rises to a height h₁, then

If after n collisions with the ground, the body rebounds with a velocity v_n and rises to a height h_n then

eⁿ = v_n / v_o = √hⁿ / h^o

Two Dimensional or Oblique Collision

If the initial and final velocities of colliding bodies do not lie along the same line, then the collision is called two dimensional or oblique Collision.

In horizontal direction,

m₁u₁ cos α₁ + m₂u₂ cos α₂= m₁v₁ cos β₁ + m₂v₂ cos β₂

In vertical direction.

m₁u₁ sin α₁ – m₂u₂ sin α₂ = m₁u₁ sin β₁ – m₂u₂ sin β₂

If m₁ = m₂ and α₁ + α₂ = 90°

then β₁ + β₂ = 90°

If a particle A of mass m₁ moving along z-axis with a speed u makes an elastic collision with another stationary body B of mass m₂

From conservation law of momentum

m₁u = m₁v₁ cos α + m₂v₂ cos β

O = m₁v₁ sin α – m₂v₂ sin β

The property of an object by virtue of which it cannot change its state of rest or of uniform motion along a straight line its own, is called inertia.

Inertia is a measure of mass of a body. Greater the mass of a body greater will be its inertia or vice-versa.

Inertia is of three types:

(i) Inertia of Rest When a bus or train starts to move suddenly, the passengers sitting in it falls backward due to inertia of rest.

(ii) Inertia of Motion When a moving bus or train stops suddenly, the passengers sitting in it jerks in forward direction due to inertia of motion.

(iii) Inertia of Direction We can protect yourself from rain by an umbrella because rain drops can not change its direction its own due to inertia of direction.

Force

Force is a push or pull which changes or tries to change the state of rest, the state of uniform motion, size or shape of a body.

Its SI unit is newton (N) and its dimensional formula is [MLT^-2].

Forces can be categorized into two types:

(i) Contact Forces Frictional force, tensional force, spring force, normal force, etc are the contact forces.

(ii) Action at a Distance Forces Electrostatic force, gravitational force, magnetic force, etc are action at a distance forces.

Impulsive Force

A force which acts on body for a short interval of time, and produces a large change in momentum is called an impulsive force.

Linear Momentum

The total amount of motion present in a body is called its momentum. Linear momentum of a body is equal to the product of its mass and velocity. It is denoted by p.

Linear momentum p = mu.

Its S1 unit is kg-m/s and dimensional formula is [MLT^-1].

It is a vector quantity and its direction is in the direction of velocity of the body.

Impulse

The product of impulsive force and time for which it acts is called impulse.

Impulse = Force * Time = Change in momentum

Its S1 unit is newton-second or kg-m/s and its dimension is [MLT^-1].

It is a vector quantity and its direction is in the direction of force.

Newton’s Laws of Motion

1. Newton’s First Law of Motion

A body continues to be in its state of rest or in uniform motion along a straight line unless an external force is applied on it.

This law is also called law of inertia.

Examples

(i) When a carpet or a blanket is beaten with a stick then the dust particles separate out from it.

(ii) If a moving vehicle suddenly stops then the passengers inside the vehicle bend outward.

2. Newton’s Second Law of Motion

The rate of change of linear momentum is proportional to the applied force and change in momentum takes place in the direction of applied force.

Mathematically F &infi; dp / dt

F = k (d / dt) (mv)

where, k is a constant of proportionality and its value is one in SI and CGS system.

F= mdv / dt = ma

Examples

(i) It is easier for a strong adult to push a full shopping cart than it is for a baby to push the same cart. (This is depending on the net force acting on the object).

(ii) It is easier for a person to push an empty shopping cart than a full one (This is depending on the mass of the object).

3. Newton’s Third Law of Motion

For every action there is an equal and opposite reaction and both acts on two different bodies

Mathematically F₁₂ = – F₂₁

Examples

(i) Swimming becomes possible because of third law of motion.

(ii) Jumping of a man from a boat onto the bank of a river.

(iii) Jerk is produced in a gun when bullet is fired from it.

(iv) Pulling of cart by a horse.

Note Newton’s second law of motion is called real law of motion because first and third laws of motion can be obtained by it.

The modern version of these laws is

(i) A body continues in its initial state of rest or motion with uniform velocity unless acted on by an unbalanced external force.

(ii) Forces always occur in pairs. If body A exerts a force on body B, an equal but opposite force is exerted by body B on body A.

Law of Conservation of Linear Momentum

If no external force acts on a system, then its total linear momentum remains conserved.

Linear momentum depends on frame of reference but law of conservation of linear momentum is independent of frame of reference.

Newton’s laws of motion are valid only in inertial frame of reference.

Weight (w)

It is a field force, the force with which a body is pulled towards the centre of the earth due to gravity. It has the magnitude mg, where m is the mass of the body and g is the acceleration due to gravity.

w = mg

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.

Rocket

Rocket is an example of variable mass following law of conservation of momentum.

Thrust on the rocket at any instant F = – u (dM / dt)

where u = exhaust speed of the burnt and dM / dt = rate 0f gases combustion of fuel.

Velocity of rocket at any instant is given by u = v_o + u log_e (M_o / M )

where, v_o = initial velocity of the rocket,

M_o = initial mass of the rocket and

M = present mass of the rocket.

If effect of gravity is taken into account then speed of rocket

u = v_o + u log_e (M_o / M) – gt

Friction

A force acting on the point of contact of the objects, which opposes the relative motion is called friction.

It acts parallel to the contact surfaces.

Frictional forces are produced due to intermolecular interactions acting between the molecules of the bodies in contact.

Friction is of three types:

1. Static Friction

It is an opposing force which comes into play when one body tends to move over the surface of the other body but actual motion is not taking place.

Static friction is a self adjusting force which increases as the applied force is increased,

2. Limiting Friction

It is the maximum value of static friction when body is at the verge of starting motion.

Limiting friction (f_s) = μ_sR

where μ_s, = coefficient of limiting friction and R = normal reaction.

Limiting friction do not depend on area of contact surfaces but depends on their nature, i.e., smoothness or roughness.

If angle of friction is θ, then coefficient of limiting friction

μ_s = tan θ

3. Kinetic Friction

If the body begins to slide on the surface, the magnitude of the frictional force rapidly decreases to a constant value f_k kinetic friction.

Kinetic friction, f_k = μ_k N

where μ k = coefficient of kinetic friction and N = normal force.

Kinetic friction is of two types:

(a) Sliding friction

(b) Rolling friction

As, rolling friction < sliding friction, therefore it is easier to roll a body than to slide.

Kinetic friction (f_k) = μ_k R

where μ_k = coefficient of kinetic friction and R = normal reaction.

Angle of repose or angle of sliding It is the minimum angle of inclination of a plane with the horizontal, such that a body placed on it, just begins to slide down.

If angle of repose is a. and coefficient of limiting friction is μ, then

μ_s = tan α

Motion on an Inclined Plane

When an object moves along an inclined plane then: different forces act on it like normal reaction of plane, friction force acting in opposite direction of motion etc. Different relations for the motion are given below.

Normal reaction of plane

R = mg cos θ

and net force acting downward on the block.

F = mg sin θ – f

Acceleration on inclined plane a = g (sin θ – μ cos θ)

When angle of inclination of the plane from horizontal is less than the angle of repose (α), then

(i) minimum force required to move the body up the inclined plane

f₁ = mg (sin θ + μ cos θ)

(ii) minimum force required to push the body down the inclined plane

f₂ = mg (μ cos θ – sin θ) J

Tension

Tension force always pulls a body.

Tension is a reactive force. It is not an active force.

Tension across a massless pulley or frictionless pulley remain constant.

Rope becomes slack when tension force becomes zero.

Motion of Bodies in Contact

(i) Two Bodies in Contact If F force is a applied on object of mass m₁ then acceleration of the bodies

a = F / (m₁ + m₂)

Contact force on m₁ = m₁a = m₁F / (m₁ + m₂)

Contact force on m₂ = m₂a = m₂F / (m₁ + m₂)

(ii) Three Bodies in Contact If F force is applied an object of mass m₁, then acceleration of the bodies = F / (m₁ + m₂ + m₃)

Contact force between m₁ and m₂

F₁ = (m₂ + m₃) F / (m₁ + m₂ + m₃)

Contact force between m₂ and m₃

F₂ = m₃ F / (m₁ + m₂ + m₃)

(iii) Motion of Two Bodies, One Resting on the Other

(a) The coefficient of friction between surface of A and B be μ. If a force F is applied on the lower body A. then common acceleration of two bodies

a = F / (M + m)

Pseudo force acting on block B due to the accelerated motion

f’= ma

The pseudo force tends to produce a relative motion between bodies A and B and consequently a frictional force

f = μ N = μmg is developed. For equilibrium

ma ≤ μ mg or a ≤ μg

(b) Let friction is also present between the ground surface and body A Let the coefficient of friction between the given surface and body A is μ₁ and the coefficient of friction between the surfaces of bodies A and B is μ₂ If a force F is applied on the lower body A

Net accelerating force = F – f_A = F – μ₁(M + m)g

∴ Net acceleration

a = F – μ₁(M + m)g / (M + m) = F / (M + m) – μ g

Pseudo force acting on the block B

f’ = ma

The pseudo force tends to produce a relative motion between the bodies A and B are consequently a frictional force f_B = μ mg is developed. For equilibrium

ma le; μ₂ mg or a ≤ μ ₂ g

If acceleration produced under the the effect of force F is more than μ₂g, then two bodies will not move together.

(iv) Motion of Bodies Connected by Strings

Acceleration of the system a = F / (m₁ + m₂ + m₃)

Tension in string T₁ = F

T₂ = ( m₂ + m₃ ) a = (m₂ + m₃) F / (m₁ + m₂ + m₃)

T₃ = m₃a = m₃F / (m₁ + m₂ + m₃)

Pulley Mass System

(i) When unequal masses m₁ and m₂ are suspended from a pulley

(m₁ > m₂)

m₁g – T = m₁a, and T – m₂g = m₂a

On solving equations, we get

a = ((m₁ – m₂) / (m₁ + m₂)) * g

T = 2m₁m₂ / (m₁ + m₂) * g

(ii) When a body of mass m₂ is placed frictionless horizontal surface, then

Acceleration a = m₁g / (m₁ + m₂)

Tension in string T = m₁m₂g / (m₁ + m₂)

(iii) When a body of mass m2 is placed on a rough horizontal surface, then

Acceleration a = ((m₁ – μm₂) / (m₁ + m₂)) * g

Tension in string T = (m₁m₂(1 + μ) / (m₁ + m₂)) * g

(iv) When two masses m₁ and m₂ are connected to a single mass M as shown in figure, then

m₁g – T₁ = m₁a …..(i)

T₂ – m₂g = m₂a ……(ii)

T₁ – T₂ = Ma …….(iii)

Acceleration a = ((m₁ – m₂ / (m₁ + m₂ + M)) * g

Tension T₁ = (2m₂ + M / (m₁ + m₂ + M) * m₁g

T₂ = (2m_a + M / (m₁ + m₂ + M) * m₂g

(v) Motion on a smooth inclined plane, then

m₁g – T = m₁a …..(i)

T – m₂g sin θ = m₂a ……(ii)

Acceleration a = ((m₁ – m₂ sin θ/ (m₁ + m₂)) * g

Tension T = m₁m₂(1 + sin θ) g / (m₁ + m₂)

(vi) Motion of two bodies placed on two inclined planes having different angle of inclination, then

Acceleration a = (m₁ sin θ₁ – m₂ sin θ₂) g / m₁ + m₂

Tension T = (m₁m₂ / m₁ + m₂) * (sin θ₁ – sin θ₂) g

When any object is thrown from horizontal at an angle θ except 90°, then the path followed by it is called trajectory, the object is called projectile and its motion is called projectile motion.

If any object is thrown with velocity u, making an angle θ, from horizontal, then

• Horizontal component of initial velocity = u cos θ.
• Vertical component of initial velocity = u sin θ.
• Horizontal component of velocity (u cos θ) remains same during the whole journey as no acceleration is acting horizontally.
• Vertical component of velocity (u sin θ) decreases gradually and becomes zero at highest point of the path.
• At highest point, the velocity of the body is u cos θ in horizontal direction and the angle between the velocity and acceleration is 90°.

Important Points & Formulae of Projectile Motion

1. At highest point, the linear momentum is mu cos θ and the kinetic energy is (1/2)m(u cos θ)².
2. The horizontal displacement of the projectile after t seconds
   x = (u cos θ)t
3. The vertical displacement of the projectile after t seconds
   y = (u sin θ) t — (1/2)gt²
4. Equation of the path of projectile
5. The path of a projectile is parabolic.
6. Kinetic energy at lowest point = (1/2) mu²
7. Linear momentum at lowest point = mu
8. Acceleration of projectile is constant throughout the motion and it acts vertically downwards being equal to g.
9. Angular momentum of projectile = mu cos θ x h, where h denotes the height.
10. In case of angular projection, the angle between velocity and acceleration varies from 0° < θ < 180°.
11. The maximum height occurs when the projectile covers a horizontal distance equal to half of the horizontal range, i.e., R/2.
12. When the maximum range of projectile is R, then its maximum height is R/4.

Time of flight It is defined as the total time for which the projectile remains in air.

Maximum height It is defined as the maximum vertical distance covered by projectile.

Horizontal range It is defined as the maximum distance covered in horizontal distance.

Note

(i) Horizontal range is maximum when it is thrown at an angle of 45° from the horizontal

(ii) For angle of projections and (90° – 0) the horizontal range is same.

Projectile Projected from Some Heights

1. When Projectile is Projected Horizontally

Initial velocity in vertical direction = 0

Time of flight T = √(2H/g)

Horizontal range x = uT = u √(2H/g)

Vertical velocity after t seconds

v_y = gt (u_y = 0)

Velocity of projectile after t seconds

If velocity makes an angle φ, from horizontal, then

Equation of the path of the projectile

2. When Projectile Projected Downward at an Angle with Horizontal

Initial velocity in horizontal direction = u cos θ

Initial velocity in vertical direction = u sin θ

Time of flight can be obtained from the equation,

Horizontal range x = (u cos θ) t

Vertical velocity after t seconds

v_y = u sin θ + gt

Velocity of projectile after t seconds

3. When Projectile Projected Upward at an Angle with Horizontal

Initial velocity in horizontal direction = u cos θ

Initial velocity in vertical direction = u sin θ

Time of flight can be obtained from the equation

Horizontal range x = (u cos θ)t

Vertical velocity after t seconds, v_y = (- u sin θ) + gt

Velocity of projectile after t seconcil

4. Projectile Motion on an Inclined Plane

When any object is thrown with velocity u making an angle α from horizontal, at a plane inclined at an angle β from horizontal, then

Initial velocity along the inclined plane = u cos (α – β)

Initial velocity perpendicular to the inclined plane

For angle of projections a and (90° – α + β), the range on inclined plane are same.

Circular Motion

Circular motion is the movement of an object in a circular path.

1. Uniform Circular Motion

If the magnitude of the velocity of the particle in circular motion remains constant, then it is called uniform circular motion.

2. Non-uniform Circular Motion

If the magnitude of the velocity of the body in circular motion is n constant, then it is called non-uniform circular motion.

Note A special kind of circular motion is when an object rotates around itself. This can be called spinning motion.

Variables in Circular Motion

(i) Angular Displacement Angular displacement is the angle subtended by the position vector at the centre of the circular path.

Angular displacement (Δθ) = (ΔS/r)

where Δs is the linear displacement and r is the radius. Its unit is radian.

(ii) Angular Velocity The time rate of change of angular displacement (Δθ) is called angular velocity.

Angular velocity (ω) = (Δθ/Δt)

Angular velocity is a vector quantity and its unit is rad/s.

Relation between linear velocity (v) and angular velocity (ω) is given by

v = rω

(iii) Angular Acceleration The time rate of change of angular velocity (dω) is called angular acceleration.

Its unit is rad/s² and dimensional formula is [T^-2].

Relation between linear acceleration (a) and angular acceleration (α).

a = rα

where, r = radius

Centripetal Acceleration

In circular motion, an acceleration acts on the body, whose direction is always towards the centre of the path. This acceleration is called centripetal acceleration.

Centripetal acceleration is also called radial acceleration as it acts along radius of circle.

Its unit is in m/s² and it is a vector quantity.

Centripetal Force

It is that force which complex a body to move in a circular path.

It is directed along radius of the circle towards its centre.

For circular motion a centripetal force is required, which is not a new force but any force present there can act as centripetal force.

where, m = mass of the body, c = linear velocity,

ω = angular velocity and r = radius.

Work done by the centripetal force is zero because the centripetal force and displacement are at right angles to each other.

Examples of some incidents and the cause of centripetal force involved.

Kinematical Equations in Circular Motion

Relations between different variables for an object executing circular motion are called kinematical equations in circular motion.

where, ω₀ = initial angular velocity, ω = final angular velocity,

α = angular acceleration, θ = angular displacement and t = time.

Centrifugal Force

Centrifugal force is equal and opposite to centripetal force.
Under centrifugal force, body moves only along a straight line.

It appears when centripetal force ceases to exist.

Centrifugal force does not act on the body in an inertial frame but arises as pseudo forces in non-inertial frames and need to be considered.

Turning at Roads

If centripetal force is obtained only by the force of friction between the tyres of the vehicle and road, then for a safe turn, the coefficient of friction (µ_s) between the road and tyres should be,

where, v = the velocity of the vehicle and r = radius of the circular path.

If centripetal force is obtained only by the banking of roads, then the speed (a) of the vehicle for a safe turn

v = √rg tan θ

If speed of the vehicle is less than √rg tan θ than it will move inward (down) and r will decrease and if speed is more than √rg tan θ, then it will move outward (up) and r will increase.

In normal life, the centripetal force is obtained by the friction force between the road and tyres as well as by the banking of the roads.

Therefore, the maximum permissible speed for the vehicle is much greater than the optimum value of the speed on a banked road. When centripetal force is obtained from friction force as well as banking of roads, then maximum safe value of speed of vehicle

When a cyclist takes turn at road, he inclined himself from the vertical, slower down his speed and move on a circular path of larger radius.

If a cyclist inclined at an angle θ, then tan θ = (v²/rg)

where, v = speed of the cyclist, r = radius of path and g = acceleration due to gravity.

Motion in a Vertical Circle

(i) Minimum value of velocity at the highest point is √gr

(ii) The minimum velocity at the bottom required to complete the circle

v_A = √5gr

(iii) Velocity of the body when string is in horizontal position

v_B = √3gr

(iv) Tension in the string

• At the top T_c = 0,
• At the bottom T_A = 6 mg
• When string is horizontal T_B = 3 mg

(v) When a vehicle is moving over a convex bridge, then at the maximum height, reaction (N₁) is N₁ = mg – (mv²/r)

(vi) When a vehicle is moving over a concave bridge, then at the lowest point, reaction (N₂) is

N₂ = mg + (mv²/r)

(vii) When a car takes a turn, sometimes it overturns. During the overturning, it is the inner wheel which leaves the ground first.

(viii) A driver sees a child in front of him during driving a car, then it, better to apply brake suddenly rather than taking a sharp turn to avoid an accident.

Non-uniform Horizontal Circular Motion

In non-uniform horizontal circular motion, the magnitude of the velocity of the body changes with time.

In this condition, centripetal (radial) acceleration (a_R) acts towards centre and a tangential acceleration (a_T) acts towards tangent. Both acceleration acts perpendicular to each other.

Resultant acceleration

where, α is angular acceleration, r = radius and a = velocity.

Conical Pendulum

It consists of a string OA whose upper end 0 is fixed and bob is tied at the other free end. The string traces the surface of the cone, the arrangement is called a conical pendulum.

Time period of conical pendulum,

If an object changes its position with respect to its surroundings with time, then it is called in motion.

Rest

If an object does not change its position with respect to its surroundings with time, then it is called at rest.

[Rest and motion are relative states. It means an object which is at rest in one frame of reference can be in motion in another frame of reference at the same time.]

Point Mass Object An object can be considered as a point mass object, if the distance travelled by it in motion is very large in comparison to its dimensions.

Types of Motion

1. One Dimensional Motion

If only one out of three coordinates specifying the position of the object changes with respect to time, then the motion is called one dimensional motion.

For instance, motion of a block in a straight line motion of a train along a straight track a man walking on a level and narrow road and object falling under gravity etc.

2. Two Dimensional Motion

If only two out of three coordinates specifying the position of the object changes with respect to time, then the motion is called two dimensional motion.

A circular motion is an instance of two dimensional motion.

3. Three Dimensional Motion

If all the three coordinates specifying the position of the object changes with respect to time, then the motion is called three dimensional motion.

A few instances of three dimension are flying bird, a flying kite, a flying aeroplane, the random motion of gas molecule etc.

Distance

The length of the actual path traversed by an object is called the distance.

It is a scalar quantity and it can never be zero or negative during the motion of an object.

Its unit is metre.

Displacement

The shortest distance between the initial and final positions of any object during motion is called displacement. The displacement of an object in a given time can be positive, zero or negative.

It is a vector quantity.

Its unit is metre.

Speed

The time rate of change of position of the object in any direction is called speed of the object.

Speed (v) = Distance travelled (s) / Time taken (t)

Its unit is m/s.

It is a scalar quantity.

Its dimensional formula is [M^oT^-1].

Uniform Speed

If an object covers equal distances in equal intervals of time, then its speed is called uniform speed.

Non-uniform or Variable Speed

If an object covers unequal distances in equal intervals of time, then its speed is called non-uniform or variable speed.

Average Speed

The ratio of the total distance travelled by the object to the total time taken is called average speed of the object.

Average speed = Total distanced travelled / Total time taken

If a particle travels distances s₁, s₂, s₃ , … with speeds v₁, v₂, v₃, …, then

Average speed = s₁ + s₂ + s₃ + ….. / (s₁ / v₁ + s₂ / v₂ + s₃ / v₃ + …..)

If particle travels equal distances (s₁ = s₂ = s) with velocities v₁ and v₂, then

Average speed = 2 v₁ v₂ / (v₁ + v₂)

If a particle travels with speeds v₁, v₂, v₃, …, during time intervals t₁, t₂, t₃,…, then

Average speed = v₁t₁ + v₂t₂ + v₃t₃ +… / t₁ + t₂ + t₃ +….

If particle travels with speeds v₁, and v₂ for equal time intervals, i.e., t₁ = t₂ = t₃, then

Average speed = v₁ + v₂ / 2

When a body travels equal distance with speeds V₁ and V₂, the average speed (v) is the harmonic mean of two speeds.

2 / v = 1 / v₁ + 1 / v₂

Instantaneous Speed

When an object is travelling with variable speed, then its speed at a given instant of time is called its instantaneous speed.

Instantaneous speed =

Velocity

Tlie rate of change of displacement of an object in a particular direction is called its velocity.

Velocity = Displacement / Time taken

Its unit is m/s.

Its dimensional formula is [M^oT^-1].

It is a vector quantity, as it has both, the magnitude and direction.

The velocity of an object can be positive, zero and negative.

Uniform Velocity

If an object undergoes equal displacements in equal intervals of time, then it is said to be moving with a uniform velocity.

Non-uniform or Variable Velocity

If an object undergoes unequal displacements in equal intervals of time, then it is said to be moving with a non-uniform or variable velocity.

Relative Velocity

Relative velocity of one object with respect to another object is the time rate of change of relative position of one object with respect to another object.

Relative velocity of object A with respect to object B

V_AB = V_A – V_B

When two objects are moving in the same direction, then

When two objects are moving in opposite direction, then

When two objects are moving at an angle, then

and tan β = v_B sin θ / v_A – v_B cos θ

Average Velocity

The ratio of the total displacement to the total time taken is called average velocity.

Average velocity = Total displacement / Total time taken

Acceleration

The time rate of change of velocity is called acceleration.

Acceleration (a) = Change in velocity (Δv) / Time interval (Δt)

Its unit is m/s²

Its dimensional formula is [M^oLT^-2].

It is a vector quantity.

Acceleration can be positive, zero or negative. Positive acceleration means velocity increasing with time, zero acceleration means velocity is uniform while negative acceleration (retardation) means velocity is decreasing with time.

If a particle is accelerated for a time t₁ with acceleration a₁ and for a time t₂ with acceleration a₂, then average acceleration

a_av = a₁t₁ + a₂t₂ / t₁ + t₂

Different Graphs of Motion

Displacement – Time Graph

Note Slope of displacement-time graph gives average velocity.

Velocity – Time Graph

Note Slope of velocity-time graph gives average acceleration.

Acceleration – Time

Equations of Uniformly Accelerated Motion

If a body starts with velocity (u) and after time t its velocity changes to v, if the uniform acceleration is a and the distance travelled in time t in s, then the following relations are obtained, which are called equations of uniformly accelerated motion.

(i) v = u + at

(ii) s = ut + at²

(iii) v² = u² + 2as

(iv) Distance travelled in nth second.

S_n = u + a / 2(2n – 1)

If a body moves with uniform acceleration and velocity changes from u to v in a time interval, then the velocity at the mid point of its path

√u² + v² / 2

Motion Under Gravity

If an object is falling freely (u = 0) under gravity, then equations of motion

(i) v = u + gt

(ii) h = ut + gt²

(iii) V² = u² + 2gh

Note If an object is thrown upward then g is replaced by – g in above three equations.

It thus follows that

(i) Time taken to reach maximum height

t_A = u / g = √2h / g

(ii) Maximum height reached by the body

h_max = u² / 2g

(iii) A ball is dropped from a building of height h and it reaches after t seconds on earth. From the same building if two ball are thrown (one upwards and other downwards) with the same velocity u and they reach the earth surface after t, and t2 seconds respectively, then

t = √t₁t₂

(iv) When a body is dropped freely from the top of the tower and another body is projected horizontally from the same point, both will reach the ground at the same time.