Revision: Properties of Solids and Liquids JEE Main Properties of Solids and Liquids

The internal restoring force per unit area of a body is called stress.

OR

The internal restoring force acting per unit area of a deformed body is called stress.

• SI Unit: N/m² (pascal, Pa)
   Dimensions: [M¹L⁻¹T⁻²]

The strain is defined as the ratio of change in dimensions of the body to its original dimensions.

Strain = `"change in dimensions"/"original dimensions"`

When equal normal forces are applied on every surface of a body causing a change in volume, the restoring force opposing this change per unit area is called hydraulic stress (also called volume stress).

When there is a decrease in the length or compression of the body due to the applied force, the stress produced is called compressive stress.

When there is an increase in the length or extension of the body in the direction of the applied force, the stress produced is called tensile stress.

The angular displacement of the surface in direct contact with the applied shear stress from its original position is called shear strain: τ = W/L = tan ⁡θ.

The ratio of change in length of the body to its initial length is called longitudinal strain: ε = ΔL/L.

Strain is defined as the ratio of the change in dimensions of the body to its original dimensions.

OR

The ratio of change in configuration to the original configuration is called strain.

• It has no unit and no dimensions (pure ratio).

The ratio of change in volume of the body to its original volume is called volume strain: ΔV/V.

The modulus of elasticity of a material is the ratio of stress to the corresponding strain. It is defined as the slope of the stress-strain curve in the elastic deforming region and depends on the nature of the material.

\[\frac {stress}{strain}\] = Constant

The constant is called the modulus of elasticity.

OR

The constant ratio of stress to strain within the elastic limit is called the Modulus of Elasticity.

A graph drawn by taking tensile strain along the x-axis and tensile stress along the y-axis, obtained by gradually increasing the load on a metal wire suspended vertically from a rigid support until the wire breaks, and measuring the elongation produced during each step.

In case of some materials like vulcanized rubber, when the stress applied on a body decreases to zero, the strain does not return to zero immediately. The strain lags behind the stress. This lagging of strain behind the stress is called elastic hysteresis.

"Young’s modulus is the ratio of longitudinal stress to longitudinal strain."

OR

The ratio of tensile (or compressive) stress to the longitudinal strain is called Young's Modulus of Elasticity, denoted by Y.

"Young’s modulus is the ratio of longitudinal stress to longitudinal strain."

OR

The ratio of tensile (or compressive) stress to the longitudinal strain is called Young's Modulus of Elasticity, denoted by Y.

"Shear modulus or modulus of rigidity: It is defined as the ratio of shear stress to shear strain within elastic limits."

OR

The ratio of shearing stress to the corresponding shearing strain in a material is called the Shear Modulus or Modulus of Rigidity, denoted by G.

"Bulk modulus is defined as the ratio of volume stress to volume strain."

OR

The ratio of hydraulic stress to the corresponding hydraulic strain (change in volume) is called the Bulk Modulus, denoted by B.

Within elastic limit, the ratio of lateral strain to the linear strain is called the Poisson's ratio.

The maximum stress that a material can withstand is called the Ultimate Tensile Strength (Point D).

The reciprocal of the bulk modulus is called compressibility: k = \[\frac {1}{B}\].

The point on the stress-strain curve up to which Hooke's Law is valid is called the proportional limit (Point A).

The stress at the yield point (end of elastic behavior and start of plastic deformation) is called the yield strength.

The point at which the material breaks and failure of the material takes place is called the fracture point (Point E).

SI unit of pressure is the pascal (Pa) or Nm⁻²
One Pascal: When a force of one newton acts normally on an area of one square metre (1 m²) then the pressure acting on the surface acting on the surface is called one Pascal.

When a liquid flows such that particles passing through a given point have different velocities from the predecessor, such a flow is called turbulent flow.

When a liquid flows such that each particle passing through a specific point follows the exact same path at the same speed as the particle before it, this type of flow is called streamline flow or steady flow.

The maximum constant velocity acquired by a body while falling freely through a viscous medium is called the terminal velocity V_T.

The property of a fluid by virtue of which it opposes the relative motion between its different layers, with the force that comes into play, is called viscosity; and that force is called the viscous force.

where η is the coefficient of viscosity.

The rate of change of velocity (dv) with distance (dx) measured from a stationary layer is called velocity gradient.

∴ Velocity gradient = `(dv)/dx`

The coefficient of viscosity of a liquid is defined as the viscous force acting tangentially per unit area of a liquid layer having a unit velocity gradient in a direction perpendicular to the direction of flow of the liquid.

The angle between tangents drawn at the point of contact to the liquid surface and the solid surface inside the liquid is called the angle of contact for a pair of solid and liquid. It is denoted by θ.

Surface tension is defined as the force per unit length acting at right angles to an imaginary line drawn on the free surface of the liquid. 

When a liquid is in contact with a solid, the angle between the tangent drawn to the free surface of the liquid and the surface of solid at the point of contact measured inside the liquid is called the angle of contact.

Surface tension is defined as the force acting on a unit length of an imaginary line drawn on the free surface of the liquid, the direction of the force being perpendicular to the line so drawn and acting parallel to the surface.

The difference of pressure between the two sides of a liquid surface, which arises in equilibrium because the pressure inside a bubble or drop is greater than outside, is called excess pressure.

The potential energy is greater for molecules at the surface film as compared to molecules well inside the liquid. This extra energy of the molecule on the surface layer of a liquid is called the surface energy of the liquid.

A thin film of liquid near its surface having thickness equal to the molecular range of attraction is called surface film.

The property of a liquid due to which its free surface tries to have minimum surface area and behaves as if it were under tension somewhat like a stretched elastic membrane is called surface tension.

OR

The force acting along the surface of a liquid per unit length is called surface tension.

The work per unit area done by the force that creates a new surface is called surface energy.

OR

The energy required to increase the surface area of a liquid is called surface energy.

An imaginary sphere drawn round a molecule (taken as centre) with a radius equal to the range of molecular attraction is called the sphere of influence of that molecule.

When cohesive forces are stronger than adhesive forces (e.g., mercury in glass), the curved liquid surface formed where cos θ is negative and the liquid level is lower is called a convex meniscus.

When adhesive forces are stronger than cohesive forces, the curved liquid surface formed when the liquid is in contact with a solid is called a concave meniscus.

The angle between the surface of the solid and the tangent drawn to the surface of the liquid at the point of contact on the side of liquid is called the angle of contact of that liquid with that solid.

OR

The angle enclosed between the tangents to the liquid surface and the solid surface inside the liquid, both the tangents being drawn at the point of contact of the liquid with the solid, is called the angle of contact.

The rise or fall of liquid in a narrow tube due to surface tension is called capillary action.

"Heat is energy in transit. When two bodies at different temperatures are brought in contact, they exchange heat."

OR

The form of energy which is exchanged among various bodies or a system on account of temperature difference is called heat.

• Units: joule (J), calorie (cal), BTU (British Thermal Unit)

"Temperature is a physical quantity that defines the thermodynamic state of a system."

OR

The degree of hotness or coldness of a body, whose natural flow is from higher temperature to lower temperature, is called temperature.

• SI unit: kelvin (K) | Scalar quantity

One mole of any substance is the amount of that substance which contains the Avogadro number (NA) of particles (such as atoms or molecules).

The temperature at which pure water freezes at 1 atm pressure is called the ice point.

Thermometry is the branch of physics dealing with temperature measurement. It relies on the principle that certain physical properties of materials change continuously and predictably with temperature.

The temperature at which pure water boils and vapourises into steam at 1 atm pressure is called the steam point or boiling point.

An adiabatic wall is an ideal partition that completely prevents heat transfer between two systems. In diagrams, it is shown as a thick, cross-hatched (slanting lines) region.

A diathermic wall is a partition that freely allows heat to flow between two systems. It is shown as a thin dark line in diagrams. A thin copper sheet is a good example.

When two bodies at different temperatures are brought into contact through a diathermic wall, heat flows from the hotter body to the cooler one. This continues until both reach the same temperature, at which point heat flow stops. This state is called thermal equilibrium.

One kelvin = 1/273.16 of the difference between absolute zero and the triple point of water.

The lowest theoretically possible temperature (0 K = −273.15 °C), where ideal gas molecules have zero kinetic energy.

OR

The lowest attainable temperature, obtained by plotting the relation between pressure of the gas vs its temperature, where all lines for different gases cut the temperature axis at the same point (−273.15°C), is called the absolute zero of temperature.

A hypothetical gas whose molecules have no volume and exert no intermolecular forces; obeys PV = μRT exactly.

The unique temperature & pressure at which solid, liquid, and gas phases of a substance coexist in equilibrium.

OR

The temperature where the solid, liquid, and gas state of a material co-exist in equilibrium, and this occurs only at a unique temperature and pressure, is called the triple point.

The temperature scale where −273.15°C corresponds to 0 K, i.e., the temperature at which the pressure of a gas would become zero, is called the absolute temperature (0 K).

The SI absolute temperature scale starting at absolute zero. Written as K (no degree symbol °).

A constant in the ideal gas equation; R = 8.31 J mol⁻¹ K⁻¹.

Extending a graph line beyond the measured data to predict values.

The increase in length per unit original length of a rod (at 0°C) per unit rise in temperature is called the coefficient of linear expansion.

1. Consider a metallic rod of length l₀ fixed between two rigid supports at T °C.
2. If the temperature of rod is increased by ΔT, length of the rod would become, l = l₀ (1 + αΔT) Where, α is the coefficient of linear expansion of the material of the rod.
3. But the supports prevent the expansion of the rod. As a result, rod exerts stress on the supports. Such stress is termed as thermal stress.

The increase in volume of a body per unit original volume (at 0°C) per unit rise in temperature is called the coefficient of cubical expansion.

The increase in the dimensions (length, area, or volume) of a body due to an increase in its temperature is called thermal expansion. Conversely, a decrease in temperature causes contraction.

OR

The increase in the dimensions of a body due to an increase in its temperature is called thermal expansion.

OR

When matter changes its shape, area and volume in response to a change in temperature (i.e., an object expands and becomes larger due to a change in its temperature), this is called thermal expansion.

The change in area per unit original surface area of a two-dimensional body (at 0°C) per unit rise in temperature is called the coefficient of superficial expansion.

The quantity of heat needed to raise the temperature of the whole body by 1°C (or 1 K) is called heat capacity.

OR

The amount of heat ΔQΔQ supplied to a substance to change its temperature from T to T + ΔT, per unit mass per unit degree change in temperature, is called specific heat:

The amount of heat required to raise the temperature of one mole of a substance through a unit degree Celsius or Kelvin is called molar heat capacity.

The heat capacity of a body is the quantity of heat required to raise its temperature by 1°C. It depends upon the mass and the nature of the body.

The specific heat capacity of a substance is the amount of heat energy required to raise the temperature of unit mass of that substance through 1°C (or 1 K).

OR

Heat capacity of a body when expressed for the unit mass is called the specific heat capacity of the substance of that body.

OR

The amount of heat energy required to raise the temperature of a unit mass of an object by 1 °C is called the specific heat of that object.

OR

The amount of heat per unit mass absorbed or given out by a substance to change its temperature by one unit (one degree), i.e., 1°C or 1 K, is called specific heat capacity.

OR

The quantity of heat required to raise the temperature of a unit mass of a gas by one degree, whose exact value depends upon the mode of heating the gas and can range from zero to infinity or even be negative, is called the specific heat capacity of a gas.

Calorimetry is the science of measuring heat exchange during physical or chemical processes. The word comes from the Latin calor (heat) + Greek metron (measure).

OR

An experimental technique for the quantitative measurement of heat exchange is called calorimetry.

A calorimeter is a cylindrical vessel which is used to measure the amount of heat gained (or lost) by a body when it is mixed with another body or substance.

The quantity of heat required to convert unit mass of a substance from its liquid state to vapour state, at its boiling point without any change in its temperature is called its latent heat of vapourization (L_v).

The quantity of heat required to convert unit mass of a substance from its solid state to the liquid state, at its melting point, without any change in its temperature, is called its latent heat of fusion (L_f).

OR

The heat energy absorbed at constant temperature during the transformation of solid into liquid is called the latent heat of fusion. The amount of heat energy absorbed at constant temperature by unit mass of a solid to convert into liquid phase is called the specific latent heat of fusion.

The heat energy absorbed (or liberated) in change of phase is not externally manifested by any rise or fall in temperature, it is called the latent heat.

OR

Latent heat is the quantity of heat energy required to change the state of unit mass of a substance from one phase to another, at constant temperature and constant pressure.

OR

The coefficient of thermal conductivity of a material is defined as the quantity of heat that flows in one second between the opposite faces of a cube of side 1 m, the faces being kept at a temperature difference of 1°C (or 1 K).

Solid substances that conduct heat easily are called good conductors of heat.

Substances that do not conduct heat easily are called bad conductors of heat.

Conduction is the process by which heat flows from the hot end to the cold end of a solid body without any net bodily movement of the particles of the body.

OR

The process by which heat flows from the hot end to the cold end of a solid body without any net bodily movement of the particles of the body is called conduction.

Convection is the process by which heat is transmitted through a substance from one point to another due to the actual bodily movement of the heated particles of the substance.

OR

The process by which heat is transmitted through a substance from one point to another due to actual bodily movement of the heated particles of the substance is called convection.

OR

The mode of heat transfer by actual motion of matter (bulk transport of fluid) from the source of heat, which occurs only in fluids, is called convection.

The transfer of heat energy from one place to another via emission of EM energy (in a straight line with the speed of light) without heating the intervening medium is called radiation.

OR

The transfer of heat energy from one place to another via emission of EM energy (in a straight line with the speed of light) without heating the intervening medium is called radiation.

The heating-up of the earth’s atmosphere due to trapped infrared rays reflected from the earth's surface by atmospheric gases is called the greenhouse effect.

The constant maximum velocity acquired by a body while falling through a viscous fluid is called terminal velocity.

The phenomenon in which a liquid rises in a capillary tube when the angle of contact is acute, or falls when the angle of contact is obtuse, due to the interplay of pressure caused by the liquid column and pressure difference due to surface tension, is called capillary ascent (or descent).

The phenomenon where a liquid in a capillary tube either ascends or descends relative to the surrounding liquid when a tube of very narrow bore is dipped in it is called capillarity.

The rise or fall of level of liquid in a capillary tube is called capillary action or capillarity.

The vertical height attained by a liquid in a capillary tube at equilibrium, which is independent of the shape of the capillary provided the radius of meniscus remains the same, is called the capillary rise height (h).

A tube with a hole of very small diameter is called a capillary tube or capillary.

The mathematical expression for Young's modulus (Y) is:

Y = \[\frac{MgL}{\pi r^2l}\] or \[\frac {FL}{AΔL}\]

Where:

• Y = Young’s Modulus
• M = Mass of the load attached
• g = Acceleration due to gravity
• L = Original length of the wire
• r = Radius of the wire cross-section
• l = Extension or elongation produced in the wire

The mathematical expression for Young's modulus (Y) is:

Y = \[\frac{MgL}{\pi r^2l}\] or \[\frac {FL}{AΔL}\]

Where:

• Y = Young’s Modulus
• M = Mass of the load attached
• g = Acceleration due to gravity
• L = Original length of the wire
• r = Radius of the wire cross-section
• l = Extension or elongation produced in the wire

The formula for modulus of rigidity is:

η = \[\frac{\text{Shear Stress}}{\text{Shear Strain}}=\frac{F/A}{\theta}=\frac{F}{A\cdot\theta}\]

Where:

• η = Modulus of rigidity (Pa or N/m²)
• F = Tangential force applied (N)
• A = Cross-sectional area on which force acts (m²)
• θ = Shear strain = Δl/l (in radians)
• Δl = Displacement of the upper surface relative to the lower surface (m)
• l = Original height of the block (m)

SI Unit: Pascal (Pa) or N/m²​

Dimensional Formula: M¹L⁻¹T⁻²

The mathematical representation of Bulk Modulus (K) is:

K = \[\frac{\text{Volume Stress}}{\text{Volume Strain}}\]

K = \[\frac{dP}{\left(\frac{dV}{V}\right)}\] = V \[\frac {dP}{dV}\]

Where:

• K: Bulk Modulus
• dP: Change in pressure (Volume Stress)
• dV: Change in volume
• V: Original volume

\[\sigma=\frac{\text{Lateral strain}}{\text{Linear strain}}=\frac{\frac{d}{D}}{\frac{\Delta l}{l}}=\frac{d\cdot l}{D\cdot\Delta l}\]

Where:

• σ = Poisson's ratio
• l = original length of the wire
• ∆l = increase or decrease in length of the wire
• D = original diameter of the wire
• d = corresponding change in diameter of the wire

Important Note: Poisson's ratio has no unit. It is dimensionless.

Case 1: θ < 90° (Concave Meniscus)

Q = mcΔT

Where:

• Q = Heat absorbed or released (in joules)
• m = Mass of the substance (in kg)
• c = Specific heat capacity (J/kg·K)
• ΔT = Change in temperature (T_final−T_initial)

\[E_k=\frac{3}{2}k_BT\]

Where:

• E_k = Average kinetic energy of the molecules (in joules)
• k_B = Boltzmann constant = 1.380649 × 10⁻²³ J/K
• T = Absolute temperature (in kelvin)

Master Conversion Formula:

\[\frac{T_F-32}{180}=\frac{T_C}{100}\] = \[\frac {T_K−273.15}{100}\]

Master Conversion Formula:

\[\frac {T_C}{100}\] = \[\frac {(T_{F}-32)}{180}\] = \[\frac {(T_{K}-273.15)}{100}\]

Specific heat capacity c = \[\frac{\text{Heat capacity of body } C'}{\text{Mass of the body } m}\]

or

Specific heat capacity c = \[\frac{Q}{m\times\Delta t}\]

C = M × c = Q/(nΔT)

Unit: J/mol · K

\[Q=mc\Delta T\]

Q = m × L

where,

Q = Heat energy absorbed or released during phase change
m = Mass of the substance undergoing phase change
L = Specific Latent Heat (characteristic of the substance & process)

SI Units = J kg⁻¹

Proportionality Form

\[\frac{dT}{dt}\propto(T-T_0)\]

Introducing the constant of proportionality C:

\[\frac{dT}{dt}=C\left(T-T_0\right)\]

T = Temperature of the body at time t
T₀ = Temperature of the surroundings (constant)
C = Constant of proportionality
\[\frac {dT}{dt}\] = Rate of fall of temperature (rate of cooling)

v = \[\frac{2}{9}\cdot\frac{r^2(\rho-\sigma)g}{\eta}\]

where:

• v = terminal velocity
• r = radius of the body
• ρ = density of the body
• σ = density of the fluid
• g = acceleration due to gravity
• η = coefficient of viscosity of the fluid

For two different liquids in the same tube:

\[\frac{h_1}{h_2}=\frac{\rho_2T_1}{\rho_1T_2}\]

For the same liquid in tubes of different radii:

h₁​r₁​ = h₂​r₂​

Principle: Pressure due to liquid column = Pressure difference due to surface tension

h = \[\frac {2T}{Rdg}\] = \[\frac {2T cos θ}{rdg}\]

where r = radius of capillary tube and θ = angle of contact.

Hooke's Law was discovered by English scientist Robert Hooke in 1660. He first stated it as a Latin anagram: "As the extension, so the force."

Statement: For small deformations, stress is directly proportional to strain, within the elastic limit.

Key Points:

• Hooke's Law is a measure of elasticity.
• It is valid only up to the elastic limit. Beyond this, the material does not return to its original shape and Hooke's Law no longer applies.
• In springs: The force needed to extend or compress a spring by distance x is proportional to that distance → F = −kx (where k is the spring constant).
• Hooke's Law is applicable only in the case of elastic deformation.

Statement: Pascal's Law states that when pressure is applied to a confined (enclosed) fluid, it is transmitted undiminished and equally in all directions throughout the fluid and to the walls of its container.

Mathematical Expression:

Key Points:

• Pressure changes by the same value at every point inside an incompressible, confined liquid.
• Used in hydraulic machines where a small force on a small area produces a large force on a large area.
• Applications: Hydraulic lift, hydraulic brake, hydraulic press, hydraulic jack.

Statement:

"According to this theorem, the total energy (pressure energy, potential energy and kinetic energy) per unit volume or mass of an incompressible and non-viscous fluid in steady flow through a pipe remains constant throughout the flow, provided there is no source or sink of the fluid along the length of the pipe."

Mathematical Form:

For unit volume of liquid flowing through a pipe:

\[P+\rho gh+\frac{1}{2}\rho v^2\] = constant

where:

• P = pressure energy per unit volume
• ρ = density of the fluid
• g = acceleration due to gravity
• h = height of the fluid (potential energy term)
• v = velocity of the fluid (kinetic energy term)

Applications of Bernoulli's Theorem:

• Speed of efflux
• Venturi tube
• Lifting up of aeroplane
• Working of an atomizer
• Blowing off of roofs by stormy wind

Statement: Stokes' Law describes the force of viscosity exerted on a spherical object as it moves through a fluid. The viscous drag on a spherical body of radius r, moving with velocity v, in a medium of viscosity η is given by:

Key Points:

• The negative sign in F = −ηA\[\frac {dv}{dx}\] shows that viscous force opposes the direction of motion.
• At terminal velocity, upward viscous force + buoyant force = weight of the body.
• Terminal velocity formula derived from Stokes' Law:
  v₀ = \[\frac{2gr^2(\rho-\sigma)}{9\eta}\]
• Terminal velocity is proportional to r² — larger spheres fall faster.
• The coefficient of viscosity η has dimensional formula [ML⁻¹T⁻¹].
• SI unit of viscosity: Ns/m² (or Pa·s); CGS unit: dyne·s/cm² = poise.

T = `F/L`

where F = Force (N), L = Length (m)

= SI unit of T = `N/m`

Surface tension can also be written as

T = `W/A`

where W = Work (J), A = Area (m²)

= SI unit of T = `J/m^2`

We know

1J=1N×1m

So,

`J/m^2 = (N * m)/m^2 = N/m`

Both units are the same

`1N/m equiv 1J/m^2`

If system A is in thermal equilibrium with system C, and system B is also in thermal equilibrium with system C, then systems A and B are in thermal equilibrium with each other.

When pressure is constant, the ratio of volume to temperature of a gas remains constant.

When temperature is constant, the product of pressure and volume of a gas remains constant.

When volume is constant, the ratio of pressure to temperature of a gas remains constant.

Statement: When different parts of an isolated system are at different temperatures, heat transfers from the part at higher temperature to the part at lower temperature. The heat lost by the hot object is equal to the heat gained by the cold object, provided no heat is allowed to escape to the surroundings.

(For liquid in calorimeter: m₁c₁Δθ + m_cc_cΔθ)

Key Points:

• A system is said to be isolated if no exchange of heat occurs between the system and its surroundings.
• Calorimetry literally means measurement of heat.
• Energy supplied by heater = VIt (voltage × current × time).
• This principle is based on the Law of Conservation of Energy.

Statement: In steady-state heat flow by conduction in a bar with ends maintained at different temperatures T_C and T_D, the heat flow is proportional to the temperature difference and the area of cross-section A, and inversely proportional to the length L.

Also written as:

Where K is the thermal conductivity of the material.

Key Points:

• Gases are poor conductors; liquids have intermediate conductivities; solids are generally good conductors.
• The greater the value of K, the more rapidly the material conducts heat.

Statement: All bodies emit radiant energy depending on their temperature. The heat emitted (H) by a body is given by:

Where:

• σ = Stefan-Boltzmann constant
• e = Emissivity (for perfect radiator, e = 1)
• A = Area of the body
• T = Temperature (in Kelvin)

Key Points:

• Black bodies absorb and emit more radiant energy than bodies of lighter colors.
• Thermal radiation is partially reflected and partially absorbed when it falls on other bodies.

The wavelength (λ_m​) for which the emissive power of a blackbody is maximum is inversely proportional to the absolute temperature of the blackbody:

With increase in temperature, λ_m​ decreases (shifts towards shorter wavelengths). Also, the energy E_max​ emitted at λ_m​ increases with the fifth power of temperature, i.e., E_max ∝ T⁵.

Statement: The rate of loss of heat \[\frac {dT}{dt}\] of the body is directly proportional to the difference of temperature (T − T₀) of the body and the surroundings, provided the difference in temperatures is small.

Mathematical Form:

Graphical Representation:

• Graph of rate of cooling \[\left(\frac{dT}{dt}\right)\] vs (T − T₀) → straight line through origin.

• Graph of Temperature T vs time t → exponential decay curve (temperature drops steeply at first, then gradually).

(b) There is no gravitational force acting downwards. However, when the starting velocity is 20 m/s, the viscous force, which is directly proportional to velocity, becomes maximum and tends to accelerate the ball upwards.

\[\text{ When the ball falls under gravity, }\]

\[\text{ neglecting the density of air: } \]

\[\text{ Mass of the sphere = m }\]

\[\text{ Radius = r }\]

\[\text{ Viscous drag coeff . }= \eta\]

\[\text{Terminal velocity is given by}: \]

\[\text{ mg  }= 6\pi\eta r v_T \]

\[ \Rightarrow \frac{6\pi\eta r v_T}{m} = g . . . (1)\]

\[\text{ Now, at terminal velocity, the acceleration of the ball due to the viscous force is given by: } \]

\[a = \frac{6\pi\eta r v_T}{m}\]

\[\text{ Comparing equations (1) and (2), we find that : } \]

\[ \text{ a = g }\]

Thus, we see that the initial acceleration of the ball will be 9.8 ms - 2  .

(c) The velocity of the ball will decrease with time because of the upward viscous drag. As the force of viscosity is directly proportional to the velocity of the ball, the acceleration due to the viscous force will also decrease.

(d) When all the kinetic energy of the ball is radiated as heat due to the viscous force, the ball comes to rest. 

• OA (Proportional Region): Stress ∝ Strain; material behaves elastically; Hooke's Law valid.
• Point A: Proportional limit — end of Hooke's Law.
• Region AB: Non-linear elastic region; material still returns to original shape.
• Point B (Elastic/Yield Limit): End of elastic behavior; start of plastic deformation.
• Region BD: Permanent (plastic) deformation; material does not return to original shape.
• Point D: Ultimate Tensile Strength — maximum stress the material can bear.
• Point E: Fracture point — material breaks.

• Hydraulic Press — Two cylinders (C & D) filled with liquid; small force applied on piston P₁ (smaller area A₁) is converted into a very large upward force on piston P₂ (larger area A₂), since A₂ > A₁.
• Hydraulic Lift — Works on Pascal's Law to lift or support heavy objects such as cars and trucks using liquid pressure.
• Hydraulic Brakes — Small force on the brake pedal is instantly transmitted equally through brake fluid to all cylinders, producing a large thrust on the wheels to stop the vehicle.

• Flow is streamline when velocity is low (parallel, orderly layers).
• Flow turns turbulent when velocity exceeds critical velocity (disordered, mixing layers).
• At low velocity → dye filament stays parallel in pipe (laminar).
• At high velocity → dye filament breaks and spreads (turbulent).

• Aerofoil / Airplane Wing: Air moves faster over the curved top surface → lower pressure above, higher below → upward dynamic lift.
• Spray / Atomizer: High-speed air over a tube creates low pressure → liquid rises and is expelled as droplets.
• Blowing off Roofs: High-velocity wind above roof creates low pressure → atmospheric pressure inside lifts the roof off.
• Magnus Effect (Spinning Ball): A spinning ball drags air, creating pressure difference between upper and lower sides → ball moves in a curved path.
• Venturimeter: Used to measure flow rate of liquid through pipes using pressure difference between wide and narrow sections.
• Attraction between Two Boats: Water between boats moves faster → pressure decreases → boats are pulled toward each other.

• A highly soluble impurity increases surface tension, while a partially soluble impurity (e.g., detergent) decreases it; a waterproofing agent increases it.
• Surface tension decreases with increase in temperature, given by T = T₀(1 − αθ), where T₀​ is surface tension at 0°C and α is the temperature coefficient.
• When a soap bubble is charged (positively or negatively), force acts outward on the surface, increasing its radius — thus electrification always decreases surface tension.

• Surface tension depends only on the nature of liquid and is independent of area of surface or length of line considered.
• Surface tension of a liquid decreases with rise of temperature; it is zero at boiling point and vanishes at critical temperature.
• Due to surface tension, a drop or bubble tends to contract, which increases internal pressure — this difference between inside and outside pressure is called excess pressure.
• For a drop and bubble in liquid: Δp = \[\frac {2T}{R}\]​; for a bubble in air: Δp = \[\frac {4T}{R}\]​(two free surfaces).

• Surface tension causes liquid drops and bubbles to be spherical (minimum surface area for given volume = sphere).
• Air bubble in liquid → 1 liquid-air interface → excess pressure = 2T/r.
• Soap bubble → 2 surfaces → excess pressure = 4T/r.

• Gases expand linearly with temperature, making them useful for thermometers. This consistent behaviour suggests the existence of a lowest temperature limit.
• Absolute zero (−273.15 °C or 0 K) is the temperature where an ideal gas would have zero pressure. It is the lowest possible temperature.
• The Kelvin scale begins at absolute zero and uses the triple point of water (273.16 K) as a reference point. It is the SI temperature scale.
• The ideal gas equation (PV = μRT) combines all gas laws into a single relationship among pressure, volume, and temperature. It works best for gases at low pressure and high temperature.

• Heat energy absorbed (Q) depends on: mass (m), rise in temperature (Δt), and specific heat capacity (c), i.e., Q ∝ m × Δt × c.
• Heat capacity (C') and specific heat capacity (c) are related by: C′ = m × c.

• A calorimeter is an insulated device used to measure heat transfer; measurement of specific heat of a substance is carried out using it.
• Principle of Calorimetry: Heat lost by hot body = Heat gained by cold body, which represents the law of conservation of heat energy.
• In the method of mixtures, a heated sample is placed in the calorimeter and the temperature change is measured to calculate specific heat using the formula Q = msΔt.
• Specific heat of a substance depends on the nature of the substance; water is preferred in calorimetry due to its high specific heat, allowing it to absorb large amounts of heat with minimal temperature change.
• For accurate results, the sample must be transferred quickly into the calorimeter and stirred well to ensure uniform heat distribution.

• Formula: Q = mL. Specific latent heat L has SI unit J kg⁻¹.
• Temperature stays constant during any phase change. Heat energy goes into breaking or forming intermolecular bonds, not into raising kinetic energy.
• Latent Heat of Fusion (water): L_f = 3.33 × 10⁵ J kg⁻¹ = 80 cal/g. Heat needed to melt 1 kg of ice at 0°C.
• Latent Heat of Vaporisation (water): L_v = 22.6 × 10⁵ J kg⁻¹ = 540 cal/g. Heat is needed to convert 1 kg of water to steam at 100°C.
• L_v ≫ L_f because vaporisation requires complete molecular separation and work against atmospheric pressure during expansion.
• All latent heat values depend on atmospheric pressure. Standard values quoted at 1 atm. Increasing pressure raises the boiling point (pressure cooker effect).

• Heat can be transferred in three ways — conduction, convection, and radiation.
• Conduction transfers heat through solids; molecules vibrate but do not move from their positions.
• Convection transfers heat through liquids and gases; molecules physically move from place to place.
• Both conduction and convection require a material medium; radiation does not.
• Radiation travels through electromagnetic waves at a speed of 3×10⁸ ms⁻¹.
• Conduction is the slowest process, convection is rapid, and radiation is the fastest mode of heat transfer.
• The energy received from the Sun is an example of heat transfer by radiation.

• The transfer of heat from the hot part to the cold part of an object is called conduction of heat.
• Conduction takes place through solid substances only — it requires a medium.
• Heat travels by molecular collisions: fast-vibrating molecules pass energy to slower neighbours.
• Copper conducts heat faster than aluminium, which conducts faster than steel.
• Conduction of heat through a substance depends on the property of that substance.
• Good conductors: silver, copper, aluminium, brass — all metals.
• Bad conductors: wood, cloth, air, paper — most non-metals.
• Good conductors of heat are also good conductors of electricity, and bad conductors of heat are also bad conductors of electricity.

• Convection occurs only in fluids (liquids and gases) — not in solids.
• In conduction, molecules vibrate but stay in place.
• In convection, molecules physically move from one place to another.
• Heating reduces density → hot fluid rises; cool fluid sinks → a convection current is set up.
• Convection currents transfer heat to the entire mass of the fluid.
• Potassium permanganate makes convection currents visible as magenta-coloured streams.

• When water is heated from the top, its density decreases, and it stays at the top. Since hot water cannot sink, convection does not occur and the bottom remains cool.
• Radiation is the transfer of heat without a medium — through electromagnetic waves.
• Heat from the Sun reaches us through radiation across the vacuum of space.
• All objects above 0 K emit thermal radiation (electromagnetic waves).
• Radiation is a two-step process: thermal energy → EM waves → thermal energy.
• Black or dark surfaces absorb more heat radiation; absorption also depends on the intrinsic properties of the substance.
• An infrared camera uses the radiation emitted by objects to see at night — useful for military surveillance.
• Copper is an excellent conductor; plastic is a bad conductor (insulator).
• Heat readily conducts through metals (copper and steel) but not through non-metals (wood and plastic).

• The greenhouse effect is a naturally occurring phenomenon that heats Earth's surface. Without it, Earth's temperature would be -18°C instead of 15°C.
• Greenhouse gases are transparent to solar radiation but retain and reflect back long-wave heat radiation. Main gases — CO₂ (60%), CH₄ (20%), CFCs (14%), N₂O (6%).
• Earth's surface re-emits heat as infrared radiation. Greenhouse gases like CO₂ and CH₄ absorb this and return heat to Earth's surface — causing the greenhouse effect.
• Rising CO₂ due to the burning of fossil fuels and deforestation intensifies the greenhouse effect, causing global warming.
• Global warming leads to melting of polar ice, rising sea levels, changes in rainfall patterns and loss of biodiversity.

• A hot body loses heat to its surroundings in the form of heat radiation.​
• The rate of cooling is directly proportional to the temperature difference between the body and its surroundings.
• The cooling curve (T vs t) shows rapid initial cooling that gradually slows down.
• Plotting \[\frac {dT}{dt}\] vs (T−T₀) gives a straight line through the origin, confirming Newton's law.​
• Mathematically: dT/dt = C(T − T₀), where C is the constant of proportionality.​
• The rate of cooling is proportional to — not independent of — the temperature difference. A 4× drop in temperature difference produces a 4× drop in cooling rate.