CSS Physics Paper-I 2023 Solved

Question 2: Gradient, Acceleration and Curl at a Point

Q2(a): Gradient of a Scalar Function

The gradient of a scalar function is a vector that gives the maximum rate of increase of the scalar function and points in the direction in which the scalar function increases most rapidly.

If φ(x,y,z) is a scalar field, then its gradient is:

In Cartesian coordinates:

Therefore:

Hence:

Physical Significance of Gradient

The physical significance of gradient is that it gives the direction and rate of the steepest increase of a scalar field. For example, if φ represents temperature, then ∇φ points in the direction in which temperature rises most rapidly. If V is electric potential, then the electric field is related to potential by:

The negative sign shows that the electric field points in the direction of decreasing potential.

Q2(b): Acceleration and Its Cartesian Components

Acceleration is the time rate of change of velocity. If position vector is:

Then velocity is:

Acceleration is:

Therefore, in Cartesian components:

where:

Q2(c): Curl of A at (1, -1, 1)

Given vector field:

So:

The curl of A is:

Now calculate each component.

i-component

j-component

k-component

Therefore:

At the point (1, -1, 1), we have x=1, y=-1, z=1.

Question 3: Rotational Kinetic Energy and Inertia

Q3(a): Rotational Kinetic Energy

Rotational kinetic energy is the energy possessed by a body due to its rotational motion. A rigid body rotating with angular speed ω can be treated as a collection of particles. Each particle has linear speed:

The kinetic energy of one particle is:

Substitute v=rω:

For the whole rigid body:

But:

Therefore:

Formulas for Disc, Hoop and Sphere

Q3(b): Inertia in Physics

Inertia is the property of a body by which it resists any change in its state of rest or uniform motion. In linear motion, mass is the measure of inertia. In rotational motion, moment of inertia is the measure of resistance to change in rotational motion.

Moment of inertia depends on both mass and distribution of mass about the axis of rotation:

Rotational Inertia of Solid Cylinder

For a solid cylinder of mass M and radius R about its central axis of symmetry:

Rotational Inertia of Hollow Cylinder

For a thin hollow cylinder of mass M and radius R about its central axis:

If the hollow cylinder is thick with inner radius R1 and outer radius R2, then:

Q3(c): Angular Speed of Watch Hands

Angular speed is:

Second Hand

The second hand completes one revolution in 60 s.

Minute Hand

The minute hand completes one revolution in 60 min = 3600 s.

Hour Hand

The hour hand completes one revolution in 12 h = 43200 s.

Question 4: Relativity, Length Contraction and Antenna Mass Equivalent

Q4(a): Physics Before Relativity

Before relativity, physics was based mainly on Newtonian mechanics and Maxwell’s electromagnetism. Newtonian mechanics assumed absolute space and absolute time. This meant that lengths, time intervals and simultaneity were considered the same for all observers. The Galilean transformation was used to relate measurements between inertial frames.

However, Maxwell’s equations predicted that light travels with a fixed speed c. This created a problem because Galilean transformation suggested that measured speed should depend on the motion of the observer. The failure to detect ether and the results of experiments such as the Michelson-Morley experiment showed that the old view was incomplete.

How Einstein Developed Special Relativity

Einstein proposed special relativity in 1905 based on two postulates:

1. The laws of physics are the same in all inertial frames.
2. The speed of light in vacuum is constant for all inertial observers, independent of the motion of source or observer.

These postulates led to Lorentz transformation, time dilation, length contraction, relativity of simultaneity and mass-energy equivalence.

Mass-Energy Interchangeability

Einstein showed that mass and energy are not separate quantities. Mass is a form of energy. The rest energy of a body is:

This equation means that even a small mass corresponds to a very large amount of energy because c² is extremely large.

If mass decreases by Δm, the released energy is:

This relation explains nuclear energy, binding energy, pair production and annihilation. For example, in nuclear reactions, a small mass defect appears as a large amount of energy.

Q4(b): Relativity of Length

According to Einstein’s special theory of relativity, length is not absolute. The measured length of an object depends on the relative motion between the object and the observer.

The length measured in the frame in which the object is at rest is called proper length L0. If the object moves with speed v relative to an observer, then the length measured parallel to the direction of motion is contracted:

where:

Since γ ≥ 1, the moving length L is less than or equal to L0. This effect is called length contraction.

Important Points About Length Contraction

1. Length contraction occurs only along the direction of relative motion.
2. Dimensions perpendicular to motion remain unchanged.
3. The effect becomes significant only when speed is comparable to the speed of light.
4. The proper length is always the longest length.

Q4(c): Mass Equivalent of Energy from Antenna

Given:

Energy radiated is:

Using mass-energy relation:

Take:

Now:

Conversion:

Question 5: Capillary Rise, Fluids and Pool-Bottom Pressure

Q5(a): Capillarity

Capillarity is the rise or fall of a liquid in a narrow tube due to surface tension and adhesive/cohesive forces. If adhesive force between liquid and tube is greater than cohesive force within the liquid, the liquid rises. If cohesive force is greater, the liquid is depressed.

Derivation of Capillary Rise

Consider a capillary tube of radius r dipped in a liquid of density ρ and surface tension T. Let the liquid rise to height h. The angle of contact is θ.

The upward vertical component of surface tension around the circumference of the tube is:

The weight of the liquid column is:

The volume of liquid column is:

So weight is:

At equilibrium:

Therefore:

Cancel πr from both sides:

Solving for h:

Q5(b): Fluids and Their Characteristics

A fluid is a substance that can flow and continuously deform under the action of shear stress. Liquids and gases are fluids.

Important Characteristics of Fluids

1. No fixed shape: Fluids take the shape of the container.
2. Ability to flow: Fluids cannot resist shear stress permanently.
3. Pressure normal to surfaces: At rest, fluids exert pressure perpendicular to any surface.
4. Density: Fluids have mass per unit volume, written as ρ=m/V.
5. Viscosity: Real fluids resist relative motion between layers.
6. Compressibility: Gases are highly compressible, while liquids are nearly incompressible.
7. Surface tension: Liquids show surface tension due to molecular forces at the surface.

Q5(c): Pressure at the Bottom of the Pool

Given:

Pressure at the bottom is:

Substitute values:

Calculate hydrostatic pressure:

Now:

Question 6: Wave Energy, Longitudinal Waves and Molecular Oscillator Work

Q6(a): Energy in a Travelling Wave

Consider a sinusoidal wave travelling through a medium:

The transverse velocity of a particle of the medium is:

Kinetic energy per unit volume is proportional to the square of particle velocity:

So:

The potential energy density for a harmonic wave is also proportional to strain squared. For a sinusoidal travelling wave, it has the same average value as kinetic energy density:

Therefore:

Total energy density is:

The time average of cos² is 1/2. Hence average total energy density is:

Since average kinetic and potential parts are equal:

Q6(b): Longitudinal Waves Through Solids

Longitudinal waves can pass through solids because solids have elasticity. In a longitudinal wave, particles oscillate parallel to the direction of wave propagation. Solids can undergo compression and rarefaction because they have restoring forces between their particles.

The velocity of longitudinal waves depends mainly on elasticity and density. In a thin rod, the speed is:

where Y is Young’s modulus and ρ is density.

For a bulk longitudinal wave in a solid, the speed is:

where K is bulk modulus, G is shear modulus and ρ is density.

Parameters Affecting Velocity

1. Elastic modulus: Greater elasticity increases wave speed.
2. Density: Greater density decreases wave speed.
3. Nature of medium: Different solids have different elastic constants.
4. Temperature and structure: These may affect elastic properties and therefore wave speed.

Q6(c): Work Required to Stretch CO Molecule

The CO molecule may be treated as a harmonic oscillator. For a harmonic oscillator:

So:

Work required to stretch the molecule by displacement x is stored as potential energy:

Reduced Mass of CO Molecule

Carbon has atomic mass approximately 12 u, and oxygen has atomic mass approximately 16 u.

Since:

Therefore:

Calculate Force Constant

Given:

Now:

Calculate Work

Given stretch:

Work is:

In electron volts:

Question 7: Work by Gas, Intermolecular Forces and Final Temperature

Q7(a): Work Done by Gas and on Gas

Consider an ideal gas enclosed in a cylinder with a movable piston. If the piston moves through a small distance dx under pressure P, and area of piston is A, then force exerted by gas is:

Small work done by gas is:

Since:

Therefore:

Total work done by gas is:

Work done on gas is the negative of work done by gas:

Why Pressure Force Is Non-Conservative

A conservative force has work that depends only on initial and final states. But for gas expansion or compression, work depends on the path followed in the P-V diagram. For example, work in isothermal expansion differs from work in adiabatic expansion even between similar volume limits.

In a cyclic process:

This means net work over a closed path can be non-zero. Therefore, pressure force in thermodynamics is non-conservative.

Q7(b): Intermolecular Forces

Intermolecular forces are forces of attraction or repulsion between molecules. They are weaker than chemical bonds but control many physical properties such as boiling point, viscosity, surface tension, melting point and real gas behavior.

Main Types of Intermolecular Forces

1. London dispersion forces: Weak attractive forces caused by temporary dipoles. They exist in all molecules and atoms.
2. Dipole-dipole forces: Attractive forces between permanent dipoles of polar molecules.
3. Hydrogen bonding: A strong dipole-dipole interaction involving hydrogen bonded to highly electronegative atoms such as oxygen, nitrogen or fluorine.
4. Ion-dipole forces: Attraction between an ion and a polar molecule, important in solutions.
5. Short-range repulsive forces: At very small separations, electron clouds repel each other strongly.

In real gases, intermolecular attraction reduces effective pressure, while finite molecular volume reduces available volume. This leads to van der Waals equation:

Q7(c): Number of Moles and Final Temperature of Oxygen

Given initial data:

Use ideal gas equation:

So:

Substitute values:

Now final data:

Using ideal gas equation again:

So:

Substitute values:

Convert to Celsius:

Question 8: Short Notes

Q8(a): Kepler’s Law of Periods

Kepler’s law of periods, also called Kepler’s third law, states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

For a circular orbit, the gravitational force provides centripetal force:

Cancel m:

So:

For circular motion:

Therefore:

Rearrange:

Thus:

For elliptical orbits, r is replaced by semi-major axis a:

Q8(b): Michelson Interferometer

The Michelson interferometer is an optical instrument used to split a light beam into two perpendicular beams and then recombine them to produce interference. It is used to measure small distances, wavelength differences, refractive indices and optical path changes.

Construction

A Michelson interferometer consists of:

1. A monochromatic light source.
2. A half-silvered glass plate or beam splitter.
3. Two mirrors placed at right angles.
4. A compensating plate.
5. A telescope or screen for observing interference fringes.

Working

The beam splitter divides the incident light into two beams. One beam travels toward mirror M1, and the other travels toward mirror M2. After reflection, the two beams return and recombine. If there is a path difference between the two beams, interference fringes are formed.

The optical path difference is approximately:

where L1 and L2 are the distances from beam splitter to the two mirrors.

If one mirror is moved by distance d, the optical path changes by 2d. If N fringes cross the field of view:

Therefore:

Uses

1. Measurement of wavelength of light.
2. Measurement of very small length changes.
3. Determination of refractive index.
4. Study of spectral line separation.
5. Historical role in the Michelson-Morley experiment.

Q8(c): Young’s Double Slit Experiment

Young’s double slit experiment demonstrates the wave nature of light through interference. A monochromatic light source illuminates two narrow slits separated by distance d. The slits act as coherent sources, and their waves overlap on a screen at distance D.

The path difference between light waves reaching a point on the screen is:

For small angles:

So:

Bright Fringes

Constructive interference occurs when:

Therefore:

Dark Fringes

Destructive interference occurs when:

Therefore:

Fringe Width

The distance between two consecutive bright or dark fringes is:

Importance

Young’s experiment proves that light has wave nature. It also provides a method for measuring wavelength of light and demonstrates the principle of superposition.