← Legal Metrology
Past Questions — 2010
Which of the following law of thermodynamics leads to the inference that it is not possible to convert whole of the heat into work continuously?
A)
Zeroth
B)
First
C)
Second
D)
Third
Second law prohibits complete continuous conversion of heat to work, defines entropy direction.
The internal energy of a perfect gas is
A)
partly kinetic and partly potential
B)
wholly potential
C)
wholly kinetic
D)
depends on the ratio of two specific heats
1. Definition of a Perfect Gas: A perfect gas (often used interchangeably with an ideal gas in this context) is a theoretical gas composed of point masses that move randomly and collide elastically.
2. Intermolecular Forces: The fundamental assumption of a perfect/ideal gas is that there are no intermolecular forces of attraction or repulsion between the gas molecules.
3. Potential Energy: In thermodynamics, internal potential energy arises from the forces of attraction or repulsion between molecules. Since these forces are assumed to be zero for a perfect gas, the internal potential energy is zero.
4. Kinetic Energy: The internal energy of a gas consists of the sum of the kinetic energies of its molecules (translational, rotational, and vibrational). For a perfect gas, the internal energy is solely a function of temperature and consists entirely of this kinetic energy.
5. Conclusion: Because there is no potential energy component, the internal energy of a perfect gas is wholly kinetic.
Comparison with other options:
- Option (a) describes a real gas, where intermolecular forces exist.
- Option (b) is incorrect as gases possess kinetic energy due to molecular motion.
- Option (d) is incorrect because while the specific heat ratio (gamma) relates to the degrees of freedom, the nature of the energy itself remains kinetic.
In Rayleigh scattering, the amount of scattered light is proportional to
A)
1/λ
B)
1/λ^2
C)
1/λ^3
D)
1/λ^4
Rayleigh scattering occurs when light is scattered by particles that are much smaller than the wavelength of the light (such as gas molecules in the atmosphere). According to the Rayleigh scattering law, the intensity of the scattered light (I) is inversely proportional to the fourth power of the wavelength (λ). Mathematically, this is expressed as I ∝ 1/λ⁴. This explains why shorter wavelengths (blue light) are scattered much more efficiently than longer wavelengths (red light), which is why the sky appears blue during the day and the sun appears red at sunset. Options a, b, and c are incorrect because they do not follow the derived physical law for scattering by small particles.
Deviation produced in a grating is independent of
A)
wavelength of light
B)
grating constant or element
C)
refractive index of grating material
D)
None of these
The deviation (diffraction angle) produced by a diffraction grating is governed by the grating equation: d sin(θ) = nλ, where 'd' is the grating element (the distance between two adjacent slits), 'θ' is the angle of diffraction (deviation), 'n' is the order of the spectrum, and 'λ' is the wavelength of the incident light.
1. Wavelength (λ): The equation shows that sin(θ) is directly proportional to λ. Therefore, deviation depends on the wavelength of light (Option a is incorrect).
2. Grating constant/element (d): The equation shows that sin(θ) is inversely proportional to d. Therefore, deviation depends on the grating constant (Option b is incorrect).
3. Refractive Index: A diffraction grating works on the principle of interference of light waves originating from multiple slits. Unlike a prism, where deviation is caused by refraction (and thus depends on the refractive index of the material), a grating's deviation is a result of diffraction and interference. The material's refractive index does not enter the grating equation. Thus, the deviation is independent of the refractive index of the grating material (Option c is correct).
Phase difference between two successive Fresnel’s Half Period Zones (H.P.Z.) is
A)
Zero
B)
π/2
C)
π
D)
2π
Fresnel's Half Period Zones (H.P.Z.) are defined as concentric rings of a wavefront such that the distance from the source to the boundary of each zone increases by exactly half a wavelength (λ/2) compared to the previous zone.
1. Path Difference: By definition, the path difference between the boundaries of two successive H.P.Z.s is Δx = λ/2.
2. Phase Difference Formula: The relationship between path difference (Δx) and phase difference (φ) is given by the formula: φ = (2π / λ) * Δx.
3. Calculation: Substituting Δx = λ/2 into the formula: φ = (2π / λ) * (λ / 2) = π radians.
Because the phase difference is π (or 180 degrees), the waves originating from successive zones interfere destructively. This is why the total amplitude at a point is the alternating sum of the amplitudes of the individual zones.
- Option (a) Zero: This would imply constructive interference and identical phase.
- Option (b) π/2: This represents a quarter-cycle shift.
- Option (d) 2π: This represents a full cycle, which would mean the zones are in phase, contradicting the definition of 'Half Period' zones.
In Rayleigh scattering, the degree of scatterimg is proportional to fourth power of wavelength, but the size of the scattering medium’s particles should be such that
A)
size of particles is at least four times the wavelength used
B)
size of particles is 1/4th the wavelength of light
C)
size of particles should be less than the wavelength of light used
D)
size should be equal to wavelength
Rayleigh scattering occurs when light interacts with particles that are significantly smaller than the wavelength of the incident light. Specifically, the condition for Rayleigh scattering is that the particle radius (r) must be much smaller than the wavelength (λ), typically expressed as r << λ. Under this condition, the intensity of the scattered light is inversely proportional to the fourth power of the wavelength (I ∝ 1/λ⁴), which explains why shorter wavelengths (blue light) are scattered more than longer wavelengths (red light), making the sky appear blue. If the particles were equal to or larger than the wavelength, the scattering would follow different laws, such as Mie scattering (where scattering is less dependent on wavelength) or geometric optics (where particles are much larger than the wavelength). Therefore, option (c) is the correct requirement for the Rayleigh regime.
Orbital velocity of a satellite revolving round the earth is independent of the
A)
mass of the earth
B)
mass of the satellite
C)
radius of the earth
D)
radius of the orbit
To find the orbital velocity of a satellite, we equate the gravitational force providing the centripetal force required for circular motion.
1. Gravitational Force: F = (G * M * m) / r²
2. Centripetal Force: F = (m * v²) / r
Where:
- G is the universal gravitational constant
- M is the mass of the Earth
- m is the mass of the satellite
- v is the orbital velocity
- r is the radius of the orbit (distance from the center of the Earth)
Equating the two:
(m * v²) / r = (G * M * m) / r²
Dividing both sides by 'm' and multiplying by 'r':
v² = (G * M) / r
v = sqrt(GM / r)
From this formula, it is evident that the orbital velocity (v) depends on the mass of the central body (M) and the radius of the orbit (r). It does NOT depend on the mass of the satellite (m), as the mass of the satellite cancels out during the derivation. Therefore, whether the satellite is a small cube or a massive space station, if they are in the same orbit, they must travel at the same velocity to maintain that orbit.
If the temperature of the sun is doubled, the rate of energy received on earth will be increased by a factor of roughly
A)
2
B)
4
C)
8
D)
16
The rate of energy emitted by a celestial body like the sun is governed by the Stefan-Boltzmann Law. According to this law, the total power (energy per unit time) radiated from a black body is proportional to the fourth power of its absolute temperature (T). The formula is given by: P = εσAT⁴, where P is the power, ε is the emissivity, σ is the Stefan-Boltzmann constant, A is the surface area, and T is the absolute temperature.
1. Let the initial temperature of the sun be T₁ and the initial rate of energy received on earth be P₁.
2. If the temperature is doubled, the new temperature T₂ = 2T₁.
3. The new rate of energy emitted (and consequently received on earth, assuming distance remains constant) is P₂ ∝ (T₂)⁴.
4. Substituting T₂ = 2T₁, we get P₂ ∝ (2T₁)⁴ = 16(T₁)⁴.
5. Therefore, P₂ / P₁ = 16.
The rate of energy received on earth will increase by a factor of 16. Options a, b, and c are incorrect because they represent linear, squared, and cubed relationships, whereas radiative heat transfer follows a T⁴ relationship.
The running of a fan makes us comfortable during summer, because it
A)
decreases the temperature of air
B)
increases the thermal conductivity of air
C)
increase the rate of evaporation of perspiration
D)
cuts off the thermal radiation reaching us
Step-by-step reasoning: 1. A fan does not cool the air; it simply moves the air around. Therefore, it does not decrease the ambient temperature (eliminating option a). 2. Thermal conductivity is a physical property of the air itself and is not changed by the movement of the fan (eliminating option b). 3. The human body cools itself through the evaporation of sweat (perspiration). Evaporation is a cooling process because it absorbs latent heat from the skin. 4. The rate of evaporation depends on the concentration of water vapor in the air immediately surrounding the skin. A fan blows away the saturated air layer (boundary layer) and replaces it with drier air, which significantly increases the rate of evaporation. 5. This increased evaporation removes heat from the body more quickly, making us feel cooler. 6. A fan does not block thermal radiation from the sun or surroundings (eliminating option d).
If the temperature of the source is increased, the efficiency of the Carnot’s engine
A)
increases
B)
decreases
C)
remains constant
D)
first increases and then remains constant
Carnot efficiency η=1-Tc/Th, increasing Th raises η.
At 0K which of the following properties of a gas will be zero?
A)
Kinetic energy
B)
Potential energy
C)
Mass
D)
Density
1. Definition of Absolute Zero: 0K (Kelvin) is known as absolute zero, the lowest possible temperature theoretically achievable.
2. Kinetic Theory of Gases: According to the kinetic molecular theory, the temperature of a gas is a direct measure of the average kinetic energy of its molecules. The relationship is expressed as KE = (3/2)kT, where k is the Boltzmann constant and T is the absolute temperature.
3. Analysis at 0K: When T = 0K, the mathematical product (3/2)k(0) equals zero. This implies that at absolute zero, the translational motion of the molecules ceases, and the kinetic energy becomes zero (classically).
4. Evaluating other options:
- Potential energy (b) does not become zero; in fact, intermolecular forces (van der Waals forces) still exist, and potential energy depends on the position of molecules relative to each other.
- Mass (c) is an intrinsic property of matter and remains constant regardless of temperature.
- Density (d) is mass per unit volume. While a gas may liquefy or solidify at 0K (increasing density), the density itself does not become zero; it actually increases as the volume decreases.
For a rigid sphere of radius a the scattering cross-section is given by
A)
2 pa2
B)
3 pa2
C)
4 pa2
D)
pa2
In the context of atomic and nuclear physics, the scattering of particles by a rigid sphere (hard sphere scattering) is a fundamental problem. For a hard sphere of radius 'a', the total scattering cross-section (σ) is calculated based on the impact parameter. In the classical limit, any particle with an impact parameter b < 2a will strike the sphere and be scattered. However, the standard result for the total cross-section of a hard sphere in the low-energy (s-wave) limit of quantum mechanics, or the effective area presented by the sphere for scattering, is given by σ = 4πa². This is exactly four times the geometric cross-section (πa²) of the sphere. This phenomenon occurs because the wave nature of the particle leads to diffraction-like scattering, effectively doubling the radius of the interaction in the cross-section calculation (σ = π(2a)² = 4πa²). Note: In the options provided, 'p' is used as a typographical representation of the mathematical constant π (pi).
The dimensions of the coefficient of thermal conductivity are
A)
ML-1 T-2K-1
B)
ML-2 T-3 K-1
C)
ML-1 T-1 K-1
D)
ML T-3 K-1
k from Fourier law: [W/mK] → M L T^-3 K^-1.
Energy is not carried by
A)
transverse progressive wave
B)
longitudinal progressive wave
C)
stationary wave
D)
electromagnetic wave
To determine which wave does not carry energy, we must distinguish between progressive waves and stationary waves:
1. **Progressive Waves (Options a, b, and d):** A progressive wave (whether transverse, longitudinal, or electromagnetic) is a wave that travels through a medium or space, transporting energy from one point to another. For example, sound waves (longitudinal) and light waves (electromagnetic) carry energy from the source to the receiver.
2. **Stationary Waves (Option c):** Also known as standing waves, these are formed by the superposition of two identical progressive waves traveling in opposite directions. In a stationary wave, the energy is 'trapped' between the nodes. While the particles of the medium vibrate, there is no net transfer of energy from one position to another along the wave. The energy oscillates between kinetic and potential forms within the antinodes, but the average energy flux across any plane is zero.
Therefore, energy is not carried (transported) by a stationary wave.
As a consequence of Brewster’s law for polarization, it follows that the angle of polarization depends upon
A)
the wavelength of light employed
B)
plane of polarization’s orientation
C)
the amplitude of light employed
D)
None of these
Brewster's Law states that the tangent of the polarizing angle (Brewster's angle, θp) is equal to the ratio of the refractive indices of the two media involved. Mathematically, it is expressed as: tan(θp) = n2 / n1, where n2 is the refractive index of the denser medium and n1 is the refractive index of the rarer medium (usually air, where n1 ≈ 1).
Analyzing the options:
1. Wavelength of light (a): While the refractive index (n) of a material can vary slightly with wavelength (a phenomenon called dispersion), Brewster's Law itself defines the angle based on the refractive index. In the context of standard physics problems regarding the 'consequence' of the law, the angle is a function of the material properties (refractive index), not the wavelength of the light source itself.
2. Plane of polarization's orientation (b): The plane of polarization is a result of the light hitting the surface at the Brewster angle; the angle does not depend on the orientation of the plane.
3. Amplitude of light (c): The refractive index and the resulting angle of polarization are independent of the intensity or amplitude of the incident light.
Since the angle of polarization depends solely on the refractive indices of the two media, and none of the provided options (wavelength, orientation, or amplitude) are the primary determinants defined by Brewster's Law, the correct option is 'None of these'.
The degree of freedom of a triatomic gas is
A)
1
B)
2
C)
6
D)
8
The degree of freedom (f) of a molecule is the number of independent ways in which it can possess energy. For a triatomic gas, we must consider both translational and rotational degrees of freedom.
1. Translational Degrees of Freedom: Every molecule, regardless of its shape, can move in three independent directions (x, y, and z axes). Thus, translational f = 3.
2. Rotational Degrees of Freedom:
- For a linear triatomic molecule (like CO2), there are 2 rotational degrees of freedom (rotation about the axis of the molecule does not count as it has negligible moment of inertia).
- For a non-linear triatomic molecule (like H2O), there are 3 rotational degrees of freedom (rotation about all three principal axes).
3. Total Degrees of Freedom (at room temperature, ignoring vibrations):
- Linear: 3 (trans) + 2 (rot) = 5
- Non-linear: 3 (trans) + 3 (rot) = 6
Comparing these results to the given options:
- Option (a) 1 and (b) 2 are too low for any gas molecule.
- Option (d) 8 would only be possible if vibrational modes were included (which usually occur at very high temperatures).
- Option (c) 6 corresponds to a non-linear triatomic molecule, which is the standard assumption for a general 'triatomic gas' unless 'linear' is specified. Therefore, 6 is the most appropriate answer among the choices provided.
Which one of the following relationships between the acceleration a and displacement y of a particle represents simple harmonic motion?
A)
a = 0.5y
B)
a = 400y^2
C)
a = -20y
D)
a = -3y^2
Simple Harmonic Motion (SHM) is defined as a type of periodic motion where the restoring force (and consequently the acceleration) is directly proportional to the displacement from the equilibrium position and is always directed opposite to the displacement. Mathematically, the condition for SHM is expressed as: a ∝ -y, or a = -ω²y, where 'a' is acceleration, 'y' is displacement, and 'ω' is the angular frequency (a positive constant).
Analyzing the options:
- Option (a) a = 0.5y: The acceleration is proportional to displacement but has a positive sign. This represents an unstable equilibrium where the particle would accelerate away from the center, not oscillate.
- Option (b) a = 400y²: The acceleration is proportional to the square of the displacement. This is non-linear motion and does not satisfy the SHM condition.
- Option (c) a = -20y: This fits the form a = -ω²y (where ω² = 20). The acceleration is proportional to the displacement and acts in the opposite direction (restoring force), which is the defining characteristic of SHM.
- Option (d) a = -3y²: While the sign is negative, the acceleration depends on the square of the displacement, making it non-linear and therefore not SHM.
A particle is moving in a circle with uniform speed. Its motion is
A)
periodic and simple harmonic
B)
periodic but not simple harmonic
C)
aperiodic
D)
None of the above
1. **Analysis of Periodicity**: A particle moving in a circle with uniform speed returns to its starting position after a fixed time interval (the time period T = 2πr/v). Since the motion repeats itself at regular intervals of time, it is by definition 'periodic'.
2. **Analysis of Simple Harmonic Motion (SHM)**: For a motion to be Simple Harmonic, it must satisfy two conditions: (i) the motion must be oscillatory (back and forth about a mean position), and (ii) the restoring force (or acceleration) must be directly proportional to the displacement from the mean position and directed towards it (F = -kx).
3. **Comparison**: In uniform circular motion, the particle does not oscillate back and forth; it moves continuously in one direction along the circumference. While the projection of this motion onto any diameter of the circle is SHM, the actual 2D motion of the particle itself is not. The acceleration (centripetal acceleration) is always directed toward the center, but the displacement is measured along the arc or as a vector from the center, which does not satisfy the linear restoring force requirement of SHM.
4. **Conclusion**: The motion is periodic because it repeats, but it is not simple harmonic because it is not an oscillation satisfying the SHM differential equation.
Which of the following is correct?
A)
Both sound and light waves in air are longitudinal
B)
Both sound and light waves in air are transverse
C)
Sound waves in air are transverse and light waves are longitudinal
D)
Sound waves in air are longitudinal and light waves are transverse
To determine the correct option, we must analyze the nature of sound and light waves:
1. **Sound Waves in Air**: Sound is a mechanical wave that requires a medium (like air, water, or solids) to travel. In air, sound travels as a series of compressions (high pressure) and rarefactions (low pressure) that move in the same direction as the wave's propagation. This characteristic—where the particle displacement is parallel to the direction of energy transport—defines a **longitudinal wave**.
2. **Light Waves**: Light is an electromagnetic wave consisting of oscillating electric and magnetic fields. These fields oscillate perpendicular (at right angles) to the direction of the wave's propagation. This characteristic—where the oscillation is perpendicular to the direction of travel—defines a **transverse wave**. Light does not require a medium and can travel through a vacuum.
**Analysis of Options:**
- Option (a) is incorrect because light is not longitudinal.
- Option (b) is incorrect because sound in air is not transverse.
- Option (c) is incorrect because it swaps the properties of both waves.
- Option (d) is correct as it accurately identifies sound in air as longitudinal and light as transverse.
A laser beam consists of
A)
light material particles
B)
electrons
C)
highly coherent photons
D)
cosmic rays
A laser (Light Amplification by Stimulated Emission of Radiation) produces a beam of light that differs from ordinary light in three primary ways: it is monochromatic (single wavelength), directional (collimated), and coherent.
1. **Photons**: Light is composed of elementary particles called photons. Since a laser is a beam of light, it consists of photons.
2. **Coherence**: In a laser, the photons are 'coherent,' meaning they are in phase with each other both in time and space. This is achieved through stimulated emission, where an incoming photon triggers an excited electron to drop to a lower energy level, releasing a second photon that is identical in frequency, phase, and direction to the first.
Why other options are incorrect:
- **a. light material particles**: Light is electromagnetic radiation, not composed of 'material particles' (which usually refers to matter with mass like atoms or molecules).
- **b. electrons**: Electrons are charged subatomic particles. While electrons are involved in the process of creating the laser (energy transitions), the beam itself is made of light (photons), not electrons. An electron beam is found in a CRT or electron microscope, not a laser.
- **d. cosmic rays**: Cosmic rays are high-energy particles (mostly protons) originating from outer space, unrelated to the artificial generation of laser light.
Standing waves can be produced
A)
on a string clamped at both the ends
B)
on a string clamped at one end and free at the other
C)
when two identical waves with a phase difference of p are moving in the same direction
D)
None of the above
A standing wave (or stationary wave) is formed by the superposition of two waves of the same frequency and amplitude traveling in opposite directions.
1. Analysis of Option (a): When a string is clamped at both ends, a wave traveling toward one end is reflected back. The incident wave and the reflected wave travel in opposite directions and superimpose, creating nodes (points of zero displacement) at the clamped ends and antinodes in between. This is the classic condition for producing standing waves (e.g., a guitar string).
2. Analysis of Option (b): While standing waves can technically be produced on a string clamped at one end and free at the other (creating a node at the fixed end and an antinode at the free end), in the context of standard physics examinations, the most fundamental and primary example of standing waves is the string clamped at both ends. However, if this were a 'multiple correct' question, (b) would also be physically possible. In a single-choice format, (a) is the definitive textbook example.
3. Analysis of Option (c): For standing waves to form, waves must move in OPPOSITE directions. If two identical waves move in the same direction, they simply undergo interference (constructive or destructive) resulting in a single traveling wave with a different amplitude, not a standing wave.
Therefore, option (a) is the most accurate and standard answer.
The velocity of sound in air is independent of change in
A)
pressure
B)
density
C)
temperature
D)
humidity
The velocity of sound in a gas (like air) is given by the formula v = sqrt(γRT/M), where γ is the adiabatic index, R is the universal gas constant, T is the absolute temperature, and M is the molar mass.
1. Temperature (Option c): The formula shows that v ∝ sqrt(T). Therefore, the velocity of sound is highly dependent on temperature.
2. Humidity (Option d): Humidity changes the composition of air (replacing nitrogen/oxygen with water vapor). Since water vapor has a lower molar mass (M) than dry air, increasing humidity decreases M, which increases the velocity of sound.
3. Density (Option b): According to the general formula v = sqrt(B/ρ) (where B is the bulk modulus and ρ is density), velocity depends on density. While density and pressure are related, the velocity specifically depends on the density of the medium.
4. Pressure (Option a): For an ideal gas at a constant temperature, if the pressure increases, the density increases proportionally (Boyle's Law). Since both the bulk modulus (B) and the density (ρ) increase by the same factor, their ratio remains constant. Therefore, the velocity of sound in an ideal gas is independent of pressure, provided the temperature remains constant.
According to which theorem, the sum of pressure head, velocity head and gravitational head is constant?
A)
Archimedes’ principle
B)
Pascal’s law
C)
Bernoulli’s theorem
D)
Torricelli’s law
Bernoulli's theorem states that for an incompressible, non-viscous fluid undergoing steady flow, the total mechanical energy remains constant along a streamline. This energy is expressed as the sum of three components:
1. Pressure Head (P/ρg): The energy per unit weight due to static pressure.
2. Velocity Head (v²/2g): The energy per unit weight due to the kinetic energy of the fluid.
3. Gravitational/Elevation Head (z): The energy per unit weight due to the height above a reference plane.
The mathematical expression is: P/ρg + v²/2g + z = constant.
Why other options are incorrect:
- Archimedes’ principle: Relates to the buoyant force acting on a body immersed in a fluid (equal to the weight of the displaced fluid).
- Pascal’s law: States that pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and to the walls of the containing vessel.
- Torricelli’s law: A specific application of Bernoulli's theorem used to calculate the speed of efflux of a liquid from an opening in a tank (v = sqrt(2gh)).
An empty vessel is filled with water, its frequency
A)
increases
B)
decreases
C)
remains the same
D)
None of these
This question refers to the natural frequency of vibration of a vessel (like a glass or a bowl) when it is struck. The natural frequency (f) of a vibrating body is inversely proportional to the square root of its mass (m), as expressed by the general formula f ∝ √(k/m), where k is the stiffness. When an empty vessel is filled with water, the effective mass of the vibrating system increases. Because the mass increases while the stiffness of the vessel walls remains relatively constant, the frequency of vibration decreases. This is why the pitch of the sound produced by a glass decreases as you fill it with water.
Which of the following phenomena gives evidence of the molecular structure of matter?
A)
Brownian motion
B)
Diffusion
C)
Evaporation
D)
All the above
The molecular structure of matter implies that matter is composed of tiny, discrete particles (atoms or molecules) that are in constant, random motion. Several physical phenomena provide evidence for this:
1. **Brownian Motion**: The random, zig-zag movement of microscopic particles (like pollen grains) suspended in a fluid. This occurs because the particles are being bombarded from all sides by the invisible, fast-moving molecules of the fluid, providing direct evidence of the existence of molecules.
2. **Diffusion**: The process by which particles of one substance spread into another (e.g., perfume spreading in a room). This happens because molecules are in constant motion and move from regions of higher concentration to lower concentration, proving that matter is not a continuous medium but made of moving particles.
3. **Evaporation**: The process where molecules at the surface of a liquid gain enough kinetic energy to overcome intermolecular forces and escape into the gas phase. This demonstrates that molecules have varying speeds and that there are forces of attraction between individual molecules.
Since all three phenomena are direct consequences of the molecular nature of matter, the correct option is 'All the above'.