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Past Questions — 2010
If A is the areal velocity of a planet of mass M, its angular momentum is
A)
M
B)
2MA
C)
A^2M
D)
AM^2
In celestial mechanics, Kepler's Second Law states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This is a direct consequence of the conservation of angular momentum.
1. Let L be the angular momentum of the planet of mass M.
2. The formula for angular momentum is L = M(r × v), where r is the position vector and v is the velocity vector.
3. The magnitude of the areal velocity (A) is defined as the rate at which area is swept out: A = dA/dt.
4. Geometrically, the area of a small triangle swept by the radius vector in time dt is dA = 1/2 |r × (v dt)|.
5. Therefore, the areal velocity A = dA/dt = 1/2 |r × v|.
6. Substituting the expression for angular momentum (L = M |r × v|), we get: A = L / (2M).
7. Rearranging this formula to solve for angular momentum L, we get: L = 2MA.
Comparing this result to the options:
- Option (a) M is just mass.
- Option (b) 2MA matches our derivation.
- Option (c) A^2M and (d) AM^2 are dimensionally incorrect for angular momentum.
Joule’s mechanical equivalent of heat relates heat energy to
A)
kinetic energy
B)
potential energy
C)
internal energy
D)
None of these
James Prescott Joule's experiments in the mid-19th century demonstrated that mechanical work could be converted into heat. Specifically, he showed that a specific amount of mechanical work (such as the falling of a weight turning a paddle wheel in water) resulted in a proportional increase in the temperature of the substance. In thermodynamics, the energy that is stored within a system—which manifests as the temperature increase of the substance—is defined as its 'internal energy'. Therefore, Joule's mechanical equivalent of heat (J) is the proportionality constant that relates the amount of mechanical work performed on a system to the increase in its internal energy. While internal energy consists of microscopic kinetic and potential energies of molecules, the macroscopic relationship established by Joule is specifically between work/heat and the total internal energy of the system.
In a given process on an ideal gas, dW = 0 and dQ < 0. Then for the gas
A)
the temperature will decrease
B)
the volume will increase
C)
the pressure will remain constant
D)
the temperature will increase
1. **First Law of Thermodynamics**: The first law is expressed as dU = dQ - dW (where dU is the change in internal energy, dQ is the heat added to the system, and dW is the work done by the system).
2. **Given Conditions**:
- dW = 0: This indicates an isochoric process (constant volume), as work done by a gas is given by W = ∫P dV. If dW = 0, then dV = 0.
- dQ < 0: This indicates that heat is being removed from the system (cooling).
3. **Applying the First Law**: Substituting the given values into the equation:
dU = dQ - 0
Since dQ < 0, it follows that dU < 0. This means the internal energy of the gas is decreasing.
4. **Ideal Gas Relation**: For an ideal gas, the internal energy (U) is a function of temperature (T) only. The relationship is given by dU = nCv dT.
5. **Conclusion**: Since dU < 0 and nCv is always positive, dT must be negative (dT < 0). Therefore, the temperature of the gas will decrease.
**Analysis of Options**:
- (a) Correct: As derived, dT < 0.
- (b) Incorrect: dW = 0 implies dV = 0, so volume remains constant.
- (c) Incorrect: For an ideal gas at constant volume, if temperature decreases, pressure must also decrease (P = nRT/V).
- (d) Incorrect: This contradicts the finding that dU < 0.
The snow on the mountain does not melt all at once when it is heated by the sun because
A)
it becomes very hard
B)
it reflects most of the heat from the sun
C)
it has a low specific heat capacity
D)
it has a high latent heat of fusion
The process of melting (changing from solid ice to liquid water) is a phase transition. During a phase transition, the temperature of the substance remains constant even though heat is being added. The amount of heat energy required to change 1 kg of a substance from solid to liquid without changing its temperature is called the 'Latent Heat of Fusion'. Water/ice has an exceptionally high latent heat of fusion (approximately 334 kJ/kg). This means a significant amount of thermal energy from the sun must be absorbed by the snow before it can actually melt. Because of this high energy requirement, the snow melts slowly and gradually rather than all at once, acting as a thermal buffer.
Why other options are incorrect:
- Option (a): Hardness is a mechanical property and does not dictate the rate of phase change.
- Option (b): While snow does have a high albedo (reflects sunlight), the question asks why it doesn't melt 'all at once' when it *is* heated. Reflection prevents heat absorption, but the slow melting of the absorbed heat is due to latent heat.
- Option (c): Specific heat capacity refers to the energy needed to raise the temperature of a substance. Once the snow reaches its melting point (0°C), specific heat is no longer the governing factor; latent heat is.
In case of linearly polarized light, the magnitude of the electric field vector
A)
is parallel to the direction of propagation
B)
does not change with time
C)
increases and decreases linearly with time
D)
varies periodically with time
Linearly polarized light is an electromagnetic wave where the electric field vector (E) oscillates in a single fixed plane perpendicular to the direction of propagation. The mathematical expression for the electric field of a linearly polarized wave is E = E₀ sin(kz - ωt), where E₀ is the amplitude, k is the wave number, and ω is the angular frequency.
1. Analysis of Option (a): The electric field vector is always perpendicular (transverse) to the direction of propagation in electromagnetic waves, not parallel. Thus, (a) is incorrect.
2. Analysis of Option (b): The electric field is an oscillating wave; its magnitude changes constantly as the wave passes a point. It is not constant. Thus, (b) is incorrect.
3. Analysis of Option (c): The magnitude varies sinusoidally (following a sine or cosine function), not linearly. A linear change would imply a triangle wave, which is not the nature of light waves. Thus, (c) is incorrect.
4. Analysis of Option (d): Because the electric field follows a sinusoidal function (sin(ωt)), it repeats its value at regular intervals of time (T = 2π/ω). This is the definition of periodic variation. Therefore, (d) is correct.
If vibrations of a string are to increased by a factor of 2, then tension in the string must be made
A)
half
B)
twice
C)
four times
D)
eight times
The question refers to the frequency of vibrations of a stretched string. The fundamental frequency (f) of a vibrating string is given by the formula: f = (1 / 2L) * sqrt(T / μ), where L is the length of the string, T is the tension, and μ is the linear mass density (mass per unit length). From this formula, we can see that the frequency is directly proportional to the square root of the tension: f ∝ √T. To increase the frequency (vibrations) by a factor of 2 (f' = 2f), we set up the ratio: f'/f = sqrt(T'/T). Substituting the factor of 2: 2 = sqrt(T'/T). Squaring both sides gives 4 = T'/T, which means T' = 4T. Therefore, the tension must be increased to four times its original value to double the frequency of vibrations.
Consider a system of two identical particles. One of the particles is at rest and the other has an acceleration a. The centre of mass has acceleration
A)
zero
B)
a
C)
a/2
D)
2a
The acceleration of the center of mass (a_cm) for a system of particles is given by the formula: a_cm = (m1*a1 + m2*a2 + ... + mn*an) / (m1 + m2 + ... + mn). In this specific problem: 1. The particles are identical, meaning m1 = m2 = m. 2. One particle is at rest, so its acceleration a1 = 0. 3. The other particle has an acceleration a2 = a. Substituting these values into the formula: a_cm = (m*0 + m*a) / (m + m) = (m*a) / (2m) = a/2. Therefore, the acceleration of the center of mass is half the acceleration of the moving particle.
A moving bullet hits a solid target resting on a frictionless surface and gets embedded in it. What is conserved in this process?
A)
Momentum and kinetic energy
B)
Kinetic energy alone
C)
Momentum alone
D)
Neither momentum nor kinetic energy
1. Analysis of Momentum: The problem states the target is on a frictionless surface. Since there are no external horizontal forces acting on the system (bullet + target), the total linear momentum of the system is conserved according to Newton's Second Law. Therefore, the momentum before the collision equals the momentum after the collision.
2. Analysis of Kinetic Energy: The bullet 'gets embedded' in the target. This describes a perfectly inelastic collision. In a perfectly inelastic collision, the maximum amount of kinetic energy is converted into other forms of energy, such as heat, sound, and the work required to deform the target as the bullet penetrates it.
3. Conclusion: While momentum is conserved due to the absence of external forces, kinetic energy is not conserved because it is dissipated as internal energy during the embedding process.
Comparing this to the options:
- Option (a) is incorrect because kinetic energy is not conserved.
- Option (b) is incorrect because kinetic energy is lost and momentum is conserved.
- Option (c) is correct because only momentum is conserved.
- Option (d) is incorrect because momentum is conserved.
The difference between Cp and Cv
A)
is equal to R
B)
is equal to 2R
C)
is equal to R/2
D)
depends upon the atomicity of the gas molecules
The relationship between the specific heat at constant pressure (Cp) and the specific heat at constant volume (Cv) for an ideal gas is governed by Mayer's Relation.
1. Definition: Cp is the heat required to raise the temperature of one mole of gas by one degree at constant pressure, while Cv is the heat required at constant volume.
2. Thermodynamic Process: At constant volume, all heat added goes into increasing the internal energy (ΔU). At constant pressure, heat added must increase the internal energy AND perform work (PΔV) to expand the gas.
3. Mathematical Derivation: For one mole of an ideal gas, the work done during expansion is W = PΔV = RT. Therefore, the energy required at constant pressure is the energy required at constant volume plus the work done: Cp = Cv + R.
4. Result: Rearranging this gives Cp - Cv = R.
Regarding other options:
- Option b (2R) and c (R/2) are mathematically incorrect based on the first law of thermodynamics.
- Option d is a common distractor; while the individual values of Cp and Cv depend on the atomicity (degrees of freedom) of the gas, their difference (Cp - Cv) is always equal to the universal gas constant R for any ideal gas, regardless of whether it is monatomic, diatomic, or polyatomic.
In a Fresnel biprism experiment, the two positions of lens give separation between the slits as 16 cm and 9 cm respectively. What is the actual distance of separation?
A)
12.5 cm
B)
12 cm
C)
13 cm
D)
14 cm
In a Fresnel Biprism experiment, the distance between the two virtual sources (slits), denoted as 'd', cannot be measured directly. To find 'd', the displacement method using a convex lens is employed. According to the displacement method for finding the distance between two sources, the actual distance 'd' is the geometric mean of the two positions of the lens that produce distinct images of the sources.
Given:
Position 1 (d1) = 16 cm
Position 2 (d2) = 9 cm
The formula for the actual separation 'd' is:
d = √(d1 * d2)
Substituting the given values:
d = √(16 * 9)
d = √144
d = 12 cm
Therefore, the actual distance of separation between the slits is 12 cm. Option (b) is correct, while options (a), (c), and (d) are incorrect as they do not satisfy the geometric mean relationship required by the displacement method.
In an interference pattern, points of minimum intensity are perfectly dark, and in a diffraction pattern, points of minimum intensity are
A)
also perfectly dark
B)
uniformly distributed
C)
not perfectly dark
D)
None of these
In an interference pattern (specifically Young's Double Slit Experiment), the minima occur when two coherent waves of equal amplitude interfere destructively. If the amplitudes are exactly equal, they cancel each other out completely, resulting in perfectly dark fringes (zero intensity). In contrast, a diffraction pattern (such as single-slit diffraction) is essentially the result of interference between an infinite number of secondary wavelets originating from different parts of the same wavefront. Because the wavelets are distributed across the slit, the cancellation at the minima is not absolute; some light from other parts of the aperture still reaches the screen. Therefore, the minima in a diffraction pattern are 'not perfectly dark' compared to the ideal minima of a two-source interference pattern.
If the Galilean transformations were correct then the aberration angle would be given by
A)
tanθ = v/c
B)
sinθ = v/c
C)
cosθ = v/c
D)
None of these
The stellar aberration of light is the apparent motion of celestial objects caused by the motion of the observer. Under the assumption of Galilean transformations (classical velocity addition), the velocity of light relative to the observer is the vector sum of the velocity of light from the source and the velocity of the observer. Let 'c' be the speed of light and 'v' be the velocity of the Earth (observer) perpendicular to the line of sight. According to Galilean relativity, the resultant velocity vector forms a right-angled triangle where the hypotenuse is the speed of light 'c' and the base is the velocity 'v'. The angle of aberration θ is the angle between the true direction and the apparent direction. In this classical framework, the tangent of the angle is the ratio of the transverse velocity to the longitudinal velocity: tanθ = v/c. (Note: In Special Relativity, the formula is slightly different due to the constancy of the speed of light, but the question specifically asks for the result if Galilean transformations were correct).
According to special theory of relativity a particle cannot travel with the speed of light because its
A)
velocity will soon be infinite
B)
mass will be infinite
C)
mass will reduced to zero
D)
None of the above
According to the Special Theory of Relativity, the relativistic mass (m) of a particle moving at velocity (v) is given by the formula: m = m₀ / sqrt(1 - v²/c²), where m₀ is the rest mass and c is the speed of light. As the velocity (v) of a particle approaches the speed of light (c), the term (1 - v²/c²) approaches zero. Dividing the rest mass by a value approaching zero causes the relativistic mass (m) to approach infinity. Since an infinite amount of energy would be required to accelerate a particle with infinite mass further, it is physically impossible for any particle with a non-zero rest mass to reach or exceed the speed of light. Option (a) is incorrect because velocity is capped at 'c'. Option (c) is incorrect because mass increases, not decreases, with speed.
What type of vibrations are produced in a sitar wire?
A)
Progressive transverse
B)
Progressive longitudinal
C)
Stationary transverse
D)
Stationary longitudinal
To determine the type of vibrations in a sitar wire, we analyze two characteristics: the direction of vibration and the nature of the wave propagation.
1. **Transverse vs. Longitudinal**: In a string instrument like a sitar, the wire is plucked. The particles of the string move perpendicular (at right angles) to the direction of the wave propagation along the length of the string. This is the definition of a transverse wave. Longitudinal waves (like sound waves in air) involve particles moving parallel to the direction of propagation, which does not happen in a stretched string.
2. **Stationary (Standing) vs. Progressive**: A sitar wire is fixed at both ends. When plucked, waves travel to the ends and reflect back. The superposition of the incident wave and the reflected wave of the same frequency and amplitude creates a 'standing wave' or 'stationary wave'. In a stationary wave, certain points (nodes) remain motionless while others (antinodes) vibrate with maximum amplitude, and the wave does not transport energy forward along the string.
Therefore, the vibrations in a sitar wire are stationary transverse waves.
A reflecting telescope utilizes
A)
a concave mirror
B)
a convex mirror
C)
a prism
D)
a plano convex lens
A reflecting telescope (also known as a Newtonian telescope) uses a curved mirror to gather and focus light from a distant object. Specifically, it utilizes a concave mirror (the primary mirror) because concave mirrors are converging mirrors; they collect parallel rays of light from a distant star or planet and reflect them to a single focal point.
Why other options are incorrect:
- b. A convex mirror: These are diverging mirrors and would spread the light rays apart, making it impossible to form a focused image of a distant object.
- c. A prism: Prisms are used for dispersing light into its constituent colors or deviating light, not for the primary gathering and focusing of light in a telescope.
- d. A plano convex lens: While lenses are used in refracting telescopes, a 'reflecting' telescope is defined by its use of mirrors rather than lenses for the primary objective to avoid chromatic aberration.
Height of geostationary satellite above the earth is approximately
A)
3,600 km
B)
42,000 km
C)
30,000 km
D)
36,000 km
Based on Mizo customary knowledge / general exam pattern.
Yield point in fatigue loading as compared to static loading is
A)
same
B)
higher
C)
lower
D)
none of them
In materials science and strength of materials, 'fatigue' refers to the progressive and localized structural damage that occurs when a material is subjected to cyclic loading. The key characteristic of fatigue is that failure (or yielding/fracture) occurs at a stress level significantly lower than the ultimate tensile strength or the yield strength measured under static (monotonic) loading. This is because cyclic stresses cause microscopic cracks to initiate and propagate over time, weakening the material's internal structure. Therefore, the effective stress level at which a material fails under fatigue loading is lower than the yield point observed during a standard static load test. Option (a) is incorrect because fatigue specifically reduces the load-bearing capacity. Option (b) is incorrect because cyclic loading never increases the strength of a material relative to its static yield point.
Crater wear takes place in a single point cutting tool at
A)
flank
B)
side rake
C)
face
D)
tip
In metal cutting using a single point tool, tool wear is generally categorized into two main types: Crater Wear and Flank Wear.
1. **Crater Wear**: This occurs on the rake face (the top surface of the tool) where the chip slides over the tool. It is caused by the high temperature and chemical diffusion between the tool material and the chip. This wear forms a concave 'crater' or depression on the face of the tool, typically just behind the cutting edge.
2. **Flank Wear**: This occurs on the flank face (the side of the tool that rubs against the finished surface of the workpiece). It is primarily caused by abrasion.
Analyzing the options:
- (a) Flank: This is where flank wear occurs, not crater wear.
- (b) Side rake: While the rake face is involved, 'face' is the standard terminology for the surface where the crater forms.
- (c) Face: Correct. The rake face is the specific area where the chip flows and where crater wear develops.
- (d) Tip: While the tip is the point of contact, the crater extends back along the face, not just at the tip.
Sulphur in pig iron tends to make it
A)
hard
B)
soft
C)
ductile
D)
malleable
In the context of pig iron and steel metallurgy, sulphur is considered an impurity. Sulphur reacts with iron to form iron sulphide (FeS). These sulphide inclusions typically concentrate at the grain boundaries of the metal. This presence of sulphur increases the hardness and brittleness of the material. Specifically, sulphur contributes to a phenomenon known as 'hot shortness' (brittleness at high temperatures), which makes the metal difficult to work or forge. Because it inhibits the sliding of grain boundaries and increases internal stress/resistance to deformation, it makes the pig iron harder and more brittle.
Why other options are incorrect:
- b. soft: Sulphur increases brittleness and hardness; it does not soften the metal.
- c. ductile: Ductility is the ability to be drawn into wires. Sulphur significantly reduces ductility by promoting brittle fracture.
- d. malleable: Malleability is the ability to be hammered into sheets. Sulphur makes the metal brittle, meaning it will crack rather than deform under compression, thus reducing malleability.
The transverse section of a helical gear is identical to
A)
bevel gear
B)
spur gear
C)
worm gear
D)
none of them
A helical gear is essentially a spur gear with teeth that are cut at an angle (the helix angle) relative to the axis of the gear. When you take a transverse section (a cross-section perpendicular to the axis of the gear), the curvature of the teeth and the profile of the gear appear exactly as they would in a spur gear. In a spur gear, the teeth are parallel to the axis, so any section perpendicular to the axis is identical to the gear's basic profile. Therefore, the transverse section of a helical gear is identical to that of a spur gear.
Comparison with other options:
- Bevel gears have conical shapes used for intersecting shafts, so their sections are not identical to helical transverse sections.
- Worm gears consist of a worm (screw) and a worm wheel, which have entirely different geometries and tooth profiles compared to the transverse section of a helical gear.
Ferrous alloys are usually cast by
A)
hot chamber machine
B)
cold chamber machine
C)
die casting machine
D)
none of them
To solve this question, we must analyze the casting methods mentioned in the options relative to the properties of ferrous alloys (such as steel and cast iron).
1. **Ferrous Alloys Characteristics**: Ferrous alloys have very high melting points (e.g., Steel melts around 1370°C to 1540°C).
2. **Die Casting (Options a, b, and c)**: Die casting involves forcing molten metal into a reusable metal mold (die) under high pressure. This process is primarily used for non-ferrous metals with lower melting points, such as Zinc, Aluminum, and Magnesium.
3. **Hot Chamber Machine (Option a)**: In this process, the melting pot is integrated into the machine. It is used for metals with low melting points (like Zinc) because high-melting-point metals would melt and destroy the pump and valves of the machine.
4. **Cold Chamber Machine (Option b)**: Here, the metal is melted in a separate furnace and poured into the chamber. While it can handle slightly higher temperatures than hot chamber machines (e.g., Aluminum), it is still not suitable for the extreme temperatures of ferrous alloys, as the molten steel would cause rapid thermal fatigue and erosion of the metal dies.
5. **Conclusion**: Ferrous alloys are typically cast using **Sand Casting**, **Investment Casting**, or **Centrifugal Casting**, where the molds can withstand the extreme heat. Since none of the die casting options (hot chamber, cold chamber, or general die casting) are suitable for standard ferrous alloy casting, the correct answer is 'none of them'.
The maximum strain energy that can be stored in a body is known as
A)
impact energy
B)
resilience
C)
proof resilience
D)
toughness
In the study of Strength of Materials, strain energy is the energy stored in a body due to deformation.
1. **Resilience**: This is the capacity of a material to absorb energy when it is deformed elastically and then, upon unloading, to have this energy recovered. It is the strain energy stored up to any point within the elastic limit.
2. **Proof Resilience**: This is the maximum energy that can be absorbed by a body per unit volume without permanent deformation. In other words, it is the strain energy stored when the material is stressed up to its elastic limit (or proportional limit). Since it is the 'maximum' energy stored within the elastic range, it is the correct answer.
3. **Toughness**: This is the total energy absorbed by a material up to the point of fracture. It includes both elastic and plastic deformation. While it is a 'maximum' energy, it refers to the energy until failure, not the stored energy typically associated with resilience.
4. **Impact Energy**: This refers to the energy absorbed by a material during a high-strain rate loading (like a Charpy or Izod test), which is a measure of toughness rather than a specific property of stored strain energy.
Therefore, the maximum strain energy that can be stored (specifically within the elastic limit) is known as proof resilience.
The cutting tool in a milling machine is mounted on
A)
tool holder
B)
arbor
C)
spindle
D)
column
In a milling machine, the cutting tool (specifically the milling cutter) is mounted on a shaft called the arbor. The arbor is a precision-ground shaft that holds the cutter and is driven by the machine's spindle to provide the necessary rotational motion for material removal.
Breakdown of options:
- a. Tool holder: While general tools are held by holders, in the specific context of milling (especially horizontal milling), the primary mounting shaft is the arbor.
- b. Arbor: Correct. It is the specific component designed to hold and rotate the milling cutter.
- c. Spindle: The spindle is the main rotating shaft of the machine. While the arbor is inserted into or driven by the spindle, the tool itself is mounted on the arbor.
- d. Column: The column is the main vertical structural support of the milling machine and does not hold the cutting tool.
Which of the following has no unit?
A)
Kinematic viscosity
B)
Surface tension
C)
Strain
D)
Elasticity
To determine which quantity has no unit, we analyze the dimensions and definitions of each option:
1. **Kinematic Viscosity (a):** It is defined as the ratio of dynamic viscosity to density. Its SI unit is square meters per second (m²/s) or Stokes (St). It has dimensions [L²/T].
2. **Surface Tension (b):** It is defined as the force per unit length acting on the surface of a liquid. Its SI unit is Newtons per meter (N/m). It has dimensions [MT⁻²].
3. **Strain (c):** In the context of Strength of Materials, strain is defined as the ratio of the change in dimension (e.g., change in length $\Delta L$) to the original dimension (e.g., original length $L$). Since it is a ratio of two identical physical quantities (length/length), the units cancel out, making it a dimensionless quantity with no unit.
4. **Elasticity (d):** Elasticity (specifically the Modulus of Elasticity) is the ratio of stress to strain. Since stress has units of pressure (N/m² or Pascals) and strain is dimensionless, the unit of elasticity is the Pascal (Pa). It has dimensions [ML⁻¹T⁻²].
Therefore, Strain is the only quantity among the options that is dimensionless and has no unit.
German silver is an alloy of
A)
silver and some impurities
B)
refined silver
C)
nickel, copper and zinc
D)
nickel and copper
German silver, also known as Nickel silver, is a misnomer because it contains no actual silver. It is a ternary alloy consisting of copper, nickel, and zinc. The typical composition is approximately 50-60% copper, 15-25% nickel, and 10-30% zinc. The addition of nickel gives the alloy a silvery-white appearance and increases its resistance to corrosion, making it useful for musical instruments, cutlery, and decorative items.
Analysis of other options:
- Option (a) and (b) are incorrect because German silver contains no silver atoms.
- Option (d) describes a binary alloy (like Cupronickel), but German silver specifically requires zinc to achieve its characteristic properties and color.