QUESTIONS
Useful constants
| Constant | Value |
|---|---|
| Universal gravitational constant, G | 6.67 × 10⁻¹¹ Nm²kg⁻² |
| Speed of light in free space | 3.0 × 10⁸ ms⁻¹ |
| Acceleration due to gravity, g | 10 ms⁻² |
| Speed of sound (air) | 340 ms⁻¹ |
| Molar gas constant, R | 8.31 Jmol⁻¹K⁻¹ |
| Linear expansivity of tungsten | 4.4 × 10⁻⁶ K⁻¹ |
| Linear expansivity of glass | 8.0 × 10⁻⁶ K⁻¹ |
| Volume expansivity of mercury | 1.81 × 10⁻⁴ K⁻¹ |
| Bulk modulus of water | 2.0 × 10⁹ Nm⁻² |
| Density of water | 1000 kgm⁻³ |
SECTION A — Answer ALL
- Determine the dimension of permittivity of free space (ε₀).
- Assume two forces of magnitudes √2N, √5N act at a point at angle 10° to each other. Find the resultant force.
- What are conservative and non-conservative forces? Give one example for each. (b) A force of magnitude (î+ĵ+2k̂)N acts on a particle of mass 5 kg through a distance of (4î−ĵ+10k̂)m. Find the work done.
- Calculate the total amount of heat released if 1 kg of steam at 107°C is converted into ice at −20°C.
5a. State the zeroth law of thermodynamics and write the mathematical expression for the first law, stating the meaning of each term. (b) In a thermodynamic system, internal energy decreases by 400 J while the system does 250 J of work. Calculate the net heat taken in by the system. - One mole of an ideal gas occupying 0.022400 m³ is heated from 0°C to 100°C at constant pressure 1.01×10⁵ Nm⁻². Calculate the work done by the gas.
- A steel wire of length 6.33 cm and cross-section 1.07×10⁻⁶ m² is stretched by an external load of 29 N. Calculate the Young modulus of the steel.
- What are free and damped oscillatory motions? Illustrate graphically. (b) A pendulum bob of mass 20 g and length 33 cm is displaced through 54°. Find the potential energy stored.
- Briefly explain the term "standing waves." Determine whether a wave of wavelength 4 mm propagated in air at 300 ms⁻¹ is audible to humans.
- If the speed of sound in air at 0°C is 330 ms⁻¹, find the change in speed at 10°C.
SECTION B — MECHANICS (Answer ONE)
11a. (i) Define "rigid body"; write the mathematical condition(s) for a body to be in equilibrium. Hence state the three factors on which moment of inertia depends. (ii) A machine wheel accelerates uniformly from rest and rotates through 1.13 rad in the first second. Find the angle rotated in the next second. (iii) Two particles of mass m₁ and m₂ lie in a plane perpendicular to line AB at distances r₁ and r₂ (Figure 1). Find the moment of inertia of the system about axis AB, about an axis through m₁ perpendicular to the line joining m₁ and m₂, and about an axis passing through both m₁ and m₂.
11b. (i) Four particles of mass 1 kg, 2 kg, 3 kg, 4 kg are placed at vertices A, B, C, D of a 1 m square (Figure 2). Find the centre of mass. (ii) What is escape velocity? Derive its expression. (iii) Two planets have masses in ratio 1:10 and radii in ratio 2:5. Find the ratio of their escape velocities.
12a. (i) What is meant by resultant force? (ii) Four forces (25N, 20N, 30N, 35N) act at a point as in Figure 3. Find the magnitude and direction of the resultant. (iii) A particle of mass 3 kg is projected vertically upward at 29.4 m/s from a tower 34.3 m high. Find the time taken to return to the tower (g=9.8 ms⁻²).
12b. (i) State Newton's three laws of motion. Hence prove that Newton's second law can be written F=ma. (ii) A force acts on a 300 g body at rest, changing its speed to 3.77 ms⁻¹ in 1.65 s. Calculate the impulse. (iii) What is trajectory? An object is projected at 30 ms⁻¹; find the possible angles of projection to cover a range of 45 m.
SECTION C — HEAT AND PROPERTIES OF MATTER (Answer any TWO)
13a. (i) Define deforming force; distinguish elasticity from plasticity. (ii) Using Figure 4, explain regions OA, OB, BC, CD, DE. State Hooke's law and the physical meaning of the slope of OA. (iii) The bulk modulus of water is 2.3×10⁹ Nm⁻². Calculate its compressibility and the pressure required to compress a water sample by 0.025%.
13b. A wire 4 m long, 300 μm diameter, is stretched 0.3 mm by a 100 N load. Calculate (i) the potential energy in the wire, (ii) the work done stretching a wire of length 2 m, cross-sectional area 1 mm² through 0.1 mm extension (Young modulus = 2.0×10¹¹ Nm⁻²).
14a. (i) Distinguish heat capacity from specific heat capacity; latent heat from latent heat of fusion. (ii) Briefly explain absolute temperature scale and triple point of water. (iii) Determine the temperature at which the Fahrenheit and Celsius scales read the same.
14b. (i) A gas at 27°C, volume 4000 cm³, pressure 100 Nm⁻² is compressed isothermally to 150 Nm⁻². Calculate the change in volume. If then heated at constant volume to 127°C, calculate the new pressure. (ii) State the assumptions of an ideal gas. Calculate the rms speed, average speed, and mean kinetic energy of CO₂ molecules (molar mass 44.0 g/mol) at 27°C.
15a. (i) What is meant by degree of freedom of gases? Find the degrees of freedom of a monatomic gas. (ii) 2 mol He mixed with 4 mol H₂: find Cv and Cp of the mixture in terms of R, and evaluate γ.
15b. (i) State and define the three modes of heat transfer. (ii) State Stefan's law of radiation with its mathematical form, defining each symbol. (iii) Sketch black-body radiation power vs wavelength curves at three temperatures, and explain the concept.
16a. (i) Define thermodynamics; state and explain the four thermodynamic processes using equations. (ii) A gas at 1200 Nm⁻² and 990.02 m³ is compressed adiabatically to 600.01 m³. Find the final pressure (γ=1.4).
16b. (i) Define entropy and briefly explain the Carnot engine. (ii) A Carnot engine operates between 227°C and 27°C, absorbing 100 J from the hot reservoir. Calculate its efficiency, work output, and heat rejected. (iii) Determine the change in entropy when 50 g of ice melts at 0°C.
SECTION D — WAVES AND VIBRATIONS (Answer any ONE)
17a. (i) Use sketch diagrams to explain the concept of standing waves. (ii) Consider a progressive wave y=6sin(0.02πx+4πt), where x and y are in metres and t is in seconds. Calculate the amplitude, wavelength, frequency, direction of propagation, maximum speed of the particle in the string, and the period of the wave.
17b. (i) Write down the relation for the normal mode frequencies (f) of transverse waves on a string of length l under tension T, fixed at both ends. Hence prove that the relation can also be written as f=√(T/ρA), where ρ and A are the density and cross-sectional area of the string. (ii) A string of length 50 cm, mass per unit length 0.01 kgm⁻¹, has a fundamental frequency of 250 Hz. What is the tension in the string?
17c. Consider a transverse wave at t=0, propagated as shown in Figure 5. If the speed of the wave is 300 ms⁻¹, write down the equation of this wave.
18a. (i) State the effect of each of the following factors on the speed of sound in air: Temperature, Pressure, and Humidity. Write down the necessary equation relating the speed for each factor. (ii) Briefly explain the terms: beat frequency and harmonics. (iii) Assume an open organ pipe of fundamental frequency 300 Hz. If the first overtone of a closed organ pipe has the same frequency as the first overtone of this open pipe, find the length of each pipe.
18b. (i) What is mechanical wave? State the characteristics of mechanical wave. (ii) A ship sails in a straight line between two transmitting stations A and B which emit radio waves of the same frequency. The signal fluctuates in intensity and passes through a minimum each time the ship travels 100 m. Calculate the frequency of the radio signals. (iii) A tube 1 m long closed at one end has its lowest resonance frequency at 86.2Hz. With the same tube open at both ends the first resonance occurs at 171Hz. Calculate the speed of sound in the column of air and the end correction for the tube. Briefly explain the underlined term.
ANSWERS
Section A
1. From Coulomb's law, ε₀ = q²/(4πFr²). Dimension: [Q]=IT, [F]=MLT⁻², [r²]=L²
[ε₀] = M⁻¹L⁻³T⁴I²
2. R = √(F₁²+F₂²+2F₁F₂cosθ) = √(2+5+2√2·√5·cos10°) = √(7+6.229) = √13.23
R ≈ 3.64 N
Direction: tanα = √5 sin10°/(√2+√5cos10°) = 0.388/3.616 → α ≈ 6.1° from the √2N force
3. Conservative forces: work done is path-independent and fully recoverable (e.g. gravity, spring force). Non-conservative forces: work depends on path and energy is dissipated (e.g. friction, air resistance).
W = F·d = (1)(4)+(1)(−1)+(2)(10) = 4−1+20 = 23 J
4. 1 kg steam 107°C → ice −20°C (using c_steam≈2010 J/kgK, L_v=2.26×10⁶ J/kg, c_water=4200 J/kgK, L_f=3.34×10⁵ J/kg, c_ice≈2100 J/kgK):
| Stage | Heat (J) |
|---|---|
| Steam 107→100°C | 14,070 |
| Condensation at 100°C | 2,260,000 |
| Water 100→0°C | 420,000 |
| Freezing at 0°C | 334,000 |
| Ice 0→−20°C | 42,000 |
| Total | ≈ 3.07 × 10⁶ J |
5a. Zeroth law: if two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other.
First law: Q = ΔU + W, where Q = heat supplied to the system, ΔU = change in internal energy, W = work done by the system.
b. ΔU = −400 J, W = 250 J → Q = ΔU+W = −400+250 = −150 J (system releases 150 J of heat)
6. V₁=0.0224 m³ at T₁=273K; T₂=373K. At constant P: V₂=V₁T₂/T₁=0.0224×373/273=0.03060 m³
ΔV = 0.00820 m³
W = PΔV = 1.01×10⁵×0.00820 = ≈ 828 J
7. Y = FL/(AΔL). With F=29N, L=0.0633m, A=1.07×10⁻⁶m²: Y = (29×0.0633)/(1.07×10⁻⁶×ΔL) — the extension ΔL is required to obtain a numerical value.
8. Free oscillation: system oscillates at its natural frequency with amplitude remaining constant (no energy loss). Damped oscillation: amplitude progressively decreases with time due to resistive forces.
PE = mgL(1−cosθ) = 0.02×10×0.33×(1−cos54°) = 0.02×10×0.33×0.4122 = ≈ 0.0272 J
9. Standing wave: formed by the superposition of two waves of equal frequency and amplitude travelling in opposite directions, producing fixed nodes and antinodes.
f = v/λ = 300/0.004 = 75,000 Hz = 75 kHz — this exceeds the human audible range (20 Hz–20 kHz), so it is NOT audible (ultrasonic).
10. Using v ≈ v₀+0.61θ: v₁₀ = 330+0.61(10) = 336.1 m/s, an increase of ≈ 6.1 m/s
Section B
11a. (i) Rigid body: a body in which the distance between any two particles remains constant under applied forces.
Equilibrium: ΣF = 0 and Στ = 0.
Moment of inertia depends on: mass of the body, distribution of mass about the axis, and position of the axis of rotation.
(ii) θ₁=½α(1)²=1.13 → α=2.26 rad/s²
θ(0–2s)=½(2.26)(2)²=4.52 rad; angle in 2nd second = 4.52−1.13 = 3.39 rad
(iii) I_AB = m₁r₁²+m₂r₂²
I (axis through m₁, perpendicular to line joining m₁,m₂) = m₂(r₁+r₂)² (only m₂ contributes since m₁ lies on the axis)
I (axis passing through both m₁ and m₂) = 0 (both particles lie on the axis)
11b. (i) x̄=(1×0+2×1+3×1+4×0)/10=0.5, ȳ=(1×1+2×1+3×0+4×0)/10=0.3
Centre of mass = (0.5 m, 0.3 m)
(ii) Escape velocity: the minimum speed needed for an object to escape a planet's gravitational field without further propulsion.
Setting ½mv²=GMm/R → v = √(2GM/R)
(iii) v ∝ √(M/R); v₁/v₂ = √[(1/2)/(10/5)] = √0.25 = 0.5 → ratio = 1:2
12a. (i) Resultant force: the single force that produces the same effect as all the individual forces acting on a body combined (vector sum).
(ii) Method: resolve each of the four forces into x and y components using the angles shown in Figure 3, sum ΣFx and ΣFy, then R=√(ΣFx²+ΣFy²) with direction θ=tan⁻¹(ΣFy/ΣFx).
(iii) Returning to launch height: 0=29.4t−½(9.8)t² → t(29.4−4.9t)=0 → t = 6 s
12b. (i) First law: a body remains at rest or in uniform motion unless acted upon by a net external force. Second law: the rate of change of momentum of a body is proportional to the net force applied and occurs in the direction of the force. Third law: for every action there is an equal and opposite reaction.
Derivation of F=ma: F ∝ dp/dt = d(mv)/dt = m(dv/dt) = ma (for constant mass)
(ii) Impulse = mΔv = 0.3×(3.77−0) = 1.131 Ns
(iii) Trajectory: the parabolic path traced by a projectile moving under gravity.
R = u²sin2θ/g → 45 = 900sin2θ/10 → sin2θ=0.5 → 2θ=30° or 150°
θ = 15° or 75°
Section C
13a. (i) Deforming force: a force that changes the shape or size of a body.
Elasticity: the ability of a body to return to its original shape and size after the removal of a deforming force. Plasticity: undergoing permanent deformation and not returning to the original shape after the force is removed.
(ii) OA: proportional/elastic region (Hooke's law obeyed); B: elastic limit; BC: plastic deformation begins; CD: further plastic flow/yielding; DE: region approaching fracture.
Hooke's law: within the elastic limit, stress is directly proportional to strain. The slope of OA represents the Young's modulus of the material.
(iii) Compressibility k = 1/B = 1/(2.3×10⁹) = 4.35×10⁻¹⁰ Pa⁻¹
ΔP = B(ΔV/V) = 2.3×10⁹×0.00025 = 5.75×10⁵ Nm⁻²
13b. A = πr² = π(1.5×10⁻⁴)² = 7.07×10⁻⁸ m²
(i) PE = ½Fe = ½×100×3×10⁻⁴ = 0.015 J
(ii) F = YAΔL/L = (2.0×10¹¹×1×10⁻⁶×1×10⁻⁴)/2 = 10 N; Work = ½FΔL = ½×10×1×10⁻⁴ = 5×10⁻⁴ J
14a. (i) Heat capacity: the quantity of heat required to raise the temperature of a given mass of a substance by 1°C (depends on mass). Specific heat capacity: heat required to raise the temperature of 1 kg of a substance by 1°C (an intrinsic property). Latent heat: heat absorbed or released at constant temperature during a change of state. Latent heat of fusion: latent heat specifically associated with the solid-to-liquid (or liquid-to-solid) change of state.
(ii) Absolute temperature scale: a scale of temperature whose zero point (absolute zero, 0 K) corresponds to the theoretical state of minimum thermal energy (0 K = −273.15°C). Triple point of water: the unique combination of temperature and pressure at which the solid, liquid, and gaseous phases of water coexist in equilibrium (0.01°C).
(iii) C = (9/5)C+32 → C = −40 (i.e. −40°C = −40°F)
14b. (i) Isothermal: V₂=P₁V₁/P₂=100×4000/150=2666.67 cm³; ΔV = 1333.3 cm³ decrease
At constant V: P₃=P₂T₃/T₂=150×400/300 = 200 Nm⁻²
(ii) Assumptions of an ideal gas: molecules occupy negligible volume compared to the container; no intermolecular forces of attraction except during collisions; collisions are perfectly elastic; molecules are in continuous, random motion; the gas obeys PV=nRT.
v_rms=√(3RT/M)=√(3×8.31×300/0.044) ≈ 412 m/s
v_avg=√(8RT/πM) ≈ 380 m/s
Mean KE per mole = (3/2)RT = 3739.5 J/mol
15a. (i) Degree of freedom: the number of independent ways in which a molecule can possess energy. Monatomic gas: 3 (translational only).
(ii) Cv(mix) = (2×1.5R+4×2.5R)/6 = 13R/6 ≈ 2.17R
Cp = Cv+R ≈ 3.17R
γ = Cp/Cv ≈ 1.46
15b. (i) Conduction: transfer of heat through a medium via molecular/electron vibration without bulk movement of matter. Convection: transfer of heat through the bulk movement of a fluid (liquid or gas). Radiation: transfer of heat via electromagnetic waves, requiring no medium.
(ii) Stefan's law: P = σAT⁴, where P = radiated power, σ = Stefan-Boltzmann constant, A = surface area of the radiating body, T = absolute temperature.
(iii) A black body absorbs all incident radiation and its emission spectrum depends only on its temperature. As temperature increases, the peak wavelength of emission shifts to shorter wavelengths (Wien's law), and the total radiated power (area under the curve) increases sharply, proportional to T⁴.
16a. (i) Thermodynamics: the branch of physics concerned with heat, work, temperature, and energy, and the laws governing their interconversion.
Isothermal (T constant): PV = constant. Isobaric (P constant): V/T = constant; W = PΔV. Isochoric (V constant): P/T = constant; W = 0. Adiabatic (Q = 0): PVᵞ = constant.
(ii) P₂ = P₁(V₁/V₂)^γ = 1200×(990.02/600.01)^1.4 = 1200×(1.65)^1.4 ≈ 2419 Nm⁻²
16b. (i) Entropy: a measure of the degree of disorder or randomness in a system. Carnot engine: an ideal, reversible heat engine that operates between two temperature reservoirs through a cycle of two isothermal and two adiabatic processes, giving the maximum theoretically possible efficiency between those temperatures.
(ii) η = 1−Tc/Th = 1−300/500 = 0.4 (40%); W = ηQ = 40 J; Qc = Q−W = 60 J
(iii) ΔS = mL/T = (0.050×3.34×10⁵)/273 = ≈ 61.2 J/K
Section D
17a. (i) Standing wave: formed by the superposition of two identical waves travelling in opposite directions along the same medium, producing fixed points of zero displacement (nodes) and maximum displacement (antinodes).
(ii) y=6sin(0.02πx+4πt): Amplitude A = 6 m; comparing with k=0.02π gives wavelength λ = 100 m; comparing with ω=4π gives frequency f = 2 Hz; since the equation has the form (+kx+ωt), the wave travels in the negative x-direction; maximum particle speed = Aω = 6×4π = ≈ 75.4 m/s; period T = 1/f = 0.5 s
17b. (i) For a string fixed at both ends, the normal mode frequencies are f_n = nv/(2l), where v is the wave speed on the string, given by v=√(T/μ). Since μ=ρA (mass per unit length = density × cross-sectional area), substituting gives f = (n/2l)√(T/ρA)
(ii) 250 = (1/(2×0.5))√(T/0.01) → 250 = √(T/0.01) → 250²=T/0.01 → T = 625 N
17c. Reading the amplitude (A = 0.08 m) and wavelength (λ, taken from the graph in Figure 5) from the given waveform, with speed v = 300 m/s:
f = v/λ, k = 2π/λ, ω = 2πf
The equation of the wave takes the general form y = 0.08 sin(kx − ωt) m, where k and ω are evaluated from the values of λ and f read directly off Figure 5.
18a. (i) Temperature: speed of sound increases with temperature, v = v₀+0.61θ (or v ∝ √T, where T is absolute temperature). Pressure: has negligible effect on speed at constant temperature, since v=√(γP/ρ) and P/ρ remains constant for a fixed mass of gas. Humidity: speed increases with humidity, since moist air is less dense than dry air.
(ii) Beat frequency: the frequency of the periodic variation in loudness heard when two sound waves of slightly different frequencies interfere, equal to |f₁−f₂|. Harmonics: frequencies that are whole-number multiples of the fundamental frequency of a vibrating system.
(iii) First overtone of open pipe = 2f₁ = 600 Hz; setting this equal to the first overtone of the closed pipe (3f₁') gives f₁' = 200 Hz
Length of open pipe: L = v/(2f₁) = 340/600 = 0.567 m
Length of closed pipe: L = v/(4f₁') = 340/800 = 0.425 m
18b. (i) Mechanical wave: a wave that requires a material medium through which to travel. Characteristics: requires a medium for propagation; transfers energy without net transfer of matter; may be transverse or longitudinal; obeys v=fλ; exhibits reflection, refraction, diffraction, and interference.
(ii) As the ship moves, the path difference between the two signals changes by 2d for every distance d travelled; successive intensity minima occur each time this path difference changes by one wavelength: 2(100)=λ → λ=200 m
f = c/λ = (3×10⁸)/200 = 1.5×10⁶ Hz (1.5 MHz)
(iii) Closed pipe: v=4f₁(L+e) → v=4(86.2)(1+e)
Open pipe: v=2f₂(L+2e) → v=2(171)(1+2e)
344.8(1+e) = 342(1+2e) → 344.8+344.8e = 342+684e → 2.8 = 339.2e → e ≈ 0.00825 m (0.825 cm)
v = 344.8(1+0.00825) ≈ 347.6 m/s
End correction: a small additional length added to the actual (measured) length of a pipe to account for the fact that the antinode of the standing wave forms slightly beyond the open end of the pipe, due to the inertia of the air layer just outside it.