## ERQ · 12 marks · Topics: B.3 Gas laws + C.1 Simple harmonic motion
**Stem.** A student designs a "gas spring" oscillator. A horizontal glass syringe of internal cross-sectional area A = 3.0 × 10⁻⁴ m² is sealed at one end and contains an ideal gas at atmospheric pressure P₀ = 1.01 × 10⁵ Pa, occupying an initial volume V₀ = 6.0 × 10⁻⁵ m³ at room temperature T = 293 K. A frictionless plunger of mass m = 0.045 kg closes the syringe. The plunger is displaced inward by a small amount and released; an ultrasonic motion sensor records its displacement x(t) about equilibrium. The external pressure on the outer face of the plunger remains P₀ throughout. The student observes oscillations with a measured period of T_exp = 0.069 s, and assumes the compressions are isothermal.
### Part (a) Outline [2 marks] · AO1 · Topic: B.3
Outline what is meant by an **isothermal** change of an ideal gas, and state the relationship between pressure and volume for such a change.
### Part (b)(i) Show that [3 marks] · AO2/AO3 · Topic: B.3
For a small inward displacement x of the plunger from equilibrium (so that the gas volume becomes V₀ − Ax, with Ax ≪ V₀), show that the net force on the plunger is approximately
$$F \approx -\frac{P_0 A^2}{V_0}\,x$$
### Part (b)(ii) Calculate [3 marks] · AO2/AO3 · Topic: B.3 + C.1
Using the result in (b)(i), calculate the theoretical period of small oscillations of the plunger.
### Part (c) Explain [2 marks] · AO3 · Topic: C.1
Using your expression in (b)(i), explain why the motion of the plunger is simple harmonic for small displacements but would **not** remain simple harmonic if the plunger were pushed in by a distance comparable to V₀ / A.
### Part (d) Suggest [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR
The measured period T_exp = 0.069 s is slightly **shorter** than the value calculated in (b)(ii). Suggest one physical reason, referring to the thermodynamic behaviour of the gas during the rapid oscillation, why the experimental period is shorter than the isothermal prediction.
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## Mark Scheme
### Part (a) [2 marks]
- M1: Isothermal = process carried out at constant temperature (with the gas in thermal equilibrium / heat exchanged with surroundings to maintain T) [ECF: no]
- M2: States PV = constant / P ∝ 1/V / quotes Boyle's law for fixed amount of gas [ECF: no]
### Part (b)(i) [3 marks]
- M1: Applies isothermal condition P₀V₀ = P(V₀ − Ax), so P = P₀V₀ / (V₀ − Ax) [ECF: no]
- M2: Net force F_net = (P₀ − P)A; substitutes and obtains F_net = P₀A[1 − V₀/(V₀ − Ax)] = −P₀A · (Ax)/(V₀ − Ax) [ECF: yes]
- M3: Applies binomial / small-x approximation V₀ − Ax ≈ V₀ to obtain F ≈ −(P₀A²/V₀) x; identifies negative sign as restoring [ECF: yes]
### Part (b)(ii) [3 marks]
- M1: Recognises effective spring constant k = P₀A²/V₀ and uses T = 2π√(m/k) [ECF: yes from b(i)]
- M2: Correct substitution: k = (1.01 × 10⁵)(3.0 × 10⁻⁴)² / (6.0 × 10⁻⁵) = 1.515 × 10² N m⁻¹; T = 2π√(0.045 / 151.5) [ECF: yes]
- M3: T ≈ 0.108 s (accept 0.11 s, 2 s.f., correct unit) [ECF: yes]
### Part (c) [2 marks]
- M1: For small x, F is linear in x (F ∝ −x) with constant coefficient P₀A²/V₀, which is the defining condition for SHM [ECF: no]
- M2: For large x the exact expression F = −P₀A²x/(V₀ − Ax) is non-linear in x (denominator no longer ≈ V₀ / restoring force grows faster than linearly on compression); hence motion becomes anharmonic / asymmetric about equilibrium [ECF: no]
### Part (d) [2 marks]
- M1: At T ≈ 0.07 s the compression/expansion is too rapid for the gas to exchange heat with the syringe walls — the process is closer to **adiabatic** than isothermal [ECF: no]
- M2: For an adiabatic change PVᵞ = const, the effective stiffness becomes k_adiabatic = γP₀A²/V₀ > k_isothermal, giving a larger restoring force and therefore a **shorter** period, consistent with T_exp < T_theory [ECF: no]
### Marker notes
- (b)(i) alternative: candidates may differentiate P = P₀V₀/V with respect to x at x = 0 to obtain dP/dx = P₀A/V₀, then F = −A · dP · ... — award full marks if logic is clean.
- (b)(ii) ECF: if student omits the factor A² or uses k = P₀A/V₀, accept downstream period of ≈ 0.0010 s domain with correct method for M1 and M3.
- (d) accept alternatives only if they reduce period: e.g. "gas heats on compression raising P above isothermal value, stiffening the spring". Do NOT accept friction, air leakage, or damping — these would *increase* the period or reduce amplitude, not shorten T.