A-LevelPhysicsOscillations9702/42 · Feb/Mar 2019 · Q310 Marks

A cylindrical tube, sealed at one end, has cross-sectional area A and contains some sand. The total mass of the tube and the sand is M. The tube floats upright in a liquid of density ρ, as illustrated in Fig. 3.1. [Figure 3.1]

(a)(i)State the two forces that act on the tube immediately after its release.[1]
(a)(ii)State and explain the direction of the resultant force acting on the tube immediately after its release.[2]
(b)The acceleration a of the tube is given by the expression a = - (Apg/M)x where x is the vertical displacement of the tube from its equilibrium position. Use the expression to explain why the tube undergoes simple harmonic oscillations in the liquid.[2]
(c)(i)For a tube having cross-sectional area A of 4.5cm² and a total mass M of 0.17kg, the period of oscillation of the tube is 1.3s. Determine the angular frequency ω of the oscillations.[2]
(c)(ii)Use your answer in (i) and the expression in (b) to determine the density ρ of the liquid in which the tube is floating.[3]

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Mark scheme

Marking points from the official mark scheme (Feb/Mar 2019). The mark or mark type is shown in brackets.

[B1] mention of upthrust and weight
[B1] upthrust is greater than the weight
[B1] (resultant force is) upwards
[B1] A, ρ, g and M are constant
[B1] either acceleration ∝ displacement or acceleration ∝ displacement and (– sign indicates) a and x in opposite directions
[C1] either ω = 2π / T or ω = 2πf and f = 1 / T
[A1] ω = 2π / 1.3 = 4.8 rad s⁻¹
[C1] ω² = Aρg/m
[C1] 4.83² = (4.5 × 10⁻⁴ × ρ × 9.81)/0.17
[A1] ρ = 900 kg m⁻³

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