A-Level Physics October/November 2024 Q3(b)(ii): The density of the water is 1.0 × 10³ kg m⁻³. Calculate the area A of the circular cros…

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  The upthrust $U$ on the cylinder is equal to the weight of the water displaced. The volume of displaced water is $V = A \times h$, where $h$ is the submerged height.

  Upthrust, $U = \text{weight of displaced fluid} = \rho g V$

  Substituting $V = A h$ into the upthrust equation: $U = \rho g (A h)$

  Rearranging for the cross-sectional area $A$: $A = \frac{U}{\rho g h}$

  From the previous part of the question, the upthrust $U = 1.4$ N and the submerged height $h = 0.10$ m. The density of water $\rho = 1.0 \times 10^3$ kg m⁻³ and $g = 9.81$ N kg⁻¹.

  Substituting the values: $A = \frac{1.4}{(1.0 \times 10^3) \times 9.81 \times 0.10}$

  $A = 0.0014271...$ m²

  $A \approx 1.4 \times 10^{-3}$ m² (to 2 s.f.)

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  How the marks are awarded

  • C1 — The first C1 mark is awarded for correctly combining the formula for upthrust ($U = \rho g V$) with the formula for the volume of the submerged cylinder ($V = Ah$) to derive the expression for area, $A = \frac{U}{\rho g h}$.
  • C1 — The second C1 mark is awarded for the correct substitution of all values into the derived formula: $U=1.4$ N, $h=0.10$ m, $\rho=1.0 \times 10^3$ kg m⁻³, and $g=9.81$ N kg⁻¹.
  • A1 — The final A1 mark is for calculating the correct numerical answer of $1.4 \times 10^{-3}$ and stating the correct unit, m².

  Common mistakes

  • Forgetting to include the acceleration of free fall, g, in the calculation (i.e. calculating A = U / (ρh)).
  • Using the total height of the cylinder instead of the submerged height, h.
  • Incorrectly rearranging the formula, for example by multiplying instead of dividing (e.g. A = Uρgh).
  • Making a power-of-ten error when entering the density into the calculator, for example using 1.0 x 10⁻³ instead of 1.0 x 10³.

  Examiner tip: Always start by writing down the relevant fundamental equations, then combine and rearrange them algebraically to make your target variable the subject before substituting any numbers.

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