A-Level Physics October/November 2024 Q1(b)(i): The drag force D acting on the sphere is given by D = 6πηv where η is a property of the…

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  First, rearrange the equation to make $\eta$ the subject. The constant $6\pi$ is dimensionless and can be ignored for unit analysis.

  $D = 6\pi\eta r v$

  $\eta = \frac{D}{6\pi r v}$

  Next, determine the SI base units for each quantity in the rearranged equation.

  • Force, $D$: From $F=ma$, the units are $\text{kg} \times \text{m s}^{-2} = \text{kg m s}^{-2}$.
  • Radius, $r$: This is a length, so the unit is $\text{m}$.
  • Velocity, $v$: The units are $\text{m s}^{-1}$.

  Now, substitute these base units back into the equation for $\eta$.

  Units of $\eta = \frac{\text{units of } D}{\text{units of } r \times \text{units of } v}$

  Units of $\eta = \frac{\text{kg m s}^{-2}}{\text{m} \times \text{m s}^{-1}}$

  Finally, simplify the expression by combining the indices.

  Units of $\eta = \frac{\text{kg m s}^{-2}}{\text{m}^2 \text{s}^{-1}}$

  Units of $\eta = \text{kg m}^{1-2} \text{s}^{-2 - (-1)}$

  Units of $\eta = \text{kg m}^{-1} \text{s}^{-1}$

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  How the marks are awarded

  • C1 — Correctly identifying the SI base units of the drag force D as kg m s⁻², typically by recalling or deriving from F=ma.
  • C1 — Correctly stating the SI base units for the other variables: radius r as m, and velocity v as m s⁻¹.
  • A1 — Correctly substituting the base units into the rearranged formula and performing the algebraic simplification of the indices to arrive at the final answer of kg m⁻¹ s⁻¹.

  Common mistakes

  • Leaving the unit of force as Newtons (N) instead of converting it to its constituent SI base units (kg m s⁻²).
  • Making an algebraic error when simplifying the indices, for example calculating s⁻² / s⁻¹ as s⁻³ or s⁻².
  • Incorrectly cancelling the 'm' units, for example cancelling the single 'm' in the numerator with only one of the 'm's in the denominator, but forgetting the other.
  • Omitting one of the variables (e.g. r) from the denominator during substitution, leading to an incorrect final unit.

  Examiner tip: For any dimensional analysis problem, first algebraically rearrange the equation to make the target quantity the subject before substituting the base units for each variable.

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