| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Mechanics (Further Paper 3 Mechanics) |
| Year | 2022 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dimensional Analysis |
| Type | Find exponents with all unknowns |
| Difficulty | Standard +0.3 This is a standard dimensional analysis problem requiring students to equate dimensions and solve simultaneous equations for three exponents. While it involves multiple steps and algebraic manipulation, the method is routine and commonly practiced in Further Maths mechanics. The question is slightly above average difficulty due to the need to handle three unknowns systematically, but it follows a well-established template. |
| Spec | 6.01d Unknown indices: using dimensions |
| Answer | Marks | Guidance |
|---|---|---|
| \(MLT^{-2}\) | B1 | States correct dimensions of force |
| Answer | Marks | Guidance |
|---|---|---|
| \([G] = MLT^{-2} \times L^2M^{-2} = M^{-1}L^3T^{-2}\) | B1 | Obtains correct expression for dimensions of \(G\) |
| \(LT^{-1} = M^{-a}L^{3a}T^{-2a} \times M^b \times L^c\) leading to \(1 = 3a+c\), \(0 = b-a\), \(-1 = -2a\) | M1 | Forms dimensional analysis equation using their dimensions for \(G\) and at least two of the other three dimensions correct |
| \(a = \frac{1}{2}\), \(b = \frac{1}{2}\) | M1 | Deduces at least one of the values of \(a\), \(b\) or \(c\) correctly |
| \(c = -\frac{1}{2}\) | A1 | Obtains correct values for \(a\), \(b\) and \(c\) |
## Question 4(a):
$MLT^{-2}$ | B1 | States correct dimensions of force
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## Question 4(b):
$[G] = MLT^{-2} \times L^2M^{-2} = M^{-1}L^3T^{-2}$ | B1 | Obtains correct expression for dimensions of $G$
$LT^{-1} = M^{-a}L^{3a}T^{-2a} \times M^b \times L^c$ leading to $1 = 3a+c$, $0 = b-a$, $-1 = -2a$ | M1 | Forms dimensional analysis equation using their dimensions for $G$ and at least two of the other three dimensions correct
$a = \frac{1}{2}$, $b = \frac{1}{2}$ | M1 | Deduces at least one of the values of $a$, $b$ or $c$ correctly
$c = -\frac{1}{2}$ | A1 | Obtains correct values for $a$, $b$ and $c$
---4
\begin{enumerate}[label=(\alph*)]
\item State the dimensions of force.
4
\item The velocity of an object in a circular orbit can be calculated using the formula
$$v = G ^ { a } m ^ { b } r ^ { c }$$
where:\\
$G =$ Universal constant of gravitation in $\mathrm { Nm } ^ { 2 } \mathrm {~kg} ^ { - 2 }$\\
$m =$ Mass of the Earth in kg\\
$r =$ Radius of the orbit in metres\\
Use dimensional analysis to find the values of $a , b$ and $c$\\[0pt]
[4 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Mechanics 2022 Q4 [5]}}