| Exam Board | OCR |
|---|---|
| Module | Further Mechanics (Further Mechanics) |
| Year | 2018 |
| Session | September |
| Marks | 10 |
| Topic | Dimensional Analysis |
| Type | Derive dimensions from formula |
| Difficulty | Standard +0.3 This is a structured dimensional analysis question with clear scaffolding through five parts. Parts (i)-(ii) involve straightforward rearrangement of a given formula and basic dimensional reasoning. Parts (iii)-(v) require checking dimensional consistency, which is a standard A-level Further Maths technique. While it requires careful bookkeeping of dimensions, each step is guided and the methods are routine applications of dimensional analysis taught in the Further Mechanics module. Slightly easier than average due to the extensive scaffolding. |
| Spec | 6.01a Dimensions: M, L, T notation6.01d Unknown indices: using dimensions |
| Answer | Marks | Guidance |
|---|---|---|
| \([E] = [Fx] = MLT^{-2}L\) | M1 | Use of correct equation for, or units of, energy to find dimensions |
| \([E] = ML^2T^{-2}\) and \([\omega] = T^{-1}\) soi | A1 | |
| \(ML^2T^{-2} = [I]T^{-1}\) \(\Rightarrow\) \([I] = ML^2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(I = mr^2 \cdots\) | B1 | Must be "\(I =\)" |
| \(\cdots \times k\) | B1 | |
| where \(k\) is dimensionless | B1 |
| Answer | Marks |
|---|---|
| \([F] = MLT^{-2}, [a] = T^{-2}\) and their \([I]\) all used | M1 |
| \([F] = MLT^{-2} \neq [I]a = ML^2T^{-2}\) so the student's suggestion is incorrect | A1 |
| Answer | Marks |
|---|---|
| \([x] = L\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| E.g. The relationship could be \(Fx = p I a\) for some dimensionless quantity \(p\) | B1 | Or: there might be some other term(s) consisting of some dimensionally correct combination of quantities that need to be included |
## (i)
$[E] = [Fx] = MLT^{-2}L$ | M1 | Use of correct equation for, or units of, energy to find dimensions
$[E] = ML^2T^{-2}$ and $[\omega] = T^{-1}$ soi | A1 |
$ML^2T^{-2} = [I]T^{-1}$ $\Rightarrow$ $[I] = ML^2$ | A1 |
## (ii)
$I = mr^2 \cdots$ | B1 | Must be "$I =$"
$\cdots \times k$ | B1 |
where $k$ is dimensionless | B1 |
## (iii)
$[F] = MLT^{-2}, [a] = T^{-2}$ and their $[I]$ all used | M1 |
$[F] = MLT^{-2} \neq [I]a = ML^2T^{-2}$ so the student's suggestion is incorrect | A1 |
## (iv)
$[x] = L$ | B1 |
## (v)
E.g. The relationship could be $Fx = p I a$ for some dimensionless quantity $p$ | B1 | Or: there might be some other term(s) consisting of some dimensionally correct combination of quantities that need to be included
---One end of a non-uniform rod is freely hinged to a fixed point so that the rod can rotate about the point. When the rod rotates with angular velocity $\omega$ it can be shown that the kinetic energy $E$ of the rod is given by $E = \frac{1}{2}I\omega^2$, where $I$ is a quantity called the moment of inertia of the rod.
\begin{enumerate}[label=(\roman*)]
\item Deduce the dimensions of $I$. [3]
\item Given that the rod has mass $m$ and length $r$, suggest an expression for $I$, explaining any additional symbols that you use. [3]
\end{enumerate}
A student notices that the formula $E = \frac{1}{2}I\omega^2$ looks similar to the formula $E = \frac{1}{2}mv^2$ for the kinetic energy of a particle, with angular velocity for the rod corresponding to velocity for the particle, and moment of inertia corresponding to mass. Assuming a similar correspondence between angular acceleration (i.e. $\frac{d\omega}{dt}$) and acceleration, the student thinks that an equation for angular motion of the rod corresponding to Newton's second law for the particle should be $F = I\alpha$, where $F$ is the force applied to the rod and $\alpha$ is the resulting angular acceleration.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Use dimensional analysis to show that the student's suggestion is incorrect. [2]
\item State the dimensions of a quantity $x$ for which the equation $Fx = I\alpha$ would be dimensionally consistent. [1]
\item Explain why the fact that the equation in part (iv) is dimensionally consistent does not necessarily mean that it is correct. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Mechanics 2018 Q5 [10]}}