| Exam Board | OCR MEI |
|---|---|
| Module | Further Mechanics Minor (Further Mechanics Minor) |
| Year | 2023 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dimensional Analysis |
| Type | Verify dimensional consistency |
| Difficulty | Standard +0.3 This is a straightforward dimensional analysis question requiring recall of standard dimensions and systematic application of the technique. Parts (a)-(c) are routine bookwork and simple verification, while part (d) involves solving three simultaneous equations from dimensional analysis—a standard textbook exercise with no novel insight required. Slightly easier than average due to the guided structure. |
| Spec | 6.01a Dimensions: M, L, T notation6.01b Units vs dimensions: relationship6.01c Dimensional analysis: error checking6.01d Unknown indices: using dimensions |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (a) | F o r c e = M L T − 2 |
| V e lo c i t y = L T − 1 | B1 | 1.1 |
| D e n s i t y = M L − 3 | B1 | 1.2 |
| Answer | Marks | Guidance |
|---|---|---|
| (b) | R H S = ( L T − 1 ) 2 ( M L − 3 M 2 ) 13 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| = ( L T − 1 ) 2 ( M 3 L − 3 ) 13 = L 2 T − 2 M L − 1 = M L T − 2 = L H S | A1 | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| (c) | Model predicts that when air density is doubled, drag force | |
| should increase by factor of 3 2 . oe | B1 | 3.5b |
| Answer | Marks |
|---|---|
| (d) | M L T 2 ( M L 3 ) ( L T 1 ) ( L 2 ) − = − − |
| Answer | Marks | Guidance |
|---|---|---|
| T : 2 − = − | M1 | 3.3 |
| Answer | Marks | Guidance |
|---|---|---|
| 1 = and 𝛽 = 2 | A1ft | 3.4 |
| −3+2+2𝛾 = 1. 2𝛾 = 2 𝛾 = 1 | A1ft | 1.1 |
Question 1:
1 | (a) | F o r c e = M L T − 2 | B1 | 1.1
V e lo c i t y = L T − 1 | B1 | 1.1
D e n s i t y = M L − 3 | B1 | 1.2
[3]
(b) | R H S = ( L T − 1 ) 2 ( M L − 3 M 2 ) 13 | M1 | 1.1 | Use expressions from (a) to express
u2(m2) 1 3 dimensionally.
= ( L T − 1 ) 2 ( M 3 L − 3 ) 13 = L 2 T − 2 M L − 1 = M L T − 2 = L H S | A1 | 1.1 | Convincingly reached.
[2]
(c) | Model predicts that when air density is doubled, drag force
should increase by factor of 3 2 . oe | B1 | 3.5b | Or by contradiction
[1]
(d) | M L T 2 ( M L 3 ) ( L T 1 ) ( L 2 ) − = − −
M : 1 =
L : 3 2 1 − + + =
T : 2 − = − | M1 | 3.3 | Setting up equations in 𝛼,𝛽 𝑎𝑛𝑑 𝛾
using given equation and their
dimensions for (a)
1 = and 𝛽 = 2 | A1ft | 3.4 | Follow through from (a)
−3+2+2𝛾 = 1. 2𝛾 = 2 𝛾 = 1 | A1ft | 1.1 | Follow through from (a)
[3]1
\begin{enumerate}[label=(\alph*)]
\item State the dimensions of the following quantities.
\begin{itemize}
\item Force
\item Velocity
\item Density
\end{itemize}
A student investigating the drag force $F$ experienced by an object moving through air conjectures the formula\\
$\mathrm { F } = \mathrm { ku } ^ { 2 } \left( \rho \mathrm {~m} ^ { 2 } \right) ^ { \frac { 1 } { 3 } }$,\\
where
\begin{itemize}
\item $k$ is a dimensionless constant
\item $u$ is the air velocity relative to the moving object
\item $\rho$ is the air density
\item $m$ is the mass of the object.
\item Show that the student's formula is dimensionally consistent.
\end{itemize}
The student carries out experiments in an airflow tunnel. When the air density is doubled, the drag force is found to double as well, with all other conditions remaining the same.
\item Show that the student's formula is inconsistent with the experimental observation.
The student's teacher suggests revising the formula as\\
$\mathrm { F } = \mathrm { k } \rho ^ { \alpha } \mathrm { u } ^ { \beta } \mathrm { A } ^ { \gamma }$\\
where $m$ has been replaced by $A$, the cross-sectional area of the object. The constant $k$ is still dimensionless.
\item Use dimensional analysis to determine the values of $\alpha , \beta$ and $\gamma$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Mechanics Minor 2023 Q1 [9]}}