| Exam Board | OCR |
|---|---|
| Module | Further Mechanics AS (Further Mechanics AS) |
| Year | 2019 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dimensional Analysis |
| Type | Find exponents with all unknowns |
| Difficulty | Moderate -0.5 This is a standard dimensional analysis problem requiring systematic equation setup and solving simultaneous equations from matching dimensions. Parts (a)-(c) are routine textbook exercises, while part (d) applies the derived relationship straightforwardly. The method is well-established and requires no novel insight, making it slightly easier than average. |
| 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 |
|---|---|---|
| Answer | Mark | Guidance |
| \([\rho] = \text{ML}^{-3}\) | B1 | If M, L and T not used B0, but do not penalise further instances of non-standard notation if used consistently |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([p] = \text{MLT}^{-2}\text{L}^{-2} = \text{ML}^{-1}\text{T}^{-2}\) | B1 | If M, L and T not used B0 provided not withheld in (a) |
| \(\text{LT}^{-1} = \text{M}^\alpha \text{L}^{-\alpha}\text{T}^{-2\alpha}\text{M}^\beta\text{L}^{-3\beta}\text{L}^\gamma\) | B1ft | Or \((MLT^{-1}T^{-2})^\alpha(ML^{-3})^\beta(L)^\gamma\); Do not allow marks for using addition instead of multiplication |
| M: \(\alpha + \beta = 0\) | M1 | |
| T: \(-2\alpha = -1\) | M1 | |
| \(\alpha = \frac{1}{2},\ \beta = -\frac{1}{2}\) | A1 | Allow if M and T equations correct |
| L: \(1 = -\alpha - 3\beta + \gamma\) | M1 | |
| \(\gamma = 0\) www | A1 | SC2 for three correct values unsupported or SC1 for correct \(\alpha\) and \(\beta\) or correct \(\gamma\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Sounds of any wavelength have the same speed through the gas | E1FT | Follow from their \(\gamma\): if \(\gamma > 0\) then speed increases as wavelength increases (or \(\gamma = \frac{1}{2} \rightarrow\) speed proportional to \(\sqrt{\lambda}\)); if \(\gamma < 0\) then speed decreases as wavelength increases |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(u \propto \frac{1}{\sqrt{\rho}}\) or \(u = k\sqrt{\frac{p}{\rho}}\) oe | M1 | \(2 = \left(\frac{\rho_B}{\rho_A}\right)^{-\frac{1}{2}}\); Using their \(\beta\); Allow missing \(k\) or use of \(=\) instead of proportion symbol |
| \(\frac{1}{4}\) | A1 | Award if no working seen provided \(\beta = -\frac{1}{2}\) |
## Question 4:
### Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $[\rho] = \text{ML}^{-3}$ | B1 | If M, L and T not used B0, but do not penalise further instances of non-standard notation if used consistently |
### Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $[p] = \text{MLT}^{-2}\text{L}^{-2} = \text{ML}^{-1}\text{T}^{-2}$ | B1 | If M, L and T not used B0 provided not withheld in (a) |
| $\text{LT}^{-1} = \text{M}^\alpha \text{L}^{-\alpha}\text{T}^{-2\alpha}\text{M}^\beta\text{L}^{-3\beta}\text{L}^\gamma$ | B1ft | Or $(MLT^{-1}T^{-2})^\alpha(ML^{-3})^\beta(L)^\gamma$; Do not allow marks for using addition instead of multiplication |
| M: $\alpha + \beta = 0$ | M1 | |
| T: $-2\alpha = -1$ | M1 | |
| $\alpha = \frac{1}{2},\ \beta = -\frac{1}{2}$ | A1 | Allow if M and T equations correct |
| L: $1 = -\alpha - 3\beta + \gamma$ | M1 | |
| $\gamma = 0$ www | A1 | SC2 for three correct values unsupported or SC1 for correct $\alpha$ and $\beta$ or correct $\gamma$ |
### Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| Sounds of any wavelength have the same speed through the gas | E1FT | Follow from their $\gamma$: if $\gamma > 0$ then speed increases as wavelength increases (or $\gamma = \frac{1}{2} \rightarrow$ speed proportional to $\sqrt{\lambda}$); if $\gamma < 0$ then speed decreases as wavelength increases |
### Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| $u \propto \frac{1}{\sqrt{\rho}}$ or $u = k\sqrt{\frac{p}{\rho}}$ oe | M1 | $2 = \left(\frac{\rho_B}{\rho_A}\right)^{-\frac{1}{2}}$; Using their $\beta$; Allow missing $k$ or use of $=$ instead of proportion symbol |
| $\frac{1}{4}$ | A1 | Award if no working seen provided $\beta = -\frac{1}{2}$ |4 A student is studying the speed of sound, $u$, in a gas under different conditions.
He assumes that $u$ depends on the pressure, $p$, of the gas, the density, $\rho$, of the gas and the wavelength, $\lambda$, of the sound in the relationship $u = k p ^ { \alpha } \rho ^ { \beta } \lambda ^ { \gamma }$, where $k$ is a dimensionless constant. (The wavelength of a sound is the distance between successive peaks in the sound wave.)
\begin{enumerate}[label=(\alph*)]
\item Use the fact that density is mass per unit volume to find $[ \rho ]$.
\item Given that the units of $p$ are $\mathrm { Nm } ^ { - 2 }$, determine the values of $\alpha , \beta$ and $\gamma$.
\item Comment on what the value of $\gamma$ means about how fast sounds of different wavelengths travel through the gas.
The student carries out two experiments, $A$ and $B$, to measure $u$. Only the density of the gas varies between the experiments, all other conditions being unchanged. He finds that the value of $u$ in experiment $B$ is double the value in experiment $A$.
\item By what factor has the density of the gas in experiment $A$ been multiplied to give the density of the gas in experiment $B$ ?\\
\includegraphics[max width=\textwidth, alt={}, center]{74bada9e-60cf-4ed4-abd0-4be155b7cf81-4_659_401_269_251}
As shown in the diagram, $A B$ is a long thin rod which is fixed vertically with $A$ above $B$. One end of a light inextensible string of length 1 m is attached to $A$ and the other end is attached to a particle $P$ of mass $m _ { 1 } \mathrm {~kg}$. One end of another light inextensible string of length 1 m is also attached to $P$. Its other end is attached to a small smooth ring $R$, of mass $m _ { 2 } \mathrm {~kg}$, which is free to move on $A B$.
Initially, $P$ moves in a horizontal circle of radius 0.6 m with constant angular velocity $\omega _ { \text {rads } } { } ^ { - 1 }$. The magnitude of the tension in string $A P$ is denoted by $T _ { 1 } \mathrm {~N}$ while that in string $P R$ is denoted by $T _ { 2 } \mathrm {~N}$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Mechanics AS 2019 Q4 [11]}}