| Exam Board | OCR MEI |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2007 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dimensional Analysis |
| Type | Verify dimensional consistency |
| Difficulty | Easy -1.2 Part (a) is pure recall of standard dimensional formulas and routine verification that terms in Bernoulli's equation have matching dimensions—a textbook exercise requiring no problem-solving. Part (b) involves standard SHM with given amplitude and period, requiring only direct application of formulas. The entire question tests basic recall and straightforward application of well-practiced techniques with no novel insight required. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x6.01a Dimensions: M, L, T notation6.01b Units vs dimensions: relationship |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Velocity: \(LT^{-1}\) | B1 | |
| Acceleration: \(LT^{-2}\) | B1 | |
| Force: \(MLT^{-2}\) | B1 | |
| Density: \(ML^{-3}\) | B1 | |
| Pressure: \(ML^{-1}T^{-2}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([P] = ML^{-1}T^{-2}\) | B1 | May be implied |
| \([\frac{1}{2}\rho v^2] = ML^{-3} \cdot L^2T^{-2} = ML^{-1}T^{-2}\) | M1 A1 | Correct working shown |
| \([\rho g h] = ML^{-3} \cdot LT^{-2} \cdot L = ML^{-1}T^{-2}\) | A1 | All three terms shown equal |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Cosine curve starting at maximum \(h = 2.2\) when \(t=0\) | B1 | Correct shape |
| Period 3.49, amplitude 0.3, oscillating between 1.6 and 2.2 | B1 | Correct features labelled |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Centre: \(\frac{1.6+2.2}{2} = 1.9\), amplitude \(= 0.3\) | B1 | |
| \(\omega = \frac{2\pi}{3.49}\) | M1 | |
| \(h = 1.9 + 0.3\cos\!\left(\frac{2\pi t}{3.49}\right)\) | A1 A1 | A1 for 1.9 + 0.3cos, A1 for correct \(\omega\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(h - 1.9 = 0.3\cos(\omega t)\), so displacement from centre \(x = h - 1.9 = -0.2\) | M1 | |
| \(a = -\omega^2 x = -\left(\frac{2\pi}{3.49}\right)^2(-0.2)\) | M1 | |
| \(a = 0.647\ \text{m s}^{-2}\) (upwards, towards centre) | A1 |
# Question 1:
## Part (a)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Velocity: $LT^{-1}$ | B1 | |
| Acceleration: $LT^{-2}$ | B1 | |
| Force: $MLT^{-2}$ | B1 | |
| Density: $ML^{-3}$ | B1 | |
| Pressure: $ML^{-1}T^{-2}$ | B1 | |
## Part (a)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[P] = ML^{-1}T^{-2}$ | B1 | May be implied |
| $[\frac{1}{2}\rho v^2] = ML^{-3} \cdot L^2T^{-2} = ML^{-1}T^{-2}$ | M1 A1 | Correct working shown |
| $[\rho g h] = ML^{-3} \cdot LT^{-2} \cdot L = ML^{-1}T^{-2}$ | A1 | All three terms shown equal |
## Part (b)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Cosine curve starting at maximum $h = 2.2$ when $t=0$ | B1 | Correct shape |
| Period 3.49, amplitude 0.3, oscillating between 1.6 and 2.2 | B1 | Correct features labelled |
## Part (b)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Centre: $\frac{1.6+2.2}{2} = 1.9$, amplitude $= 0.3$ | B1 | |
| $\omega = \frac{2\pi}{3.49}$ | M1 | |
| $h = 1.9 + 0.3\cos\!\left(\frac{2\pi t}{3.49}\right)$ | A1 A1 | A1 for 1.9 + 0.3cos, A1 for correct $\omega$ |
## Part (b)(iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $h - 1.9 = 0.3\cos(\omega t)$, so displacement from centre $x = h - 1.9 = -0.2$ | M1 | |
| $a = -\omega^2 x = -\left(\frac{2\pi}{3.49}\right)^2(-0.2)$ | M1 | |
| $a = 0.647\ \text{m s}^{-2}$ (upwards, towards centre) | A1 | |
---1
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the dimensions of the following quantities.
\begin{displayquote}
Velocity\\
Acceleration\\
Force\\
Density (which is mass per unit volume)\\
Pressure (which is force per unit area)
\end{displayquote}
For a fluid with constant density $\rho$, the velocity $v$, pressure $P$ and height $h$ at points on a streamline are related by Bernoulli's equation
$$P + \frac { 1 } { 2 } \rho v ^ { 2 } + \rho g h = \mathrm { constant } ,$$
where $g$ is the acceleration due to gravity.
\item Show that the left-hand side of Bernoulli's equation is dimensionally consistent.
\end{enumerate}\item In a wave tank, a float is performing simple harmonic motion with period 3.49 s in a vertical line. The height of the float above the bottom of the tank is $h \mathrm {~m}$ at a time $t \mathrm {~s}$. When $t = 0$, the height has its maximum value. The value of $h$ varies between 1.6 and 2.2.
\begin{enumerate}[label=(\roman*)]
\item Sketch a graph showing how $h$ varies with $t$.
\item Express $h$ in terms of $t$.
\item Find the magnitude and direction of the acceleration of the float when $h = 1.7$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI M3 2007 Q1 [18]}}