| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2003 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Find period from given information |
| Difficulty | Standard +0.3 This is a standard SHM question requiring application of well-known formulas (v² = ω²(a² - x²), vₘₐₓ = ωa, aₘₐₓ = ω²a) with straightforward algebraic manipulation. Part (d) requires finding time intervals using inverse trig, which is slightly beyond routine but still a common SHM exercise. Slightly above average difficulty due to the multi-part nature and part (d), but all techniques are standard M3 material. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x4.10g Damped oscillations: model and interpret |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| \(\frac{2\pi}{\omega} = \pi \Rightarrow \omega = 2\) | B1 | |
| \(2.4^2 = 4(a^2 - 0.5^2)\) | M1 A1ft | |
| \(a = 1.3\) m | A1 | (4) |
| Answer | Marks | Guidance |
|---|---|---|
| \(v_{\max} = a\omega = 2.6\) ms\(^{-1}\) | B1 | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{arct}_{\max} = a\omega^2 = 5.2\) ms\(^{-2}\) | B1ft | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.5 = 1.3\sin 2t\) | M1 | |
| \(t = \frac{1}{2}\sin^{-1}\left(\frac{0.5}{1.3}\right)\) | M1 A1 | |
| Total time \(= 4t = 0.79\) (2 dp) | M1 A1 | (5) |
# Question 4:
## Part (a):
| Working | Marks | Notes |
|---------|-------|-------|
| $\frac{2\pi}{\omega} = \pi \Rightarrow \omega = 2$ | B1 | |
| $2.4^2 = 4(a^2 - 0.5^2)$ | M1 A1ft | |
| $a = 1.3$ m | A1 | (4) |
## Part (b):
| $v_{\max} = a\omega = 2.6$ ms$^{-1}$ | B1 | (1) |
## Part (c):
| $\text{arct}_{\max} = a\omega^2 = 5.2$ ms$^{-2}$ | B1ft | (1) |
## Part (d):
| $0.5 = 1.3\sin 2t$ | M1 | |
| $t = \frac{1}{2}\sin^{-1}\left(\frac{0.5}{1.3}\right)$ | M1 A1 | |
| Total time $= 4t = 0.79$ (2 dp) | M1 A1 | (5) |
---4. A piston $P$ in a machine moves in a straight line with simple harmonic motion about a point $O$, which is the centre of the oscillations. The period of the oscillations is $\pi \mathrm { s }$. When $P$ is 0.5 m from $O$, its speed is $2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find
\begin{enumerate}[label=(\alph*)]
\item the amplitude of the motion,
\item the maximum speed of $P$ during the motion,
\item the maximum magnitude of the acceleration of $P$ during the motion,
\item the total time, in s to 2 decimal places, in each complete oscillation for which the speed of $P$ is greater than $2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2003 Q4 [11]}}