| Exam Board | OCR |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2012 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Displacement and velocity at given time |
| Difficulty | Standard +0.8 This is a multi-part SHM question requiring understanding of period, amplitude, and phase relationships. Part (i) is standard, but parts (ii) and (iii) require careful analysis of the motion's geometry and solving transcendental equations to find multiple time instances with the same speed, which goes beyond routine application of SHM formulas. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([0.4\pi = 2\pi/n]\) | M1 | For using \(T = 2\pi/n\) |
| \(n = 5\) | A1 | |
| Distance \(OA\) is \(0.8\) m | M1, A1 | For using \(v_{\max} = n(OA)\) |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([x = 0.8\cos(5 \times 1)]\) | M1 | For using \(x = a\cos nt\) |
| \(x = 0.227\) | A1 | |
| \([\dot{x} = -0.8 \times 5\sin(5 \times 1)]\) | M1 | For using \(\dot{x} = -an\sin nt\); use of \(v^2 = n^2(a^2 - x^2)\) M1; direction needs to be shown for A1 |
| Velocity is \(3.84\text{ ms}^{-1}\) | A1 | |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(t\) and \(x\) for one point | B2 | Values of \(t\) are \(0.257, 0.372, 0.885\); \(0.4\pi - 1,\ 1 - 0.2\pi,\ 0.6\pi - 1\) |
| \(t\) and \(x\) for second point | B1 | Values of \(x\) are \(0.227, -0.227, -0.227\); ignore ref to point when \(t = 1\); can show on graph |
| \(t\) and \(x\) for third point | B1 | |
| Correctly stating precisely 3 points | B1 | sc all 3 \(x\) values B2; all 3 \(t\) values B2; one \(t\) value B1; one \(x\) value B1 |
| If B1 or B0 scored (out of first 4) on above scheme, allow, subject to max mark 2: Number of occasions is 3 | (M1), (A1) | For \(t = 1 \approx 0.8T \Rightarrow 3/4T < 1 < 4/4T\) or equiv |
| [5] |
## Question 6:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[0.4\pi = 2\pi/n]$ | M1 | For using $T = 2\pi/n$ |
| $n = 5$ | A1 | |
| Distance $OA$ is $0.8$ m | M1, A1 | For using $v_{\max} = n(OA)$ |
| **[4]** | | |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[x = 0.8\cos(5 \times 1)]$ | M1 | For using $x = a\cos nt$ |
| $x = 0.227$ | A1 | |
| $[\dot{x} = -0.8 \times 5\sin(5 \times 1)]$ | M1 | For using $\dot{x} = -an\sin nt$; use of $v^2 = n^2(a^2 - x^2)$ M1; direction needs to be shown for A1 |
| Velocity is $3.84\text{ ms}^{-1}$ | A1 | |
| **[4]** | | |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $t$ and $x$ for one point | B2 | Values of $t$ are $0.257, 0.372, 0.885$; $0.4\pi - 1,\ 1 - 0.2\pi,\ 0.6\pi - 1$ |
| $t$ and $x$ for second point | B1 | Values of $x$ are $0.227, -0.227, -0.227$; ignore ref to point when $t = 1$; can show on graph |
| $t$ and $x$ for third point | B1 | |
| Correctly stating precisely 3 points | B1 | sc all 3 $x$ values B2; all 3 $t$ values B2; one $t$ value B1; one $x$ value B1 |
| If B1 or B0 scored (out of first 4) on above scheme, allow, subject to max mark 2: Number of occasions is 3 | (M1), (A1) | For $t = 1 \approx 0.8T \Rightarrow 3/4T < 1 < 4/4T$ or equiv |
| **[5]** | | |
---6 A particle $P$ starts from rest at a point $A$ and moves in a straight line with simple harmonic motion. At time $t \mathrm {~s}$ after the motion starts, $P$ 's displacement from a point $O$ on the line is $x \mathrm {~m}$ towards $A$. The particle $P$ returns to $A$ for the first time when $t = 0.4 \pi$. The maximum speed of $P$ is $4 \mathrm {~ms} ^ { - 1 }$ and occurs when $P$ passes through $O$.\\
(i) Find the distance $O A$.\\
(ii) Find the value of $x$ and the velocity of $P$ when $t = 1$.\\
(iii) Find the number of occasions in the interval $0 < t < 1$ at which $P$ 's speed is the same as that when $t = 1$, and find the corresponding values of $x$ and $t$.
\hfill \mbox{\textit{OCR M3 2012 Q6 [13]}}