| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Vertical SHM with two strings |
| Difficulty | Challenging +1.2 This is a standard M4/FM mechanics question on damped SHM with two springs. Part (a) is routine equilibrium with Hooke's law, part (b) requires setting up forces and applying F=ma (standard technique), and part (c) tests knowledge of damping conditions (ω² > (k/2)²). While it involves multiple springs and requires careful bookkeeping, the methods are all standard textbook techniques for Further Maths mechanics with no novel insight required. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x4.10g Damped oscillations: model and interpret6.02g Hooke's law: T = k*x or T = lambda*x/l6.02i Conservation of energy: mechanical energy principle |
| Answer | Marks | Guidance |
|---|---|---|
| \(T_1 = mg + T_2\) | M1 A1 | No resultant force and use of Hooke's law Correct equation in one unknown \(\frac{3mg(AP - a)}{a} = mg + \frac{mg(3a - AP)}{a}\), \(3AP - 3a = a + 3a - AP\) |
| \(\frac{3mge}{a} = mg + \frac{mg(2a - e)}{a}\) | ||
| \(e = \frac{3a}{4} \Rightarrow AP = \frac{7a}{4}\) | A1 | Derive given result correctly. Condone verification for 3/3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(mg + T_2 - T_1 - mkv = m\ddot{s}\) | M1 A1 | Equation of motion – requires all terms but condone sign errors. o.e. Correct equation in \(T_1\) & \(T_2\). |
| \(mg + \frac{mg(\frac{3}{2}a - x)}{a} - \frac{3mg(\frac{3}{2}a + x)}{a} - mkv = m\ddot{s}\) | DM1 A1 | Use Hooke's law with extensions of the form \(ka ± x\) o.e. Correct unsimplified Given answer derived correctly |
| \(\ddot{s} + k\dot{s} + \frac{4g}{a}x = 0\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| For a damped oscillation, \(k^2 < \frac{16g}{a}\) | M1 A1 | AE will have complex roots Correctly substituted inequality |
| i.e. \(k < 4\sqrt{\frac{g}{a}}\) | A1 | Only (Q gives k>0) \(-4\sqrt{\frac{g}{a}} < k < 4\sqrt{\frac{g}{a}}\) is AO. |
**Part (a)**
| $T_1 = mg + T_2$ | M1 A1 | No resultant force and use of Hooke's law Correct equation in one unknown $\frac{3mg(AP - a)}{a} = mg + \frac{mg(3a - AP)}{a}$, $3AP - 3a = a + 3a - AP$ |
| $\frac{3mge}{a} = mg + \frac{mg(2a - e)}{a}$ | | |
| $e = \frac{3a}{4} \Rightarrow AP = \frac{7a}{4}$ | A1 | Derive given result correctly. Condone verification for 3/3 |
**(3) marks**(11)
**Part (b)**
| $mg + T_2 - T_1 - mkv = m\ddot{s}$ | M1 A1 | Equation of motion – requires all terms but condone sign errors. o.e. Correct equation in $T_1$ & $T_2$. |
| $mg + \frac{mg(\frac{3}{2}a - x)}{a} - \frac{3mg(\frac{3}{2}a + x)}{a} - mkv = m\ddot{s}$ | DM1 A1 | Use Hooke's law with extensions of the form $ka ± x$ o.e. Correct unsimplified Given answer derived correctly |
| $\ddot{s} + k\dot{s} + \frac{4g}{a}x = 0$ | A1 | |
**(5) marks**(11)
**Part (c)**
| For a damped oscillation, $k^2 < \frac{16g}{a}$ | M1 A1 | AE will have complex roots Correctly substituted inequality |
| i.e. $k < 4\sqrt{\frac{g}{a}}$ | A1 | Only (Q gives k>0) $-4\sqrt{\frac{g}{a}} < k < 4\sqrt{\frac{g}{a}}$ is AO. |
**(3) 11 marks total for Question 6**
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**TOTAL MARKS: 65**6. Two points $A$ and $B$ are in a vertical line, with $A$ above $B$ and $A B = 4 a$. One end of a light elastic spring, of natural length $a$ and modulus of elasticity $3 m g$, is attached to $A$. The other end of the spring is attached to a particle $P$ of mass $m$. Another light elastic spring, of natural length $a$ and modulus of elasticity $m g$, has one end attached to $B$ and the other end attached to $P$. The particle $P$ hangs at rest in equilibrium.
\begin{enumerate}[label=(\alph*)]
\item Show that $A P = \frac { 7 a } { 4 }$
The particle $P$ is now pulled down vertically from its equilibrium position towards $B$ and at time $t = 0$ it is released from rest. At time $t$, the particle $P$ is moving with speed $v$ and has displacement $x$ from its equilibrium position. The particle $P$ is subject to air resistance of magnitude $m k v$, where $k$ is a positive constant.
\item Show that
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + k \frac { \mathrm {~d} x } { \mathrm {~d} t } + \frac { 4 g } { a } x = 0$$
\item Find the range of values of $k$ which would result in the motion of $P$ being a damped oscillation.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 2012 Q6 [11]}}