2 \includegraphics[max width=\textwidth, alt={}, center]{71a3b842-9d31-4c25-b894-ca6d1f47d84b-2_293_875_525_635} Two light elastic strings, each of natural length \(a\) and modulus of elasticity \(2 m g\), are attached to a particle \(P\) of mass \(m\). The strings join the particle to the points \(A\) and \(B\) which are fixed and at a distance \(4 a\) apart on a smooth horizontal surface. The particle is at rest at the mid-point \(O\) of \(A B\). The particle is now displaced a small distance in a direction perpendicular to \(A B\), on the surface, and released from rest. At time \(t\), the displacement of \(P\) from \(O\) is \(x\) (see diagram). Show that $$\ddot { x } = - \frac { 4 g x } { a } \left( 1 - \frac { 1 } { 2 } \left( 1 + \frac { x ^ { 2 } } { 4 a ^ { 2 } } \right) ^ { - \frac { 1 } { 2 } } \right) .$$ Given that \(\frac { x } { a }\) is so small that \(\left( \frac { x } { a } \right) ^ { 2 }\) and higher powers may be neglected, show that the motion of \(P\) is approximately simple harmonic and state the period of the motion.

## Question 2:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $AP = \sqrt{4a^2 + x^2}$ | B1 | Find $AP$ (or $BP$) in terms of $x$ and $a$ |
| $T = 2mg(AP - a)/a$ | B1 | Hooke's Law for one string |
| $md^2x/dt^2 = -2Tx/AP$ | M1 | Equation of motion at general point |
| $d^2x/dt^2 = -(4gx/a)(1 - a/AP)$ | M1 | Combine to find $d^2x/dt^2$ |
| $= -(4gx/a)\{1 - \frac{1}{2}(1 + x^2/4a^2)^{-\frac{1}{2}}\}$ **A.G.** | A1 | **M.R.** Vertical motion loses only this A1; Motion along $AB$ loses all marks |
| Neglect $(x/a)^2$: $d^2x/dt^2 = -(4gx/a)\cdot\frac{1}{2} = -2gx/a$ | B1 | |
| $2\pi\sqrt{a/(2g)}$ or $\pi\sqrt{2a/g}$ | B1 | Period using $2\pi/\omega$ (AEF) |

**Total: 7 marks**

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\includegraphics[max width=\textwidth, alt={}, center]{71a3b842-9d31-4c25-b894-ca6d1f47d84b-2_293_875_525_635}

Two light elastic strings, each of natural length $a$ and modulus of elasticity $2 m g$, are attached to a particle $P$ of mass $m$. The strings join the particle to the points $A$ and $B$ which are fixed and at a distance $4 a$ apart on a smooth horizontal surface. The particle is at rest at the mid-point $O$ of $A B$. The particle is now displaced a small distance in a direction perpendicular to $A B$, on the surface, and released from rest. At time $t$, the displacement of $P$ from $O$ is $x$ (see diagram). Show that

$$\ddot { x } = - \frac { 4 g x } { a } \left( 1 - \frac { 1 } { 2 } \left( 1 + \frac { x ^ { 2 } } { 4 a ^ { 2 } } \right) ^ { - \frac { 1 } { 2 } } \right) .$$

Given that $\frac { x } { a }$ is so small that $\left( \frac { x } { a } \right) ^ { 2 }$ and higher powers may be neglected, show that the motion of $P$ is approximately simple harmonic and state the period of the motion.

\hfill \mbox{\textit{CAIE FP2 2012 Q2 [7]}}