2 A particle of mass \(m\) is attached to the mid-point of a light elastic string. The string is stretched between two points \(A\) and \(B\) on a smooth horizontal surface, where \(A B = 2 a\). The string has modulus of elasticity \(\lambda\) and natural length \(2 l\), where \(l < a\). The particle is in motion on the surface along a line passing through the mid-point of \(A B\) and perpendicular to \(A B\). When the displacement of the particle from \(A B\) is \(x\), the tension in the string is \(T\). Given that \(x\) is small enough for \(x ^ { 2 }\) to be neglected, show that $$T = \frac { \lambda } { l } ( a - l )$$ The particle is slightly disturbed from its equilibrium position. Show that it will perform approximate simple harmonic motion and find the period of the motion.

## Question 2:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $T = \lambda\{2\sqrt{(a^2 + x^2)} - 2l\}/2l$ | M1 A1 | Find $T$ by Hooke's Law for whole [or half] string |
| $= (\lambda/l)(a - l)$ **A.G.** | A1 | Neglect $x^2$: [M0 for motion along string] |
| $md^2x/dt^2 = -2Tx/\sqrt{(a^2 + x^2)}$ | M1 | Find eqn of motion at general point |
| $-(2T/ma)x$ or $-\{2\lambda(a-l)/alm\}x$ | M1 A1 | Neglect $x^2$ to give SHM eqn for $d^2x/dt^2$ |
| $2\pi\sqrt{\{alm/2\lambda(a-l)\}}$ | B1 | State period using $2\pi/\omega$ (A.E.F.) |
| $d^2x/dt^2 = -2\lambda x/lm$ | (B1) | **S.R.** Motion along string earns max 1 mark |

**M.R.** Vertical motion loses one A/B mark. **Total: [7]**

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2 A particle of mass $m$ is attached to the mid-point of a light elastic string. The string is stretched between two points $A$ and $B$ on a smooth horizontal surface, where $A B = 2 a$. The string has modulus of elasticity $\lambda$ and natural length $2 l$, where $l < a$. The particle is in motion on the surface along a line passing through the mid-point of $A B$ and perpendicular to $A B$. When the displacement of the particle from $A B$ is $x$, the tension in the string is $T$. Given that $x$ is small enough for $x ^ { 2 }$ to be neglected, show that

$$T = \frac { \lambda } { l } ( a - l )$$

The particle is slightly disturbed from its equilibrium position. Show that it will perform approximate simple harmonic motion and find the period of the motion.

\hfill \mbox{\textit{CAIE FP2 2011 Q2 [7]}}