4 A particle \(P\) of mass \(m\) moves along part of a horizontal straight line \(A B\). The mid-point of \(A B\) is \(O\), and \(A B = 4 a\). At time \(t , A P = 2 a + x\). The particle \(P\) is acted on by two horizontal forces. One force has magnitude \(m g \left( \frac { 2 a + x } { 2 a } \right) ^ { - \frac { 1 } { 2 } }\) and acts towards \(A\); the other force has magnitude \(m g \left( \frac { 2 a - x } { 2 a } \right)\) and acts towards \(B\). Show that, provided \(\frac { x } { a }\) remains small, \(P\) moves in approximate simple harmonic motion with centre \(O\), and state the period of this motion. At time \(t = 0 , P\) is released from rest at the point where \(x = \frac { 1 } { 20 } a\). Show that the speed of \(P\) when \(x = \frac { 1 } { 40 } a\) is \(\frac { 1 } { 80 } \sqrt { } ( 3 a g )\), and find the value of \(t\) when \(P\) reaches this point for the first time.

## Question 4:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $m\,d^2x/dt^2 = mg(2a-x)/2a - mg\{(2a+x)/2a\}^{-\frac{1}{2}}$ | M1 A1 | Find eqn. of motion at general point (wrong sign loses this A1 only) |
| $d^2x/dt^2 = g(1-x/2a) - g(1-x/4a) = -gx/4a$ | M1 A1 | Neglect $(x/a)^2$ and higher powers |
| $T = 2\pi\sqrt{4a/g}$ or $4\pi\sqrt{a/g}$ | B1 | Find period from $T = 2\pi/\omega$ |
| $v^2 = (g/4a)\{(a/20)^2 - (a/40)^2\}$ | M1 | Find $v$ from $v^2 = \omega^2(A^2 - x^2)$ |
| $v = (1/80)\sqrt{3ag}$ **A.G.** | A1 | |
| $t = \sqrt{4a/g}\cos^{-1}\frac{1}{2} = \frac{2}{3}\pi\sqrt{a/g}$ | M1 A1 | Find $t$ from $x = a\cos\omega t$ (A.E.F.) |

**Total: 9 marks**

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4 A particle $P$ of mass $m$ moves along part of a horizontal straight line $A B$. The mid-point of $A B$ is $O$, and $A B = 4 a$. At time $t , A P = 2 a + x$. The particle $P$ is acted on by two horizontal forces. One force has magnitude $m g \left( \frac { 2 a + x } { 2 a } \right) ^ { - \frac { 1 } { 2 } }$ and acts towards $A$; the other force has magnitude $m g \left( \frac { 2 a - x } { 2 a } \right)$ and acts towards $B$. Show that, provided $\frac { x } { a }$ remains small, $P$ moves in approximate simple harmonic motion with centre $O$, and state the period of this motion.

At time $t = 0 , P$ is released from rest at the point where $x = \frac { 1 } { 20 } a$. Show that the speed of $P$ when $x = \frac { 1 } { 40 } a$ is $\frac { 1 } { 80 } \sqrt { } ( 3 a g )$, and find the value of $t$ when $P$ reaches this point for the first time.

\hfill \mbox{\textit{CAIE FP2 2013 Q4 [9]}}