5 A particle of mass \(m\) moves in a straight line \(A B\) of length \(2 a\). When the particle is at a general point \(P\) there are two forces acting, one in the direction \(\overrightarrow { P A }\) with magnitude \(m g \left( \frac { P A } { a } \right) ^ { - \frac { 1 } { 4 } }\) and the other in the direction \(\overrightarrow { P B }\) with magnitude \(m g \left( \frac { P B } { a } \right) ^ { \frac { 1 } { 2 } }\). At time \(t = 0\) the particle is released from rest at the point \(C\), where \(A C = 1.04 a\). At time \(t\) the distance \(A P\) is \(a + x\). Show that the particle moves in approximate simple harmonic motion. Using the approximate simple harmonic motion, find the speed of \(P\) when it first reaches the mid-point of \(A B\) and the time taken for \(P\) to first reach half of this speed.

## Question 5:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $m\,d^2x/dt^2 = mg((a-x)/a)^{1/2} - mg((a+x)/a)^{-1/4}$ | M1 | Find equation of motion at general point |
| $\approx mg((1 - x/2a) - (1 - x/4a))$ | M1 A1 | Expand terms and approximate |
| $d^2x/dt^2 = -gx/4a$ | A1 | Simplify to give SHM eqn; **Part total: 4** |
| $v_{max}^2 = (g/4a)(0.04a)^2$ | M1 A1 | Use SHM eqn to find speed when $x = 0$ |
| $v_{max} = 0.02\sqrt{(ag)}$ or $0.0632\sqrt{a}$ | A1 | Simplify (A.E.F.) |
| $\frac{1}{2}a\omega = a\omega\sin\omega t$ (A.E.F.) | M1 A1 | Use SHM eqn to find time when $v = \frac{1}{2}v_{max}$ |
| $t = \sqrt{(4a/g)}\sin^{-1}\frac{1}{2}$ | M1 | Substitute $\omega = \sqrt{(g/4a)}$ and simplify |
| $= (\pi/3)\sqrt{(a/g)}$ (A.E.F.) | A1 | **Total: 11** |

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5 A particle of mass $m$ moves in a straight line $A B$ of length $2 a$. When the particle is at a general point $P$ there are two forces acting, one in the direction $\overrightarrow { P A }$ with magnitude $m g \left( \frac { P A } { a } \right) ^ { - \frac { 1 } { 4 } }$ and the other in the direction $\overrightarrow { P B }$ with magnitude $m g \left( \frac { P B } { a } \right) ^ { \frac { 1 } { 2 } }$. At time $t = 0$ the particle is released from rest at the point $C$, where $A C = 1.04 a$. At time $t$ the distance $A P$ is $a + x$. Show that the particle moves in approximate simple harmonic motion.

Using the approximate simple harmonic motion, find the speed of $P$ when it first reaches the mid-point of $A B$ and the time taken for $P$ to first reach half of this speed.

\hfill \mbox{\textit{CAIE FP2 2008 Q5 [11]}}