2 A small bead of mass \(m\) is threaded on a thin smooth wire which forms a circle of radius \(a\). The wire is fixed in a vertical plane. A light inextensible string is attached to the bead and passes through a small smooth ring fixed at the centre of the circle. The other end of the string is attached to a particle of mass \(4 m\) which hangs freely under gravity. The bead is projected from the lowest point of the wire with speed \(\sqrt { } ( k g a )\). Show that, when the angle between the two parts of the string is \(\theta\), the normal force exerted on the bead by the wire is \(m g ( 3 \cos \theta + k - 6 )\), towards the centre. Given that the bead reaches the highest point of the wire, find an inequality which must be satisfied by \(k\).

## Question 2:
| Answer/Working | Marks | Guidance |
|---|---|---|
| Conservation of energy: $\frac{1}{2}mv^2 = \frac{1}{2}mkga - mga(1-\cos\theta)$ | B1 | |
| $F=ma$ radially: $R + 4mg - mg\cos\theta = mv^2/a$ | M1 A1 | |
| Eliminate $v$ to find $R$: $R = mg(3\cos\theta + k - 6)$ | M1 A1 | **A.G.** |
| Find $k$ from $v \geq 0$ (or $>0$) when $\theta = \pi$: $k \geq 4$ (or $k > 4$) | M1 A1 | Total: [7] |

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2 A small bead of mass $m$ is threaded on a thin smooth wire which forms a circle of radius $a$. The wire is fixed in a vertical plane. A light inextensible string is attached to the bead and passes through a small smooth ring fixed at the centre of the circle. The other end of the string is attached to a particle of mass $4 m$ which hangs freely under gravity. The bead is projected from the lowest point of the wire with speed $\sqrt { } ( k g a )$. Show that, when the angle between the two parts of the string is $\theta$, the normal force exerted on the bead by the wire is $m g ( 3 \cos \theta + k - 6 )$, towards the centre.

Given that the bead reaches the highest point of the wire, find an inequality which must be satisfied by $k$.

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