A body consists of a uniform plane circular disc, of radius \(r\) and mass \(2m\), with a particle of mass \(3m\) attached to the circumference of the disc at the point \(P\). The line \(PQ\) is a diameter of the disc. The body is free to rotate in a vertical plane about a fixed smooth horizontal axis, \(L\), which is perpendicular to the plane of the disc and passes through \(Q\). The body is held with \(QP\) making an angle \(\beta\) with the downward vertical through \(Q\), where \(\sin \beta = 0.25\), and released from rest. Find the magnitude of the component, perpendicular to \(PQ\), of the force acting on the body at \(Q\) at the instant when it is released. [You may assume that the moment of inertia of the body about \(L\) is \(15mr^2\).] [6]

$M(Q)$, $2mgr\sin\beta + 3mg \cdot 2r\sin\beta = 15mr^2\theta$ | M1 A1 |

OR $M(Q)$, $5mg\frac{3r}{2}\sin\beta = 15mr^2\theta$ | M1 A1 |

$\square 2mg\sin\beta + 3mg\sin\beta - X = 2mr\theta + 3m2r\theta$ | M1 A1 |

OR $\square 5mg\sin\beta - X = 5m\frac{3r}{2}\theta$ | M1 A1 |

solving for X, | M1 |

$X = \frac{11mg}{60}$ | A1 |

A body consists of a uniform plane circular disc, of radius $r$ and mass $2m$, with a particle of mass $3m$ attached to the circumference of the disc at the point $P$.
The line $PQ$ is a diameter of the disc. The body is free to rotate in a vertical plane about a fixed smooth horizontal axis, $L$, which is perpendicular to the plane of the disc and passes through $Q$. The body is held with $QP$ making an angle $\beta$ with the downward vertical through $Q$, where $\sin \beta = 0.25$, and released from rest. Find the magnitude of the component, perpendicular to $PQ$, of the force acting on the body at $Q$ at the instant when it is released.

[You may assume that the moment of inertia of the body about $L$ is $15mr^2$.]
[6]

\hfill \mbox{\textit{Edexcel M5 2012 Q4 [6]}}