Challenging +1.8 This M5 question requires applying rotational dynamics with a non-uniform force distribution. Students must resolve forces, use the equation of rotational motion for a rod about an endpoint (I = 4ma²/3), and handle the centripetal acceleration of the center of mass. The trigonometric manipulation to reach the specific form 2mg|cos(θ/2)| from standard force components is non-trivial and requires insight beyond routine application of formulas. The 8-mark allocation reflects substantial working, but it's a standard M5 problem type rather than requiring novel problem-solving approaches.
A uniform rod \(PQ\), of mass \(m\) and length \(2a\), is made to rotate in a vertical plane with constant angular speed \(\sqrt{\frac{g}{a}}\) about a fixed smooth horizontal axis through the end \(P\) of the rod.
Show that, when the rod is inclined at an angle \(\theta\) to the downward vertical, the magnitude of the force exerted on the axis by the rod is \(2mg|\cos(\frac{1}{2}\theta)|\).
[8]
A uniform rod $PQ$, of mass $m$ and length $2a$, is made to rotate in a vertical plane with constant angular speed $\sqrt{\frac{g}{a}}$ about a fixed smooth horizontal axis through the end $P$ of the rod.
Show that, when the rod is inclined at an angle $\theta$ to the downward vertical, the magnitude of the force exerted on the axis by the rod is $2mg|\cos(\frac{1}{2}\theta)|$.
[8]
\hfill \mbox{\textit{Edexcel M5 Q5 [8]}}