Edexcel M3 — Question 3 8 marks

Exam BoardEdexcel
ModuleM3 (Mechanics 3)
Marks8
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TopicCircular Motion 1
TypeVertical circle – string/rod (tension and energy)
DifficultyStandard +0.8 This is a solid M3 circular motion problem requiring energy conservation to find speed, then resolving forces at the lowest point to find both acceleration components and reaction force. It involves multiple connected steps (energy equation, centripetal acceleration, force resolution) and careful consideration of the constraint that the ring moves on a circular path. More demanding than routine mechanics questions but follows standard M3 methodology without requiring novel insight.
Spec6.02i Conservation of energy: mechanical energy principle6.05f Vertical circle: motion including free fall
A smooth circular hoop of radius \(1\) m, with centre \(O\), is fixed in a vertical plane. A small ring \(Q\), of mass \(0.1\) kg, is threaded onto the hoop and held so that the angle \(QOH = 30°\), where \(H\) is the highest point of the hoop. \(Q\) is released from rest at this position. Find, in terms of \(g\), \begin{enumerate}[label=(\alph*)] \item the horizontal and vertical components of the acceleration of \(Q\) when it reaches the lowest point of the hoop; [5 marks] \item the magnitude of the reaction between \(Q\) and the hoop at this lowest point. [3 marks]
AnswerMarks Guidance
(a) Energy conserved, so \(mg(1 + \cos 30°) = \frac{1}{2}mv^2\)M1 A1
Hence \(v^2 = 2g(1 + \cos 30°) = g(2 + \sqrt{3})\)A1
\(a_r = \frac{v^2}{r} = g(2 + \sqrt{3})\) towards \(O\); \(a_x = 0\) (no horizontal force)A1 A1
(b) At bottom, \(R = \frac{mv^2}{r} + mg = mg(2 + \sqrt{3}) + mg = 0.1g(3 + \sqrt{3})\)M1 A1 A1 8 marks 
**(a)** Energy conserved, so $mg(1 + \cos 30°) = \frac{1}{2}mv^2$ | M1 A1 |
Hence $v^2 = 2g(1 + \cos 30°) = g(2 + \sqrt{3})$ | A1 |
$a_r = \frac{v^2}{r} = g(2 + \sqrt{3})$ towards $O$; $a_x = 0$ (no horizontal force) | A1 A1 |
**(b)** At bottom, $R = \frac{mv^2}{r} + mg = mg(2 + \sqrt{3}) + mg = 0.1g(3 + \sqrt{3})$ | M1 A1 A1 | 8 marks
A smooth circular hoop of radius $1$ m, with centre $O$, is fixed in a vertical plane. A small ring $Q$, of mass $0.1$ kg, is threaded onto the hoop and held so that the angle $QOH = 30°$, where $H$ is the highest point of the hoop. $Q$ is released from rest at this position. Find, in terms of $g$,

\begin{enumerate}[label=(\alph*)]
\item the horizontal and vertical components of the acceleration of $Q$ when it reaches the lowest point of the hoop; [5 marks]
\item the magnitude of the reaction between $Q$ and the hoop at this lowest point. [3 marks]
</end{enumerate}

\hfill \mbox{\textit{Edexcel M3  Q3 [8]}}
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