OCR MEI Further Mechanics Minor 2024 June — Question 5 12 marks

Exam BoardOCR MEI
ModuleFurther Mechanics Minor (Further Mechanics Minor)
Year2024
SessionJune
Marks12
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TopicCentre of Mass 1
TypeEquilibrium with applied force
DifficultyStandard +0.8 This is a multi-part Further Mechanics question requiring: (1) centre of mass calculation for a trapezium using composite shapes or integration, (2) equilibrium analysis with geometric constraints on support positions, and (3) solving a system of three equations (vertical forces and two moments) for reaction forces. While the techniques are standard for FM, the geometric setup and multi-step reasoning elevate it above routine exercises.
Spec6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces
5 A uniform lamina OABC is in the shape of a trapezium where O is the origin of the coordinate system in which the points \(\mathrm { A } , \mathrm { B }\) and C have coordinates \(( 120,0 )\), \(( 60,90 )\) and \(( 30,90 )\) respectively (see diagram). The units of the axes are centimetres. \includegraphics[max width=\textwidth, alt={}, center]{0a790ad0-7eda-40f1-9894-f156766ae46f-5_561_720_404_242} The centre of mass of the lamina lies at ( \(\mathrm { x } , \mathrm { y }\) ).
  1. Show that \(\bar { x } = 54\) and determine the value of \(\bar { y }\). The lamina is placed horizontally so that it rests on three supports, whose points of contact are at \(\mathrm { B } , \mathrm { C }\) and D , where D lies on the edge OA and has coordinates \(( d , 0 )\).
  2. Determine the range of values of \(d\) for the lamina to rest in equilibrium. It is now given that \(d = 63\), and that the lamina has a weight of 100 N .
  3. Determine the forces exerted on the lamina by each of the supports at \(\mathrm { B } , \mathrm { C }\) and D .
5 A uniform lamina OABC is in the shape of a trapezium where O is the origin of the coordinate system in which the points $\mathrm { A } , \mathrm { B }$ and C have coordinates $( 120,0 )$, $( 60,90 )$ and $( 30,90 )$ respectively (see diagram). The units of the axes are centimetres.\\
\includegraphics[max width=\textwidth, alt={}, center]{0a790ad0-7eda-40f1-9894-f156766ae46f-5_561_720_404_242}

The centre of mass of the lamina lies at ( $\mathrm { x } , \mathrm { y }$ ).
\begin{enumerate}[label=(\alph*)]
\item Show that $\bar { x } = 54$ and determine the value of $\bar { y }$.

The lamina is placed horizontally so that it rests on three supports, whose points of contact are at $\mathrm { B } , \mathrm { C }$ and D , where D lies on the edge OA and has coordinates $( d , 0 )$.
\item Determine the range of values of $d$ for the lamina to rest in equilibrium.

It is now given that $d = 63$, and that the lamina has a weight of 100 N .
\item Determine the forces exerted on the lamina by each of the supports at $\mathrm { B } , \mathrm { C }$ and D .
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Mechanics Minor 2024 Q5 [12]}}
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Q1 5 Q2 8 Q3 9 Q4 15 Q5 12 Q6 11