OCR MEI M4 2007 June — Question 1 12 marks

Exam BoardOCR MEI
ModuleM4 (Mechanics 4)
Year2007
SessionJune
Marks12
PaperDownload PDF ↗
TopicHooke's law and elastic energy
TypeElastic potential energy calculations
DifficultyChallenging +1.2 This is a multi-part mechanics problem requiring geometric analysis of string extension, energy methods, and force resolution. While it involves several steps and the energy approach to equilibrium, the techniques are standard for M4 level: Pythagoras for extension, differentiation of PE for equilibrium, and basic force resolution. The problem is structured with clear guidance through parts (i)-(iii), making it more accessible than if posed as a single open question.
Spec3.03n Equilibrium in 2D: particle under forces6.02i Conservation of energy: mechanical energy principle6.02j Conservation with elastics: springs and strings
1 A light elastic string has one end fixed to a vertical pole at A . The string passes round a smooth horizontal peg, P , at a distance \(a\) from the pole and has a smooth ring of mass \(m\) attached at its other end B . The ring is threaded onto the pole below A . The ring is at a distance \(y\) below the horizontal level of the peg. This situation is shown in Fig. 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8aab7e54-a204-481b-8f09-4bf4ca4e115d-2_462_275_557_897} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} The string has stiffness \(k\) and natural length equal to the distance AP .
  1. Express the extension of the string in terms of \(y\) and \(a\). Hence find the potential energy of the system relative to the level of P .
  2. Use the potential energy to find the equilibrium position of the system, and show that it is stable.
  3. Calculate the normal reaction exerted by the pole on the ring in the equilibrium position.
1 A light elastic string has one end fixed to a vertical pole at A . The string passes round a smooth horizontal peg, P , at a distance $a$ from the pole and has a smooth ring of mass $m$ attached at its other end B . The ring is threaded onto the pole below A . The ring is at a distance $y$ below the horizontal level of the peg. This situation is shown in Fig. 1.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{8aab7e54-a204-481b-8f09-4bf4ca4e115d-2_462_275_557_897}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}

The string has stiffness $k$ and natural length equal to the distance AP .\\
(i) Express the extension of the string in terms of $y$ and $a$. Hence find the potential energy of the system relative to the level of P .\\
(ii) Use the potential energy to find the equilibrium position of the system, and show that it is stable.\\
(iii) Calculate the normal reaction exerted by the pole on the ring in the equilibrium position.

\hfill \mbox{\textit{OCR MEI M4 2007 Q1 [12]}}
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Q1 12 Q2 12 Q3 24 Q4 24