OCR MEI M3 2013 June — Question 3 18 marks

Exam BoardOCR MEI
ModuleM3 (Mechanics 3)
Year2013
SessionJune
Marks18
PaperDownload PDF ↗
TopicSimple Harmonic Motion
TypeTwo springs/strings system equilibrium
DifficultyStandard +0.8 This is a substantial multi-part SHM question requiring: equilibrium analysis with Hooke's law, deriving the equation of motion from forces in two springs, solving the SHM differential equation with initial conditions, and calculating period/amplitude/velocity/distance. While methodical, it demands careful bookkeeping across 6 parts and understanding of when springs are in tension vs thrust. More demanding than a standard single-spring SHM problem but follows established M3 techniques without requiring novel insight.
Spec4.10f Simple harmonic motion: x'' = -omega^2 x
3 A light spring, with modulus of elasticity 686 N , has one end attached to a fixed point A . The other end is attached to a particle P of mass 18 kg which hangs in equilibrium when it is 2.2 m vertically below A .
  1. Find the natural length of the spring AP . Another light spring has natural length 2.5 m and modulus of elasticity 145 N . One end of this spring is now attached to the particle P , and the other end is attached to a fixed point B which is 2.5 m vertically below P (so leaving the equilibrium position of P unchanged). While in its equilibrium position, P is set in motion with initial velocity \(3.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards, as shown in Fig. 3. It now executes simple harmonic motion along part of the vertical line AB . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-4_721_383_726_831} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure} At time \(t\) seconds after it is set in motion, P is \(x\) metres below its equilibrium position.
  2. Show that the tension in the spring AP is \(( 176.4 + 392 x ) \mathrm { N }\), and write down an expression for the thrust in the spring BP.
  3. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 25 x\).
  4. Find the period and the amplitude of the motion.
  5. Find the magnitude and direction of the velocity of P when \(t = 2.4\).
  6. Find the total distance travelled by P during the first 2.4 seconds of its motion.
3 A light spring, with modulus of elasticity 686 N , has one end attached to a fixed point A . The other end is attached to a particle P of mass 18 kg which hangs in equilibrium when it is 2.2 m vertically below A .\\
(i) Find the natural length of the spring AP .

Another light spring has natural length 2.5 m and modulus of elasticity 145 N . One end of this spring is now attached to the particle P , and the other end is attached to a fixed point B which is 2.5 m vertically below P (so leaving the equilibrium position of P unchanged). While in its equilibrium position, P is set in motion with initial velocity $3.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ vertically downwards, as shown in Fig. 3. It now executes simple harmonic motion along part of the vertical line AB .

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-4_721_383_726_831}
\captionsetup{labelformat=empty}
\caption{Fig. 3}
\end{center}
\end{figure}

At time $t$ seconds after it is set in motion, P is $x$ metres below its equilibrium position.\\
(ii) Show that the tension in the spring AP is $( 176.4 + 392 x ) \mathrm { N }$, and write down an expression for the thrust in the spring BP.\\
(iii) Show that $\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 25 x$.\\
(iv) Find the period and the amplitude of the motion.\\
(v) Find the magnitude and direction of the velocity of P when $t = 2.4$.\\
(vi) Find the total distance travelled by P during the first 2.4 seconds of its motion.

\hfill \mbox{\textit{OCR MEI M3 2013 Q3 [18]}}
This paper (4 questions)
View full paper
Q1 18 Q2 18 Q3 18 Q4 18