Prove SHM and find period: horizontal or non-standard geometry

A particle on a horizontal surface, bead on a wire, or other non-vertical geometry attached to springs/strings, requiring proof that motion is SHM and finding the period.

4 questions · Standard +0.7

4.10f Simple harmonic motion: x'' = -omega^2 x
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Edexcel M3 2004 January Q5
12 marks Standard +0.3
5. A piston in a machine is modelled as a particle of mass 0.2 kg attached to one end \(A\) of a light elastic spring, of natural length 0.6 m and modulus of elasticity 48 N . The other end \(B\) of the spring is fixed and the piston is free to move in a horizontal tube which is assumed to be smooth. The piston is released from rest when \(A B = 0.9 \mathrm {~m}\).
  1. Prove that the motion of the piston is simple harmonic with period \(\frac { \pi } { 10 } \mathrm {~s}\).
    (5)
  2. Find the maximum speed of the piston.
    (2)
  3. Find, in terms of \(\pi\), the length of time during each oscillation for which the length of the spring is less than 0.75 m .
    (5)
Edexcel M3 Q4
11 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{45c51316-7d58-4c16-9b5f-1d7421060a88-4_332_1056_251_459} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A small smooth bead \(B\) of mass 0.2 kg is threaded on a smooth horizontal wire. The point \(A\) is on the same horizontal level as the wire and at a perpendicular distance \(d\) from the wire. The point \(O\) is the point on the wire nearest to \(A\), as shown in Fig. 2. The bead experiences a force of magnitude \(5 ( A B )\) newtons in the direction \(B A\) towards \(A\). Initially \(B\) is at rest with \(O B = 2 \mathrm {~m}\).
  1. Prove that \(B\) moves with simple harmonic motion about \(O\), with period \(\frac { 2 \pi } { 5 } \mathrm {~s}\).
  2. Find the greatest speed of \(B\) in the motion.
  3. Find the time when \(B\) has first moved a distance 3 m from its initial position.
CAIE FP2 2018 November Q5
12 marks Standard +0.8
The fixed points \(A\) and \(B\) are on a smooth horizontal surface with \(AB = 2.6\) m. One end of a light elastic spring, of natural length 1.25 m and modulus of elasticity \(0.6\) N, is attached to \(A\). The other end is attached to a particle \(P\) of mass 0.4 kg. One end of a second light elastic spring, of natural length 1.0 m and modulus of elasticity \(0.62\) N, is attached to \(B\); its other end is attached to \(P\). The system is in equilibrium with \(P\) on the surface at the point \(E\).
  1. Show that \(AE = 1.4\) m. [4]
The particle \(P\) is now displaced slightly from \(E\), along the line \(AB\).
  1. Show that, in the subsequent motion, \(P\) performs simple harmonic motion. [5]
  2. Given that the period of the motion is \(\frac{4}{\pi}\) s, find the value of \(\lambda\). [3]
Edexcel M3 2003 June Q5
13 marks Standard +0.3
A particle \(P\) of mass \(0.8\) kg is attached to one end \(A\) of a light elastic spring \(OA\), of natural length \(60\) cm and modulus of elasticity \(12\) N. The spring is placed on a smooth horizontal table and the end \(O\) is fixed. The particle \(P\) is pulled away from \(O\) to a point \(B\), where \(OB = 85\) cm, and is released from rest.
  1. Prove that the motion of \(P\) is simple harmonic with period \(\frac{2\pi}{5}\) s. [5]
  2. Find the greatest magnitude of the acceleration of \(P\) during the motion. [2]
Two seconds after being released from rest, \(P\) passes through the point \(C\).
  1. Find, to 2 significant figures, the speed of \(P\) as it passes through \(C\). [5]
  2. State the direction in which \(P\) is moving 2 s after being released. [1]