Simple harmonic motion and damped oscillationsEdexcel A-Level Further Maths: Subtopic test

1. 1
   A particle moves on a straight line, with displacement $x$ metres from a fixed point $O$ at time $t$ seconds, so that $\ddot{x}=-16x$. At $t=0$ the particle is at $x=0.5$ and is instantaneously at rest.

   (a)

   Find the period of the motion.

   [1 mark]

   • A$\frac{\pi}{2}$ s
   • B$\frac{\pi}{8}$ s
   • C$8\pi$ s
   • D$\frac{2}{\pi}$ s

   (b)

   Find the maximum speed of the particle.

   [1 mark]

   • A$0.125$ m s$^{-1}$
   • B$8$ m s$^{-1}$
   • C$2$ m s$^{-1}$
   • D$0.5$ m s$^{-1}$

   (c)

   Find the displacement of the particle from $O$ when $t=\frac{\pi}{6}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A damped oscillator has displacement $x$ at time $t$ satisfying $\ddot{x}+2c\dot{x}+25x=0$, where $c>0$ is a constant.

   (a)

   Find the value of $c$ for which the damping is critical.

   [1 mark]

   • A$2.5$
   • B$5$
   • C$10$
   • D$25$

   (b)

   Describe the motion when $c=3$.

   [1 mark]

   • AOscillations of constant amplitude
   • BA return to $x=0$ without oscillating
   • COscillations of increasing amplitude
   • DOscillations of decreasing amplitude

   (c)

   Find the general solution of the differential equation when $c=3$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle of mass 2 kg lies on a smooth horizontal surface, attached to one end of a light spring of stiffness 50 N m$^{-1}$. The other end of the spring is fixed. The spring is at its natural length when the particle is at $O$, and $x$ metres is the displacement of the particle from $O$ at time $t$ seconds.

   (a)

   Show that $\ddot{x}+25x=0$.

   [3 marks]

   (b)

   At $t=0$ the particle is at $x=0.2$ and is moving away from $O$ with speed 1.5 m s$^{-1}$. Find the amplitude of the motion.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle of mass 1 kg moves along a straight line. At time $t$ seconds its displacement from a fixed point $O$ is $x$ metres. The particle is acted on by a force of magnitude $5x$ N directed towards $O$ and a resistive force of magnitude $2\dot{x}$ N.

   (a)

   (i) Show that $\ddot{x}+2\dot{x}+5x=0$.
   (ii) Given that $x=1$ and $\dot{x}=3$ when $t=0$, find $x$ in terms of $t$.

   [6 marks]

   (b)

   (i) Explain why the motion is described as lightly damped.
   (ii) State the period of the oscillation.
   (iii) Using your answer to part (a), show that $x(t+\pi)=e^{-\pi}x(t)$.
   (iv) The resistive force is changed to $c\dot{x}$ N, where $c>0$. Find the range of values of $c$ for which the motion is not oscillatory.

   [6 marks]

   Total for question 4: 12 marks

End of questions