Damped oscillationsAQA A-Level Further Maths: Subtopic test

1. 1
   A particle moves on a straight line, with displacement $x$ m from $O$ at time $t$ s, where $x''+6x'+25x=0$.

   (a)

   Find the roots of the auxiliary equation.

   [1 mark]

   • A$-6\pm8i$
   • B$3\pm4i$
   • C$-3\pm4i$
   • D$-3\pm8i$

   (b)

   Which description of the damping is correct?

   [1 mark]

   • ALight damping
   • BCritical damping
   • CHeavy damping
   • DNo damping

   (c)

   Write down the general solution of the differential equation.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle of mass 0.5 kg moves on a straight line, with displacement $x$ m from $O$ at time $t$ s. It is acted on by a restoring force of magnitude $2|x|$ N towards $O$ and a resistive force of magnitude $\lambda v$ N, where $v$ m s$^{-1}$ is its speed and $\lambda$ is a positive constant.

   (a)

   Which differential equation models the motion?

   [1 mark]

   • A$x''+\lambda x'+4x=0$
   • B$x''+2\lambda x'+x=0$
   • C$x''+2\lambda x'-4x=0$
   • D$x''+2\lambda x'+4x=0$

   (b)

   Find the value of $\lambda$ for which the motion is critically damped.

   [1 mark]

   • A$\lambda=1$
   • B$\lambda=2$
   • C$\lambda=4$
   • D$\lambda=8$

   (c)

   State, with a reason, the type of damping when $\lambda=3$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The displacement $x$ m of a particle from $O$ at time $t$ s satisfies $x''+4x'+4x=0$. When $t=0$, $x=3$ and $x'=0$.

   (a)

   Show that the damping is critical and write down the general solution.

   [3 marks]

   (b)

   Find $x$ in terms of $t$, and show that the particle never reaches $O$ for $t>0$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle of mass 2 kg moves on a straight line, with displacement $x$ m from $O$ at time $t$ s. A restoring force of magnitude $26|x|$ N acts towards $O$, and the particle is also acted on by a resistive force of magnitude $4v$ N, where $v$ m s$^{-1}$ is its speed.

   (a)

   (i) Show that $x''+2x'+13x=0$.
   (ii) Given that $x=0$ and $x'=6$ when $t=0$, find $x$ in terms of $t$.

   [6 marks]

   (b)

   Given that $x=2e^{-t}\sin3t$:
   (i) find the time between successive passes through $O$ in the same direction;
   (ii) find the first time $t>0$ at which the particle is at its greatest displacement from $O$;
   (iii) the resistive force is changed to $cv$ N, with the same restoring force. Find the value of $c$ for which the motion is critically damped.

   [6 marks]

   Total for question 4: 12 marks

End of questions