Simple harmonic motionAQA A-Level Further Maths: Subtopic test

1. 1
   A particle moves on a straight line with displacement $x$ m from a fixed point $O$ at time $t$ s, where $x''=-16x$. When $t=0$, $x=0.3$ and the particle is at rest.

   (a)

   Find the period of the motion.

   [1 mark]

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

   (b)

   Find the maximum speed of the particle.

   [1 mark]

   • A$0.075$ m s$^{-1}$
   • B$4.8$ m s$^{-1}$
   • C$1.2$ m s$^{-1}$
   • D$0.3$ m s$^{-1}$

   (c)

   Find an expression for $x$ in terms of $t$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle moves with simple harmonic motion about a fixed point $O$, with period 2 s and amplitude 0.5 m.

   (a)

   Find the angular frequency $\omega$ of the motion.

   [1 mark]

   • A$\frac{\pi}{2}$ rad s$^{-1}$
   • B$\pi$ rad s$^{-1}$
   • C$2\pi$ rad s$^{-1}$
   • D$2$ rad s$^{-1}$

   (b)

   Find the maximum acceleration of the particle.

   [1 mark]

   • A$0.5\pi$ m s$^{-2}$
   • B$\pi^2$ m s$^{-2}$
   • C$2\pi^2$ m s$^{-2}$
   • D$0.5\pi^2$ m s$^{-2}$

   (c)

   Find the speed of the particle when it is 0.3 m from $O$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle of mass 2 kg hangs in equilibrium from a fixed point on a light vertical spring, where the tension is $T=kx$, with $x$ m the extension and $k=200$ N m$^{-1}$. The particle is pulled a further 0.05 m downwards and released from rest at $t=0$. Let $y$ m be the displacement of the particle below its equilibrium position at time $t$ s. Assume the spring stays taut.

   (a)

   Show that $y''=-100y$.

   [3 marks]

   (b)

   Find $y$ in terms of $t$, and find the maximum speed of the particle.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle moves on a straight line with displacement $x$ m from $O$ at time $t$ s, where $x''+4x=0$. When $t=0$, $x=1$ and $x'=2\sqrt3$.

   (a)

   Solve the differential equation to find $x$ in terms of $t$, and find the amplitude of the motion.

   [6 marks]

   (b)

   (i) State the period of the motion.
   (ii) Find the first time $t>0$ at which the particle is at $O$.
   (iii) Find the speed of the particle at that time, and explain why this is the greatest speed.

   [6 marks]

   Total for question 4: 12 marks

End of questions