Simple harmonic motionEdexcel International A Level Further Maths: Subtopic test

1. 1
   A particle moves in a straight line with simple harmonic motion about a centre $O$. The amplitude of the motion is $0.5$ m and the period is $2$ s.

   (a)

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

   [1 mark]

   • A$\pi$ rad s$^{-1}$
   • B$2\pi$ rad s$^{-1}$
   • C$\frac1\pi$ rad s$^{-1}$
   • D$4\pi$ rad s$^{-1}$

   (b)

   Find the maximum speed of the particle.

   [1 mark]

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

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The displacement of a particle $P$ from a fixed point $O$ on a straight line, at time $t$ seconds, is $x=0.4\cos3t$ metres.

   (a)

   Find the period of the motion.

   [1 mark]

   • A$\frac{3}{2\pi}$ s
   • B$\frac{2\pi}{3}$ s
   • C$6\pi$ s
   • D$3$ s

   (b)

   Find the acceleration of $P$ when $x=0.3$ m.

   [1 mark]

   • A$-0.9$ m s$^{-2}$
   • B$-3.6$ m s$^{-2}$
   • C$+2.7$ m s$^{-2}$
   • D$-2.7$ m s$^{-2}$

   (c)

   Find the first time at which $P$ is at $x=-0.2$ m.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle $P$ of mass $2$ kg moves on a smooth horizontal straight line. When $P$ is at displacement $x$ metres from a fixed point $O$ on the line, the only horizontal force on $P$ is directed towards $O$ and has magnitude $18x$ N.

   (a)

   Show that $P$ moves with simple harmonic motion, and find the period of the motion.

   [3 marks]

   (b)

   $P$ passes through $O$ with speed $6$ m s$^{-1}$. Find the amplitude of the motion and the speed of $P$ when it is $1.5$ m from $O$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ moves with simple harmonic motion along a straight line with centre $O$. When $P$ is $1.5$ m from $O$ its speed is $4$ m s$^{-1}$. When $P$ is $2$ m from $O$ its speed is $3$ m s$^{-1}$.

   (a)

   Find the angular frequency and the amplitude of the motion.

   [6 marks]

   (b)

   Find the time taken for $P$ to move directly from the point $1.5$ m from $O$ to the point $2$ m from $O$, on the same side of $O$.

   [6 marks]

   Total for question 4: 12 marks

End of questions