Simple harmonic motionEdexcel A-Level Further Maths: Subtopic test

1. 1
   A particle $P$ moves in a straight line with simple harmonic motion about a fixed centre $O$. The amplitude of the motion is 0.8 m and the angular frequency is $\omega=5$ rad s⁻¹.

   (a)

   Find the maximum speed of $P$.

   [1 mark]

   • A$4$ m s⁻¹
   • B$20$ m s⁻¹
   • C$0.16$ m s⁻¹
   • D$0.8$ m s⁻¹

   (b)

   Find the speed of $P$ when it is 0.4 m from $O$.

   [1 mark]

   • A$2.00$ m s⁻¹
   • B$4.00$ m s⁻¹
   • C$3.46$ m s⁻¹
   • D$4.47$ m s⁻¹

   (c)

   Given that $x=0.8\sin5t$ with $t=0$ at $O$, find the time taken for $P$ to move directly from $O$ to a point 0.4 m from $O$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle $P$ of mass 0.5 kg is attached to one end of a light elastic spring of natural length 0.4 m and modulus of elasticity 20 N. The other end of the spring is fixed to a ceiling and $P$ hangs in equilibrium. $P$ is then pulled vertically downwards a distance 0.05 m from the equilibrium position and released from rest. Take $g=9.8$ m s⁻².

   (a)

   Find the extension of the spring when $P$ is in equilibrium.

   [1 mark]

   • A$0.245$ m
   • B$0.098$ m
   • C$10.2$ m
   • D$0.049$ m

   (b)

   Find the period of the oscillations of $P$.

   [1 mark]

   • A$1.59$ s
   • B$6.28$ s
   • C$0.314$ s
   • D$0.628$ s

   (c)

   Find the maximum speed of $P$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle $P$ of mass 0.4 kg lies on a smooth horizontal table. It is attached to two identical light elastic strings, each of natural length 0.8 m and modulus of elasticity 16 N. The other ends of the strings are fixed to points $A$ and $B$ on the table, where $AB=2.4$ m, and $P$ rests in equilibrium at the midpoint $M$ of $AB$. $P$ is then displaced along $AB$ and released from rest.

   (a)

   $P$ is displaced a distance $x$ metres from $M$ towards $B$, with both strings taut. Show that $P$ moves with simple harmonic motion, and state the value of $\omega$.

   [3 marks]

   (b)

   $P$ is released from rest at a point 0.25 m from $M$. Find the maximum speed of $P$, and its speed when it is 0.15 m from $M$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ of mass 0.5 kg is attached to one end of a light elastic string of natural length 0.5 m and modulus of elasticity 12.25 N. The other end of the string is fixed to a point $O$ on a ceiling, and $P$ hangs in equilibrium at the point $E$. Take $g=9.8$ m s⁻². Air resistance may be ignored.

   (a)

   $P$ is pulled down 0.1 m from $E$ and released from rest. Find the extension of the string at $E$, and show that $P$ moves with simple harmonic motion. Find the period of the motion.

   [6 marks]

   (b)

   Instead, $P$ is pulled down 0.3 m from $E$ and released from rest. Show that the string becomes slack, find the speed of $P$ when this happens, and find the greatest height of $P$ above $E$.

   [6 marks]

   Total for question 4: 12 marks

End of questions