Oscillations of elastic strings and springsEdexcel International A Level Further Maths: Subtopic test

1. 1
   A particle $P$ of mass $0.4$ kg is attached to one end of a light spring of natural length $0.5$ m and modulus of elasticity $20$ N. The other end of the spring is fixed to a point $O$ on a smooth horizontal table, and $P$ moves on the table along the line of the spring. At time $t$ seconds the extension of the spring is $x$ metres.

   (a)

   Which equation of motion applies to $P$ while the spring is extended?

   [1 mark]

   • A$\ddot x=-40x$
   • B$\ddot x=-50x$
   • C$\ddot x=-16x$
   • D$\ddot x=-100x$

   (b)

   Find the period of the oscillations of $P$.

   [1 mark]

   • A$\frac{\pi}{50}$ s
   • B$\frac{\pi}{5}$ s
   • C$\frac{5}{\pi}$ s
   • D$\frac{2\pi}{\sqrt{40}}$ s

   (c)

   $P$ is held at rest with the spring extended by $0.3$ m and then released. Find the maximum speed of $P$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle $P$ of mass $0.5$ kg hangs in equilibrium attached to the lower end of a light spring of natural length $0.8$ m and modulus of elasticity $10$ N. The upper end of the spring is fixed to a point $A$ on a ceiling. Take $g=9.8$ m s$^{-2}$.

   (a)

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

   [1 mark]

   • A$0.49$ m
   • B$0.04$ m
   • C$0.392$ m
   • D$0.784$ m

   (b)

   $P$ is displaced vertically and released, and the string stays taut. Find the period of the resulting oscillations.

   [1 mark]

   • A$1.26$ s
   • B$0.251$ s
   • C$1.78$ s
   • D$0.796$ s

   (c)

   $P$ is pulled down $0.1$ m from the equilibrium position and released from rest. Find the greatest tension in the spring.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle $P$ of mass $0.25$ kg is attached to one end of a light elastic string of natural length $0.6$ m and modulus of elasticity $15$ N. The other end of the string is fixed to a point $O$ on a ceiling, and $P$ hangs in equilibrium vertically below $O$. $P$ is pulled down $0.04$ m from the equilibrium position and released from rest, and the string stays taut throughout the motion. Take $g=9.8$ m s$^{-2}$.

   (a)

   Show that, when $P$ is $x$ metres below the equilibrium position, $\ddot x=-100x$.

   [3 marks]

   (b)

   (i) Find the greatest speed of $P$.
   (ii) Find the time taken for $P$ to move from its lowest point to a point $0.02$ m below the equilibrium position.

   [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 $1.2$ m and modulus of elasticity $24$ N. The other end of the string is fixed to a point $O$ on a ceiling, and $P$ hangs in equilibrium vertically below $O$. $P$ is pulled down a further $0.6$ m from the equilibrium position and released from rest. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the speed of $P$ at the instant the string becomes slack.

   [6 marks]

   (b)

   Find the vertical distance between the lowest point and the highest point reached by $P$.

   [6 marks]

   Total for question 4: 12 marks

End of questions