Oscillations of elastic strings and springsEdexcel International A Level Further Maths: Mind map

Exam questions on Oscillations of elastic strings and springs

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.
   $P$ is held at rest with the spring extended by $0.3$ m and then released. Find the maximum speed of $P$.2 marks
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$^{-2}$.
   $P$ is pulled down $0.1$ m from the equilibrium position and released from rest. Find the greatest tension in the spring.2 marks
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$^{-2}$.
   Show that, when $P$ is $x$ metres below the equilibrium position, $\ddot x=-100x$.3 marks