Simple Harmonic Motion (Edexcel A Level Further Maths: Core Pure): Exam Questions

A company plans to build a new fairground ride. The ride will consist of a capsule that will hold the passengers and the capsule will be attached to a tall tower. The capsule is to be released from rest from a point half way up the tower and then made to oscillate in a vertical line.
The vertical displacement, x metres, of the top of the capsule below its initial position at time t seconds is modelled by the differential equation,

$m\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+4\frac{\mathrm{d}x}{\mathrm{d}t}+x=200\cos t,$      $t\ge 0$

where $m$ is the mass of the capsule including its passengers, in thousands of kilograms.

The maximum permissible weight for the capsule, including its passengers, is 30 000 N.

Taking the value of $g$ to be 10 ms^-2 and assuming the capsule is at its maximum permissible weight,

(i) explain why the value of $m$ is 3

(ii) show that a particular solution to the differential equation is

$x=40\sin t-20\mathrm{co}st$

(iii) hence find the general solution of the differential equation.

Using the model, find, to the nearest metre, the vertical distance of the top of the capsule from its initial position, 9 seconds after it is released.