Simple Harmonic Motion | Edexcel A Level Further Maths: Core Pure Topic Questions 2017

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8 marks

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,

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

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 30000N.

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

(a)

(i)

explain why the value of

m

is 3

(ii)

show that a particular solution to the differential equation is

$x = 40 \sin t - 20 co s t$

(iii)

hence find the general solution of the differential equation.

How did you do?

4 marks

(b)

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.

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

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