Question 1

Paper 2Medium14 marks

A small mass moves vertically on a spring. For t≥0t\geq 0 its displacement yy metres from its equilibrium position is modelled by

y(t)=0.32sin⁡(1.4t+0.30),y(t)=0.32\sin(1.4t+0.30),

where tt is measured in seconds and angles are in radians. This is a model of simple harmonic motion.

(a)

State the amplitude and period of the motion. Calculate the displacement of the mass at t=0t=0 giving your answer to three significant figures.

[3]
(b)

Show that

d2ydt2=−0.6272sin⁡(1.4t+0.30),\frac{d^2y}{dt^2}=-0.6272\sin(1.4t+0.30),

and hence show that

d2ydt2=−1.96y(t).\frac{d^2y}{dt^2}=-1.96y(t).
[3]
(c)

Determine the time of the first stationary point after t=0t=0 Use the second derivative test to classify this point.

[4]
(d)

Using your result from part (b), determine the time of the first point of inflexion after t=0t=0 State the concavity immediately before and after this point, and interpret the point in terms of the mass's motion.

[4]

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