Question

A 2kg mass is suspended vertically from a spring attached to a fixed support. The spring satisfies Hooke's law with a spring constant of k 18 N m1. No damping is present. Gravity acts on the mass with a gravitational constant of g 10 m s2. An external force of R 24 sin (wt) Newton is applied to the mass, directed downwards, where t is the time in seconds since the mass was set in motion and w is a positive real constant The mass is initially set in motion from its equilibrium position with a velocity of 4 m s1 downwards Let y(t) be the displacement in metres of the mass below its equilibrium position at time t seconds (a) Use Newton's second law to show that the displacement y(t) satisfies the following equation of motion "(t)9y(t) 12 sin(wt) (b) For the case of w 3, solve the equation of motion to find y(t) (c) Describe the long-term behaviour of the solution y(t) predicted by this model when w 3. Explain whether this behaviour is realistic (d) Describe the long-term behaviour of the solution y(t) predicted by the model when w = 2. Explain your answer but do not solve for y(t) in this case.

Answer 1

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