The position x of a mass m attached to a spring obeys the differential equation i + yi + w? x = 0 where y 2w. a) (2 marks) write down expressions for the forces on the mass due to (i) the spring, and (ii) damping. (3 marks) using a trial solution x = ae"', show that a = --y/2 ± (y2/4 - «2)1/2 b) c) (4 marks) show, by finding wd, that the solution is a damped oscillation of the form xc ace/2 cos(wat + pc) after some time to the mass is driven by applying a force that varies sinusoidally with frequency wf. the differential equation for x becomes i+ yx + w? x = ao cos(wft p). apcos(wft d) (6 marks) show by substitution that xp p) is a particular solution, provided that ap and pp satisfy certain conditions. find the two equations that they must satisfy. (nb: you need not explicitly solve for ap and pp.) e) (3 marks) draw a qualitative plot illustrating the dependence of ap on wf. indicate how your plot would change if y were increased. f) (2 marks) write down the general solution for x(t), and explain how this encompasses the initial transient behaviour of the mass when the driver is switched on, as well as the long term behaviour. the position x of a mass m attached to a spring obeys the differential equation i + yi + w? x = 0 where y 2w. a) (2 marks) write down expressions for the forces on the mass due to (i) the spring, and (ii) damping. (3 marks) using a trial solution x = ae"', show that a = --y/2 ± (y2/4 - «2)1/2 b) c) (4 marks) show, by finding wd, that the solution is a damped oscillation of the form xc ace/2 cos(wat + pc) after some time to the mass is driven by applying a force that varies sinusoidally with frequency wf. the differential equation for x becomes i+ yx + w? x = ao cos(wft p). apcos(wft d) (6 marks) show by substitution that xp p) is a particular solution, provided that ap and pp satisfy certain conditions. find the two equations that they must satisfy. (nb: you need not explicitly solve for ap and pp.) e) (3 marks) draw a qualitative plot illustrating the dependence of ap on wf. indicate how your plot would change if y were increased. f) (2 marks) write down the general solution for x(t), and explain how this encompasses the initial transient behaviour of the mass when the driver is switched on, as well as the long term behaviour.