$0\mathrm{.}30-$kg mass hangs at equilibrium from a spring suspended from a rod. Pulling the

mass down $4\mathrm{.}0$ cm from its equilibrium position requires a force of $5\mathrm{.}0$ N to stretch the

spring.

$($a$)$ Letting go of the mass after displacing it $4\mathrm{.}0$ cm from its equilibrium position,

you observe that its amplitude decays to $0\mathrm{.}50$ cm after $10$ oscillations. What is

the Q of this system?

$($b$)$ Suppose that the system is now driven with a oscillating driving force of F $($t$)=$

F$0$ sin t starting at time t $=0,$ where F$0=0\mathrm{.}50$ N and the driving frequency is

f $=2\mathrm{.}5$ Hz$.$ The initial displacement of the mass is$,$ once again, $4$ cm below its

equilibrium position.

i$.$ Find an equation for the subsequent motion of the mass and then plot it as

a function of time.

ii$.$ What is the steady state amplitude of the oscillations?

iii. What is the initial period of the oscillations?

iv$.$ What is the ultimate period of the oscillations?

v$.$ How much energy is stored in the oscillating system at steady state?

vi$.$ How much energy is dissipated per period by the system at steady state?