Question

1. A damped driven harmonic oscillator with m=12 kg, k=280 N/m, and b=75 kg/sis subjected to a driving force given by F(t) = F₀cos(ωt), where F₀=55 N.

a) What value of ω results in steady-state oscillations with maximum amplitude?

b) What is the maximum amplitude?

c) What is the phase angle?

2. An undamped, driven harmonic oscillator satisfies the equation of motion

where the driving force is switched on at t=0.

a) Assuming a solution of the form x(t) = A(ω)sin(ωt-δ), find expressions for the amplitude A(ω) and the phase angle δ.

Answer 1

1. I will try to find out the angular frequency

hence

a)

The value of for steady-state oscillations with maximum amplitude is given by

_____________________________________

b) the amplitude is given by

hence the maximum amplitude is given by

________________________________________

c) the phase angle is given by

________________________________________

2. The equation for an undamped, driven oscilator is

____________________________(1)

if we assume a solution

________________________(2)

then,

________________(3)

substituting (2) and (3) into (1) we get

_______________(4)

now, using the trigonometric identity,

into (4), we get

___________________(5)

from (5), we obtain

_______________________________________________________(6)

and

______________________________________(7)

the above equation implies,

____________________________________________________________________(8)

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