differential equations- vibration of a wine glass

i) A model for the vibration of a wine glass is x''+λx'+ω^(2)x=0,
where λ and ω are constants. Suppose that when struck the wine glass vibrates at 660 Hz. Show that
(4ω^(2)-λ(^2))^(0.5)=2640pi.

ii) If it takes about 3 seconds for the sound to die away, and this happens when
the original vibrations have reduced to 1/100 of their original level, show that λ=(2log 100)/3
and hence that λ = 3.07 and ω = 4.15×103
(both to three significant figures).

Note: For the sinusoidal movement x(t)=sin(Ωt), the frequency is defined to be Ω/2pi and is measured in Hertz.

Can someone please help as I am not sure how to go about these questions.

Reply 1

ztibor

10

Original post by ellemay96

i) A model for the vibration of a wine glass is x''+λx'+ω^(2)x=0,
where λ and ω are constants. Suppose that when struck the wine glass vibrates at 660 Hz. Show that
(4ω^(2)-λ(^2))^(0.5)=2640pi.

ii) If it takes about 3 seconds for the sound to die away, and this happens when
the original vibrations have reduced to 1/100 of their original level, show that λ=(2log 100)/3
and hence that λ = 3.07 and ω = 4.15×103
(both to three significant figures).

Note: For the sinusoidal movement x(t)=sin(Ωt), the frequency is defined to be Ω/2pi and is measured in Hertz.

Can someone please help as I am not sure how to go about these questions.

for i)

\omega =2\pi \cdot f

and

\lambda=0

(no damping)

Substitute them into the given expression.

Reply 2

ellemay96

OP

1

Original post by ztibor

for i)

\omega =2\pi \cdot f

and

\lambda=0

(no damping)

Substitute them into the given expression.

Oh thank you!

Reply 3

DFranklin

18

Original post by ztibor

for i)

\omega =2\pi \cdot f

and

\lambda=0

(no damping)I don't think this is a valid assumption: From

Original post by ellemay96

ii) If it takes about 3 seconds for the sound to die away

it's clear that there is damping to be considered here.

[I would solve the original equation in the form

x = e^{-\alpha t}\cos(A+B\omega T)

and work from there].

Reply 4

ztibor

10

Original post by DFranklin

I don't think this is a valid assumption: From

.

I assumed no damping because the wine glass has just been struck
(and these values answer the first question (give 2640pi))

Reply 5

DFranklin

18

Original post by ztibor

I assumed no damping because the wine glass has just been struck
(and these values answer the first question (give 2640pi))

The equations of motion aren't going to change between when the glass is first struct and the sound decays.

That it gives something that matches the question is no real assurance: if a question asks you to show that (4ω^(2)-λ(^2))^(0.5)=2640pi, the implication is that the value of

\lambda

is relevant (as opposed to "you can ignore lambda because it's 0").

Reply 6

ellemay96

OP

1

I still don't understand (ii).
If you find the general equation, you'll still have constants A and B. Are you able to find these constants to then go on to find λ, or do you have to do something different?

Reply 7

DFranklin

18

Original post by ellemay96

I still don't understand (ii).
If you find the general equation, you'll still have constants A and B. Are you able to find these constants to then go on to find λ, or do you have to do something different?