Ggdf
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Hi there

I'm self-teaching introductory physics and wonder if anyone could please help me work through the following problem?

Hair cells in the ear may be modelled as a mechanical system equivalent to a damped spring subject to oscillating forces ie sound waves. When hit by a sound wave, after an initial, transient motion, the hair settles down to a steady oscillation. The amplitude, A, of this oscillation depends on the frequency of the sound wave, f, measured in hertz (Hz). Figure 2 shows the amplitude of the response of two different hairs (upper and lower curve) to sounds of different frequencies.

Name:  Physics 4.jpg
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(i) From the figure, estimate the resonant frequencies of the system for the two different amounts of damping.

(ii) For the upper curve, estimate the two values of the frequency, f, where the amplitude, A, is 50% of its maximum value.

(iii) For the lower curve, estimate the two values of the frequency, f, where the amplitude, A, is 50% of its maximum value.

For i, I wonder if the resonant frequency for the larger curve is 1.2? And the resonant frequency for the smaller curve 1.19?

I'm unsure how to work out ii and iii.

Please might anyone be able to help me?

Thank you very much
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Phichi
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(Original post by Ggdf)
Hi there

I'm self-teaching introductory physics and wonder if anyone could please help me work through the following problem?

Hair cells in the ear may be modelled as a mechanical system equivalent to a damped spring subject to oscillating forces ie sound waves. When hit by a sound wave, after an initial, transient motion, the hair settles down to a steady oscillation. The amplitude, A, of this oscillation depends on the frequency of the sound wave, f, measured in hertz (Hz). Figure 2 shows the amplitude of the response of two different hairs (upper and lower curve) to sounds of different frequencies.

Image

(i) From the figure, estimate the resonant frequencies of the system for the two different amounts of damping.

(ii) For the upper curve, estimate the two values of the frequency, f, where the amplitude, A, is 50% of its maximum value.

(iii) For the lower curve, estimate the two values of the frequency, f, where the amplitude, A, is 50% of its maximum value.

For i, I wonder if the resonant frequency for the larger curve is 1.2? And the resonant frequency for the smaller curve 1.19?

I'm unsure how to work out ii and iii.

Please might anyone be able to help me?

Thank you very much
The resonant frequencies occur on each curve where the amplitude is greatest. It looks like you've got the right answer for the two, however the graph is two small to verify, seems you've got the right idea though.

At the resonant frequency the amplitude is max. You can find the amplitude of the resonant frequency using the scale on the y-axis. To find the values where the amplitude is 50% of that at the resonant frequency, just draw a horizontal line from that point on the y-axis. E.g, if your 50% amplitude is 5 (looks like this is the correct value for the big curve), draw a horizontal line at y=5, it should cross the curve at two points, record the frequencies at these points.
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Ggdf
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(Original post by Phichi)
The resonant frequencies occur on each curve where the amplitude is greatest. It looks like you've got the right answer for the two, however the graph is two small to verify, seems you've got the right idea though.

At the resonant frequency the amplitude is max. You can find the amplitude of the resonant frequency using the scale on the y-axis. To find the values where the amplitude is 50% of that at the resonant frequency, just draw a horizontal line from that point on the y-axis. E.g, if your 50% amplitude is 5 (looks like this is the correct value for the big curve), draw a horizontal line at y=5, it should cross the curve at two points, record the frequencies at these points.
I'm really grateful for your help.

I'm glad I understood the first one.

For the first graph, would I therefore do something like this?

Name:  Resonant frequency 1.jpg
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And I would take the 2 forcing frequencies where my vertical drawn lines cross the x axis?

For graph 2 would it roughly look something like this?

Name:  Resonant frequency 2.jpg
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So the frequencies might be roughly 1.96 and 1.37?

Thank you
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Phichi
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(Original post by Ggdf)
I'm really grateful for your help.

I'm glad I understood the first one.

For the first graph, would I therefore do something like this?

Image

And I would take the 2 forcing frequencies where my vertical drawn lines cross the x axis?

For graph 2 would it roughly look something like this?

Image

So the frequencies might be roughly 1.96 and 1.37?

Thank you
As long as those lines are at the half amplitude point (can't see that well), your method is exactly right. The two points are your two frequencies.
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Ggdf
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(Original post by Phichi)
As long as those lines are at the half amplitude point (can't see that well), your method is exactly right. The two points are your two frequencies.
Excellent!
Thank you very much!
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Phichi
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(Original post by Ggdf)
Excellent!
Thank you very much!
No worries
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Ggdf
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(Original post by Phichi)
No worries
Thanks again!

Do you happen to know how to work out a Q factor?

I believe it is resonant frequency/frequency width.

If I were to work it out for the bigger curve here using the resonant frequency of 1.2 and two frequencies of 1.31 and 1.12 (roughly taken from the graph), would my Q factor equation look like:

Q factor = 1.2 / (1.31-1.12) = 6.32?

Also am I correct to think that this 6.32 has no units?

I appreciate your help!
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uberteknik
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(Original post by Ggdf)
Thanks again!

Do you happen to know how to work out a Q factor?

I believe it is resonant frequency/frequency width.

If I were to work it out for the bigger curve here using the resonant frequency of 1.2 and two frequencies of 1.31 and 1.12 (roughly taken from the graph), would my Q factor equation look like:

Q factor = 1.2 / (1.31-1.12) = 6.32?

Also am I correct to think that this 6.32 has no units?

I appreciate your help!
Yes, this is correct within the error introduced by the uncertainty in reading the scale.

I would round the answer to 6.3 given the difficulty in reading the scale.

Q factor is indeed a dimensionless ratio.
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