Self-Assessment package question explanation - Proportionality

[Cundiff1080]

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Hoping someone can help me with an explanation to question 14 on the physics section of the self-assessment package...


Passage: "the fundamental tones of these strings are separated by a perfect fifth, which means the fundamental frequency of each string is 2/3 that of the higher frequency string.... fundamental frequency of a string is given by... f = ((T/p)^½) / (2L)"


Question: "By what factor would a string's tension need to be changed to raise its fundamental frequency by a perfect fifth"

Answer: 9/4
Explanation: "the relation between string frequency and tension is given in the passage as f is proportional to T^½. To raise the string frequency by a perfect fifth (a factor of 3/2), the tension must be increased by a factor of (3/2)^2 = 9/4"

I don't understand why it is 3/2 and not 2/3 like in the passage... and could you explain to me the rational of how you would set up such a problem? Thanks

 

[Jepstein30]

Hoping someone can help me with an explanation to question 14 on the physics section of the self-assessment package...


Passage: "the fundamental tones of these strings are separated by a perfect fifth, which means the fundamental frequency of each string is 2/3 that of the higher frequency string.... fundamental frequency of a string is given by... f = ((T/p)^½) / (2L)"


Question: "By what factor would a string's tension need to be changed to raise its fundamental frequency by a perfect fifth"

Answer: 9/4
Explanation: "the relation between string frequency and tension is given in the passage as f is proportional to T^½. To raise the string frequency by a perfect fifth (a factor of 3/2), the tension must be increased by a factor of (3/2)^2 = 9/4"

I don't understand why it is 3/2 and not 2/3 like in the passage... and could you explain to me the rational of how you would set up such a problem? Thanks

Just plug in numbers.

Best to use numbers easy to work with. Something like 8 and 12.

So-

8 = ((T/p)^1/2)/ 2L
12 = ((T'/p)^1/2) / 2L

only tension will be changed so reduces to

8 = T^1/2
12 = T' ^1/2

T = 64
T' = 144

T'/T= 144/64 = 9 / 4

 

[Cundiff1080]

Wait why 8 and 12, I know you said easy numbers but i assume you took 8 and multiplied by 3/2 to get 12? Why 3/2 and not 2/3?

 

[Jepstein30]

Wait why 8 and 12, I know you said easy numbers but i assume you took 8 and multiplied by 3/2 to get 12? Why 3/2 and not 2/3?

"Passage: "the fundamental tones of these strings are separated by a perfect fifth, which means the fundamental frequency of each string is 2/3 that of the higher frequency string.... fundamental frequency of a string is given by... f = ((T/p)^½) / (2L)"

Just picked 8 and 12 because they were easy numbers. 8 is 2/3 of 12 (so to get to 12, you'd multiply 3/2...).

Could pick any two numbers where one number is 2/3 the other..

​

 

[Cundiff1080]

Hmmm.... okay that makes sense, i'm gonna have to mess around with the numbers, is that the way that you always work out proportionality problems, by just plugging in some easy variables?

 

[Jepstein30]

Hmmm.... okay that makes sense, i'm gonna have to mess around with the numbers, is that the way that you always work out proportionality problems, by just plugging in some easy variables?

Pretty much. Any time I have an equation and they talk about increasing or decreasing by certain factors, I just plug in numbers. Makes it easier for me to work with concrete numbers instead of concepts in my head. Too easy to make a mistake doing some conceptual math in your head and much easier to keep it simple.

 

[Cundiff1080]

Awesome, i'm gonna try that strategy, thanks for all of your help!

 

[sciencebooks]

Hoping someone can help me with an explanation to question 14 on the physics section of the self-assessment package...


Passage: "the fundamental tones of these strings are separated by a perfect fifth, which means the fundamental frequency of each string is 2/3 that of the higher frequency string.... fundamental frequency of a string is given by... f = ((T/p)^½) / (2L)"


Question: "By what factor would a string's tension need to be changed to raise its fundamental frequency by a perfect fifth"

Answer: 9/4
Explanation: "the relation between string frequency and tension is given in the passage as f is proportional to T^½. To raise the string frequency by a perfect fifth (a factor of 3/2), the tension must be increased by a factor of (3/2)^2 = 9/4"

I don't understand why it is 3/2 and not 2/3 like in the passage... and could you explain to me the rational of how you would set up such a problem? Thanks

How I went through it: The passage doesn't say that the next higher frequency is 2/3. It's saying that the one below it is 2/3 of it. So the one below is 2/3 and the next higher frequency is 1. To get from 2/3 to 1, you multiply by 3/2. (Wow, that's really weird to put into words, lol.)

 

[wsid]

I'm confused

so if you essentially are multiplying f by 2/3 --> That means T has to also increase by 2/3, but because T is in a radical 2/3 has to be squared so you get 4/9. So if you multiply T by 4/9 you will be increasing f by a factor 2/3. I think the answer should be 4/9. Can someone explain whats wrong with my logic. If it doesn't make sense I can elaborate.