Cross sectional area of a tube = to ?

[Addallat]

pg 194 of TPR physics book states that

cross sectional area of the pipe is equal to the square of the diameter of a tube

pg 193 of TPR physics book states that

"flow speed is inversely proportional to the cross sectional area of a tube(or to the square of the radius) of the tube"
^~~~ this above statement in quotation marks can someone spell it out for me mathematically; I think this is where the source of my confusion is coming from

flow rate = 1/A = 1/r^2 = 1/d^2 ???

I'm really confused, I don't understand how both

r^2 = cross sectional area of a pipe and diameter^2 can = cross sectional area of a pipe

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[AureatEagle]

pg 194 of TPR physics book states that

cross sectional area of the pipe is equal to the square of the diameter of a tube

pg 193 of TPR physics book states that

"flow speed is inversely proportional to the cross sectional area of a tube(or to the square of the radius) of the tube"
^~~~ this above statement in quotation marks can someone spell it out for me mathematically; I think this is where the source of my confusion is coming from

flow rate = 1/A = 1/r^2 = 1/d^2 ???

I'm really confused, I don't understand how both

r^2 = cross sectional area of a pipe and diameter^2 can = cross sectional area of a pipe

The second statement is correct: the flow rate is proportional to the cross sectional area of the tube, which is exactly what it sounds like: pi*r^2.

The first statement is true, just not 100% specific, which is why you're confused. We know that cross sectional area is proportional to the square of the radius of the tube. However, since radius = diameter/2 , then technically we could also write cross sectional area = pi *(diameter/2)^2. You'll notice that if we write the equation like this, cross sectional area is still technically proportional to the square of the diameter-- it just leaves out the pi and the (1/2)^2 term.

They're saying the same thing at the end of the day.

 

[Addallat]

thanks for replying! wouldn't leaving out the 1/2 change the answer you want though? I can't see why it's okay not to divide the diameter by 2 to get the cross sectional area; doesn't doing so change your cross sectional area significantly?

 

[AureatEagle]

thanks for replying! wouldn't leaving out the 1/2 change the answer you want though? I can't see why it's okay not to divide the diameter by 2 to get the cross sectional area; doesn't doing so change your cross sectional area significantly?

Yes, it would if they were asking for an actual quantitative answer. But the question is only talking about proportionalities, which is qualitative, and omits all other constants and variables.

For example, in PV = nRT, we'd say that P is proportional to T even though there's a V, n, and R factor.

Similarly, in A = pi (d/2)^2, we'd say that A is proportional to d^2 even though there's other constant factors involved if we were to calculate an actual problem out.

 

[mehc012]

thanks for replying! wouldn't leaving out the 1/2 change the answer you want though? I can't see why it's okay not to divide the diameter by 2 to get the cross sectional area; doesn't doing so change your cross sectional area significantly?

When you're talking proportions, constants are irrelevant. For example, x α y in all of these:
x = y
x = ½y
x = 3·π·y·e·8³

They're all just forms of x = kx, which is what proportionality means

Therefore, if x α y and y α z, then x α z, because the value of k is irrelevant:
x = ky
y = jz
x = k·j·z
x α z

In your example,
x = π r²
r = ½d
x = π·½·½·r²

 

[Addallat]

when you're talking proportions, constants are irrelevant. For example, x α y in all of these:
X = y
x = ½y
x = 3·π·y·e·8³

they're all just forms of x = kx, which is what proportionality means

therefore, if x α y and y α z, then x α z, because the value of k is irrelevant:
X = ky
y = jz
x = k·j·z
x α z

in your example,
x = π r²
r = ½d
x = π·½·½·r²

you are freaking awesome! Thank you!!!