Dimensional Analysis

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Homework Statement



Pressure(water)~density(water)xg(grav. accel.)xz(depth)

Homework Equations





The Attempt at a Solution


basically i'm doing a fermi problem regarding the pressure of water on Jupiter's moon Europa.
Now i tried coming up with the equation for pressure my self by pressure ~ densityxgxmxz because i had figured the mass of the water would matter. so
[M]^1[L]^(-1)[T]^(-1)=[MxL^(-3)]^x [LxT^(-2)]^y [M]^w [L]^z
and so we get that y = 1 and the equations we are left with are -2=-3x+z and 1=x+w. And so i need to get another approach to pressure that would include density and mass so i could get a relationship between x and w; or maybe a different relationship so i could get a value for z. At this point im stuck, from the Dim. analysis that i wrote in criteria 1(from the book) it seems i need to show that pressure is not dependent on mass.
Thanks in advance
 

Answers and Replies

  • #2
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oh i forgot to say that one solution i came up with was with scaling. I would just say "if x=1 then w is 0. if x=2 then w=-1 and the relationship holds for all value. Thus we just pick x to be 1 and regard mass as unimportant for pressure. Is this a correct way to go about this?
Im assuming not as the numbers would not be the same in different cases of these relationships
 
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  • #3
BruceW
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[M]^1[L]^(-1)[T]^(-1)=[MxL^(-3)]^x [LxT^(-2)]^y [M]^w [L]^z
This is not correct on the left-hand side.
sergey90 said:
and so we get that y = 1 and the equations we are left with are -2=-3x+z and 1=x+w.
But this is correct, so I guess you just typed the equation wrong.

sergey90 said:
oh i forgot to say that one solution i came up with was with scaling. I would just say "if x=1 then w is 0. if x=2 then w=-1 and the relationship holds for all value. Thus we just pick x to be 1 and regard mass as unimportant for pressure. Is this a correct way to go about this?
Im assuming not as the numbers would not be the same in different cases of these relationships
This is basically right. You have a choice of equations which all will be dimensionally correct.
 
  • #4
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so i could just equate the dimension of pressure to a dimensional relationship involving gravitational acceleration, Volume of the water, and the area of the water and still get the dimensions of pressure,without involving the depth z. But this doesn't make sense, since we all know depth does matter. So in the first situation we have a case where the equations are dimensionally correct but the numbers will be different with the mass compared to without the mass; in the second case we have a relationship that doesn't involve depth at all and so im assuming the answer will be different. So how does one know which relationship to choose? This seems like too vague of a method to say the least.
 
  • #5
BruceW
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The idea of dimensional analysis is that it tells you the form of equations you are allowed to make from a collection of physical dimensions (like mass, length, e.t.c.)

So dimensional analysis is not vague. The vague part might be where the physicist has to use his intuition to guess what physical dimensions are important for a given situation.
 

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