# Dimensional analysis (Speed of sound)

Speed of sound

The speed of sound c in a gas depends on among other things on the pressure on the gas, the density and probably, possibly on their viscosity. Determine c-

My Variable list:

Pressure p ML^-1T^-2
Density ρ ML^-3
Speed v LT^-1

My matrix:

| | M | L | T |
| p | 1 | -1 | -2 |
| ρ | 1 | -3 | 0 |
| v | 0 | 1 | -1 |

after a couple of row reductions i got it to be:

k = P/(ρv^2) (pi=k)(k = constant)

The answer should be v = k(P/ρ)^(1/2)

I'm doing it right?

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Curious3141
Homework Helper
Speed of sound

The speed of sound c in a gas depends on among other things on the pressure on the gas, the density and probably, possibly on their viscosity. Determine c-

My Variable list:

Pressure p ML^-1T^-2
Density ρ ML^-3
Speed v LT^-1

My matrix:

| | M | L | T |
| p | 1 | -1 | -2 |
| ρ | 1 | -3 | 0 |
| v | 0 | 1 | -1 |

after a couple of row reductions i got it to be:

k = P/(ρv^2) (pi=k)(k = constant)

The answer should be v = k(P/ρ)^(1/2)

I'm doing it right?
Your answer is essentially equivalent, except you've not rearranged it to get the velocity on the LHS, as required. Remember that k is just an arbitrary dimensionless constant, if you bring it to the other side and have to take the reciprocal, just replace it with another arbitrary dimensionless constant.

But then i will get:
kv^2=P/ρ => kv=√(P/ρ) =>v = (√(P/ρ))/k which is not equal to v = k√(P/ρ)

Curious3141
Homework Helper
But then i will get:
kv^2=P/ρ => kv=√(P/ρ) =>v = (√(P/ρ))/k which is not equal to v = k√(P/ρ)
That's why I said k doesn't really "matter" - it's just a dimensionless constant. Just replace (1/k) in your expression with K (another constant). Even better, *start* with K in your derivation, and then replace 1/K with k, to get the exact same expression as the expected solution.

Remember, k (or K) is just an arbitrary dimensionless constant. If you move it around, reciprocate it, square it, etc., just replace it with another constant in the final expression to make it "neat".

Torsion

when a homogeneous circular rod is subjected to a torsional moment M it will deform. (The rod is fastened at the bottom). A measure of deformation is the angle θ with which the upper end is rotated.

Determine a relation between θ and M and the other quantities

My variable list:

Mach number M 1 (dimensionless)
Shear modulus G ML^-1T^-2
Angle θ 1 (dimensionless)
Height s L

My matrix

| |M|L|T|
|M|0|0|0|
|G|1|-1|-2|
|θ|0|0|0|
|r|0|1|0|
|s|0|1|0|

It should be θ = Φ((M/Gr^3), s/r)

Ss there something missing here ?