Dimensional analysis (Speed of sound)

[Firben]

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?

 

[Curious3141]

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.

 

[Firben]

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]

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".

 

[Firben]

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)
Radius r L
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 ?