Dimensional Analysis: Matrix Setup for M/(L^2T^2)

[princejan7]

Homework Statement

How do I set up a matrix to find the combination of ( M(L^2)/T ) and I ( L^4)
that results in units of M/ ( L^2 T^2 ) ?

Homework Equations

The Attempt at a Solution

I think it looks something like
[ 1 0
2 4
-2 0 ]
*
[ a1, a2, a3]
=
[1 -2 -2]

but the dimensions of those matrices aren't right

 

[jedishrfu]

Given:

Y = M * X

Are you trying to get a vector Y with units of measure of M / ( L^2 T^2 ) from a vector X with units of measure M(L^2)/T multiplied with matrix M? or is this a dot product?

 

[Stephen Tashi]

Homework Statement

How do I set up a matrix to find the combination of ( M(L^2)/T ) and I ( L^4)
that results in units of M/ ( L^2 T^2 ) ?

That statement of the problem isn't clear. (What would "a combination" mean in this context? ) Try stating the problem as it is actually worded.

I think it looks something like

Matrices can't be reliably displayed using ordinary typing. You can resort to LaTex https://www.physicsforums.com/help/latexhelp/ In the meantime, it might be better to use notation like [1,2,-2]^T to denote a column vector.

To have valid multiplication In your work you'd have to multiply on the left by the row vector:

$\begin{bmatrix}a_1&a_2&a_3 \end{bmatrix} \begin{bmatrix}1&0\\2&4\\-2&0 \end{bmatrix} = \begin{bmatrix} 1\\-2\\-2 \end{bmatrix}$

but I don't know if that equation is appropriate, because I don't know what problem you are solving.

Are you trying work a problem similar to the examples shown in the Wikipedia article http://en.wikipedia.org/wiki/Buckingham_π_theorem ?

 

[haruspex]

jedishfru, Steven,

princejan7 is trying to solve $\left(\frac {ML^2}T \right)^{a_1}\left(L^4\right)^{a_2} = \frac M{L^2T^2}$. This leads to the matrix equation shown...
except there is no a3, the a1, a2 should be a column vector, and either the problem has been stated incorrectly or the -2 at lower left of the matrix should be -1.
$\begin{bmatrix}1&0\\2&4\\-1&0 \end{bmatrix} \begin{bmatrix}a_1&a_2 \end{bmatrix}^T= \begin{bmatrix} 1\\-2\\-2 \end{bmatrix}$
Note: there is no solution. princejan7 , is there perhaps some third input parameter? Or is the -2 right in the matrix (which would permit a solution)?

 

[HalfLostAndSearching]

I would suggest using dimensional analysis to solve this problem. Dimensional analysis is a method used to check the consistency of equations or to convert between different units. In this case, we are trying to find the combination of ( M(L^2)/T ) and I ( L^4) that results in units of M/ ( L^2 T^2 ).

First, we need to assign variables to represent each unit. Let's use M for mass, L for length, and T for time. Then we can write the given units as (M(L^2)/T) and I(L^4). We want to find a combination of these units that results in M/(L^2 T^2).

Next, we can use the principle of dimensional homogeneity, which states that the dimensions on both sides of an equation must be equal. In this case, the dimensions of the left side are M*(L^2)/T and L^4, which can be simplified to M*L^2/T and L^4. On the right side, the dimension is M/(L^2 T^2). In order for these dimensions to be equal, we need to have L^4 on the left side, which means we will need to raise L to the fourth power in our combination.

Now, we can set up a matrix with the variables for each unit and their corresponding powers. It will look like this:

[1 2 0]
[a1 a2 a3]
[0 0 1]

The first row corresponds to the power of M, the second row corresponds to the power of L, and the third row corresponds to the power of T. The first column corresponds to the power of (M(L^2)/T), the second column corresponds to the power of I, and the third column corresponds to the power of (L^4).

We can then solve for the coefficients a1, a2, and a3 by setting the matrix equal to [1 -2 -2] and using matrix algebra to find the values.

Therefore, the combination of (M(L^2)/T) and I(L^4) that results in M/(L^2 T^2) is a1(M(L^2)/T) + a2(I) + a3(L^4).