Does the unit of a quantity change if you take square root of it?

Question

For example, I have a mass, m = 0.1kg and I square root it, giving me m = 0.316 (3s.f.) does the unit still stay as kg, or does it change?

Ilmari Karonen · Accepted Answer

As the other answers (and dmckee's comments) note, yes, if you take the square root of a dimensional quantity then you need to take the square root of the units too:

$$ sqrt{4;{rm kg}} = 2;{rm kg}^{frac12} $$

And no, I can't think of any meaningful physical interpretation for the unit ${rm kg}^{frac12}$ either.

However, in the comments you say that you were "told to plot a graph of distance against square root of mass."  What that means is simply that you should scale the mass axis non-linearly, presumably in order to more clearly show the relationship between the two quantities.  For labeling the mass axis, you basically have two choices:

label the axis $sqrt m$, with equally spaced ticks at, say, $1;{rm kg}^{frac12}, 2;{rm kg}^{frac12}, 3;{rm kg}^{frac12}, 4;{rm kg}^{frac12}, dotsc$, or
label the axis $m$, with equally spaced ticks at $1;{rm kg}, 4;{rm kg}, 9;{rm kg}, 16;{rm kg}, dotsc$.

While, technically, both of these are valid, I would strongly recommend the latter option.  Just compare these two plots and see which one you find easier to read:

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Alas, not all plotting software necessarily supports such axis labeling, or at least doesn't make it easy, which is why you sometimes see plots with funny units like ${rm kg}^{frac12}$.

Jerry Schirmer · Answer

It becomes the square root of the unit.  Think of energy:

$$E = frac{1}{2}mv^{2}$$

If I solve for $v$, I have $v = sqrt{frac{2E}{m}}$.  Since $rm 1 J = 1 kg cdot m^{2}/s^{2}$, we see that the units have to obey the square root, or we will end up with our velocity equalling something other than m/s.

innisfree · Answer

Yes, the dimension of a quantity changes if it is square-rooted. If $m$ is a mass with dimension $[m]=textrm{kg}$, $sqrt{m}$ is not a mass, but another quantity with dimension $[sqrt{m}] = textrm{kg}^{1/2}$.

More generally, if $[a] = A$ and if $[b]=B$, then $[a^n b^m] = A^nB^m$ etc.

user13267 · Answer

Take the root of the unit of area (Eg: 4 m$^2$ )
We get the unit of length (Eg: 2 m)  which is the unit for different physical quantity
So it definitely changes

Petr · Answer

Let's square root 0.1kg:

expressed in kg, we get $sqrt{0.1}approx 0.316$.
expressed in g, we get $sqrt{100}=10$.

So obviously the unit changes. If it stayed the same, we'd have $0.316mbox{kg} = 10mbox{g}$ which is clearly false.

Does the unit of a quantity change if you take square root of it?

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