If two units have the same dimensions are they always inequivalent?

1. May 24, 2013

alewisGB

If two units have the same dimensions are they always inequivalent? I was thinking, would doing dimensional analysis on units prove they are equivalent or could they have the same units but not be equivalent.

2. May 24, 2013

phinds

what do you mean by equivalent? Are 5 miles and 10 miles equivalent?

3. May 24, 2013

alewisGB

My question was relating to Units only, If two units are in SI and you do dimensional analysis on the two different units that come to the same dimensions does that mean the units are equal?
e.g Resistance Ω Is equvilant as Inductance H * frequency Hz so, A resistor could be measured as 100 H Hz or 100 Ω

4. May 24, 2013

DennisN

Let's take two velocities in the same direction, let's say to the right; vx1 = 5 m/s. vx2 = 10 m/s. Same units. Are the two velocities equivalent?

Let's take two velocities in different directions, one to the right, and one upwards; vx1 = 5 m/s. vy1 = 5 m/s. Same units and the same magnitude, but different directions. Are the two velocities equivalent?

(EDIT: I now saw new replies was posted above)

5. May 24, 2013

mikeph

Yes, but only because of the caveat that they're both SI units, in which case it's trivial.

If you have two units with dimensions of length there is no reason they must be equivalent (1 cm =/= 1 yard). However, if you say both units are SI units then they must both be metres, and they are automatically equal (x metres = y metres means x = y).

6. May 24, 2013

jbriggs444

I take the question as asking, for instance, whether torque and energy are equal since both have units of newton-meters.

7. May 24, 2013

mikeph

I didn't read it as that, because he said "the same dimension", and I understand dimension as "energy" or "torque". In this example, a single unit (Nm) can have two possible different dimensions, but the OP asked: if two units have the same dimension, are the units equal?" In your example, the two units do not have the same dimension.

8. May 24, 2013

f95toli

If I understand the question correctly the answer is no.
Example: The dimensions of surface resistance is ohm. but it is a propery of the material (the thin film), it does not become a "real" resistance until you multiply it with a lenght and divide by the width.

9. May 24, 2013

mikeph

If I understand right, in this example the first quantity you describe is resistivity, a material property, with units Ohm*metre; the second quantity is resistance, with units Ohm (resistivity*length/area).

10. May 24, 2013

bp_psy

In my opinion no. For example it doesn't always make sense to say that s^-1 is the same as Hz. Since you can use s^-1 in more ways than cycles per seconds.

11. May 24, 2013

Hush

Sometimes a ratio/product between two units of the same dimension shared a ratio/product that is dimensionless.
Obviously if such a ratio is constant, then equivalency is established.

The only risk here is misinterpreting the meaning to the question posed.
Different answers to this question imply the question is not well stated.

12. May 24, 2013

phinds

+1 on that and I notice that the OP has made no attempt at clarification so far.

13. Jun 4, 2013

MabAsakura

No, two units with the same dimensions are not always equivalent. Take stress and pressure. Both have units of (Force/Area). However, stress a consequence of a point force being applied to a surface; whilst pressure is a consequence of forces exerting itself evenly in every direction along a surface.

14. Jun 5, 2013

f95toli

Sorry, I did not see this until now.

No, sheet resistince is not the same thing as the resistivity. The latter is a "3D" property of a material, whereas the former is the property of a thin film. The two are related, but the relationship between them is not straightforward since the sheet resistance will depend on the subtrate, film qualiy, how the film grew (if it is epitaxial or polycrystalline etc). Hence, sheet resistance is the propety of a specific film and is not a "universal" material property like resistivity.

Sheet resistance is often written with the "unit" of ohms/sq (sometimes the "sq" is replaced by a square symbol) to differentiate it from "proper" resistance, but "sq" is obviously not a real unit.

15. Jun 5, 2013

stevendaryl

Staff Emeritus
There are cases where quantities "accidentally" turn out to have the same dimensions, even though they don't at all mean the same thing. For example, torque is rate of change of angular momentum. It has the same dimensions as energy: mass distance2 time-2. But they don't seem to be very related. I don't see any way to convert torque into an equivalent amount of energy (but maybe there is such a conversion).