physics dimensional consistency?
the radius of a circle inscribed in any triangle whose sides are a,b, and c is given by the following equation , in which s ia an abbreviation for (a+b+c)/2. Check this formula for dimensional consistency.
- Dr SpockLv 61 decade agoFavorite Answer
The previous responder is actually answering a DIFFERENT question, namely one about the AREA of a triangle, with dimensions L^2!
In YOUR question, the LHS (r) is a RADIUS, and thus has the dimension of a (single) length, that is to say L.
Meanwhile the RHS, properly written as √[(s-a)(s-b)(s-c)/s], to make it clear that the ' √ ' sign operates on EVERYTHING to its right, has dimension(s) of √[L*L*L / L] = √[L^2] = L.
So the equation is dimensionally consistent.
THAT'S what you wanted to know. QED
Live long and prosper.
P.S. Do NOT make the mistake of confusing the question of the DIMENSIONALITY of something (the combinations of the different dimensional essences L, M and T involved in it) with the UNITS that you choose to assign to those dimensionalities.
This seems to be a very common error in YA! answers, which may well represent widespread confusion among today's teachers. One of the main reasons for using dimensional arguments is that they get you away from using a SPECIFIC choice of units! The results that you obtain from a dimensional argument will be true in ANY unit system. And from simply a writing point of view, it is so much more EFFICIENT, CLEANER and CLEARER to write 'L' instead of 'meters' or M instead of 'kg' etc. Just notice how the single CAPITAL letters jump out at your eye instead of all the clutter of multiple, repeated lower case letters.
- Anonymous4 years ago
To check, you work out the units of both sides and see if they match. You might have to 'juggle' the units and use standard relationships such as: Force = ma, so the newton (N) is the same as 1kg x m/s² = 1kg.m/s². Work = F x d, so the joule (J) is the same as 1N x 1m = 1kg.m/s² x 1m = 1kg.m²/s². Examples: Here is a formula for the kinetic energy of moving object. Is it dimensionally consistent? Kinetic energy = 4mv³ Left side units = J = kg.m²/s² Right side units = kg.(m/s)³ = kg.m³/s³ (ignore the '4' - it has no units) You can see the left and right sides have different units; the equation is not dimensionally consistent – it must be incorrect. Here is another formula for the kinetic energy of moving object. Is it dimensionally consistent? Kinetic energy = 4mv² Left side units = J = kg.m²/s² Right side units = kg.(m/s)² = kg.m²/s² (ignore the '4' - it has no units) You can see the left and right sides have the SAME units; the equation IS dimensionally consistent - it MIGHT be correct. But in fact it is incorrect because the '4' should be ‘½'. The correct formula is: Kinetic energy = ½mv² (which is, of course, dimensionally consistent).
- 5 years ago
how do we check an equation is dimensional consistence
- 1 decade ago
Heron's Formula(for triangle with sides a, b, c):
area=sqrt( s(s-a)(s-b)(s-c) ) where s=(a+b+c)/2
meters^2=sqrt( meters*meters*meters*meters )
meters^2=sqrt( meters^4 )