# Q & A: units in equations

Q:
How do you calculate field forces when your numbers are less than one? Newton's equation doesn't appear to work when you plug in masses less than 1, since the factor of number less than 1 is less than either number, the opposite of numbers greater than one. Since there is nothing special physically about 1 kg, the equation shouldn't switch its scaling at that point, should it?
- melisa smith (age 19)
taos, NM, USA
A:
One of your remarks is very perceptive: "Since there is nothing special physically about 1 kg, the equation shouldn't switch its scaling at that point, should it?" That's exactly right.
Honestly,  I can't follow the previous questions. So let's just illustrate how this works.

Say that something is accelerating at 5 m/s2 . If this object has a mass of 3 kg, then the force, given by F=ma, is 15 kg m/s2 or 15 Newtons.  What if the mass is instead 0.3 kg? Then you'd get F=1.5 N, by exactly the same method.

So I'm not sure what  the mystery is. But your point that there's nothing special about the particular mass we chose to call 1 kg is exactly right, and also true for our other familiar units.

Mike W.

(published on 03/19/2009)

## Follow-Up #1: multiplying units

Q:
Follow up on previous question: you apply my question to Newton's equation F=ma, but I asked about fields, so I was asking about the field equation F=GMm/r2. If you put two masses less than 1kg in that equation, F scales differently than if you put two masses greater than 1 kg. For instance, if M and m are both greater than 1, then the product Mm is greater than M or m. But if M and m are both less than 1, then product Mm is less than M or m. Doesn't anyone see that as a problem?
- melisa smith
taos, NM, USA
A:
Right, no problem. The product of the masses isn't in the same type of thing as a mass, so it cannot be said to be either larger or smaller than a mass. For example, if you measure the masses in kg and get 0.3 and 0.7 kg, the product is 0.21 kg2.

Think of the area of a rectangle with sides a and b, A=ab. Here it's easy to picture. It makes no sense  to say that A is bigger or smaller than a or b. They're in different units. One's an area, the others are lengths. Say a=b= 0.1 meter. Then A=0.01m2. If you measure in cm, then a=b= 10 cm. Then A=100 cm2. The equations mean exactly the same thing in either set of units.

Mike W.

(published on 03/27/2009)