# Why do most formulas in physics have integer exponents?

## Main Question or Discussion Point

I mean why is f=ma?
why not m0.123a1.43 or some random non integers?
I hope you understand that my doubt doesn't limit just to force or energy or velocity e.t.c.
it also extends to area of a square,circle e.t.c and all other formulae
i think whole thing starts with direct proportionality.

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For the same reason that the area of a square is L2. You're taking the lengths of the sides of a square and multiplying them together, thus the same value multiplied by the same value.

The reason for whole number powers, & whole number coefficients is due to the relationship that is expressed. You need to understand how the formula is created, not simply what the formula looks like.

Furthermore, C = πd, where π is 3.14159..... an irrational number! Furthermore, look at the formula for the volume of a cylinder. A fraction is involved there. But not for mystical reasons, but because it is a way to express the relationships between parts that can be measured.

I think your question lacks depth. Learn why the formulas are what they are. Continue your studies to find situations without whole number coefficients and powers. Check out the universal gravitation equation. The gravitation constant is not a "nice" number.

C = πd, where π is 3.14159..... an irrational number!
I know why it has pi.it's because we chose the constant for area square to be one.If we chose it to be 1 for circle then area of square will have a constant=1/pi

To be correct, the area of a square is directly proportional to the length of the side, but rather, the area is directly proportional to the length of a side squared. That is, to the power of 2.

Based on your comments, I do not believe you understand the relationship between the length of a square's sides and the area of that square.
i know that A=L2 ,but why 2?
and why is it 3 for volume of a cube.
i understand

[Mentor's note: Edited to remove references to some deleted posts]

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To be correct, the area of a square is directly proportional to the length of the side, but rather, the area is directly proportional to the length of a side squared. That is, to the power of 2.

Based on your comments, I do not believe you understand the relationship between the length of a square's sides and the area of that square.
i know that A=L2 ,but why 2?
don't say that it is because L*L=L2

DaveC426913
Gold Member
don't say that it is because L*L=L2

I too am having difficulty with the source of your confusion.

Thanks DaveC! This must be too "deep" for me.

• dushyanth
i mean how did we chose a physical quantity such that it's square is equal to area of square

Thanks DaveC! This must be too "deep" for me.
That was a good one.I can't stop laughing but please stop mocking me

DaveC426913
Gold Member
The definition of a square is that both sides are the same. Instead of saying l*l, we can say l2. We cannot say the same for any other rectangular shape - it would be l*w.

Your question is like asking, why is the colour green the colour green and not the colour purple. Simply!, by definition that is what it is.

The relationship between area of a square, is found by squaring the length of a side. If you don't believe that's true, measure it! Prove it to yourself! There is nothing mystical about this relationship!

DaveC426913
Gold Member
You mentioned proportionality.

Gravitational force is proportional to m/r2. This is only a qualitative relationship.
To show a quantitative relationship, we need to add a constant: 6.677 x10-11.
So, G = g*m/r2.

F=ma is actually an oversimplification. At leas the way I was taught.
Is is more correctly F~m*a. These reads as F is proportional to m*a.
F does not equal m*a unless you use the correct units.

You mentioned proportionality.

Gravitational force is proportional to m/r2. This is only a qualitative relationship.
To show a quantitative relationship, we need to add a constant: 6.677 x10-11.
So, G = g*m/r2.

F=ma is actually an oversimplification. At leas the way I was taught.
Is is more correctly F~m*a. These reads as F is proportional to m*a.
F does not equal m*a unless you use the correct units.
i completely understand that a constant is necessary for that.i have previously mentioned a similar thing on the area of circle above

DaveC426913
Gold Member
OK. Can you reformulate your question a little more descriptively? Pick a particular formula that concerns you and describe why it does not behave the way you expect.

OK. Can you reformulate your question a little more descriptively? Pick a particular formula that concerns you and describe why it does not behave the way you expect.
thank you very much for asking.Lets say f=ma
why does mass and aceeleration have integer powers to them?

DaveC426913
Gold Member
thank you very much for asking.Lets say f=ma
why does mass and aceeleration have integer powers to them?
Put in the simplest way possible: every doubling kilogram of moving mass will double the force it applies when it hits something.

That word "every" means that F always directly and exactly varies as mass varies. And that means the exponent is 1.
If you double the mass, you don't get four times the force applied (m2) or a quarter of the force applied (m.5).

Newton's second law is an axiom. This is one of the starting points necessary to deduce the consequences of classical physics. So, in a sense, the answer to 'why' are the powers in this equation integer powers is that because that is the way Newton formulated it (actually it was Euler who wrote the law algebraically - Newton only used geometric and paragraph proofs).

Your question is similar to asking of Euclid why a straight line segment can be drawn between two points.

every doubling kilogram of moving mass will double the force it applies when it hits something

every doubling kilogram of moving mass will double the force it applies when it hits something why not 1.5 times or some other non integer

That would not change the fact that the exponents are 1! that changes the proportionality constant. Pick a different system of units (or base your units on this) and that can be true.

DaveC426913
Gold Member
every doubling kilogram of moving mass will double the force it applies when it hits something why not 1.5 times or some other non integer
Does this make any sense though?

Say you had a one kg cannonball in a bag, and swinging created a force of f.
Then you added a second one kg cannonball to the bag. Would you exepct that the second ball (which is, in every way identical to the first) would only add .5f?

Does this make any sense though?

Say you had a one kg cannonball in a bag, and swinging created a force of f.
Then you added a second one kg cannonball to the bag. Would you exepct that the second ball (which is, in every way identical to the first) would only add .5f?
can i apply the same to velocity in k.e=mv2/2

That would not change the fact that the exponents are 1! that changes the proportionality constant.
no.

DaveC426913
Gold Member
can i apply the same to velocity in k.e=mv2/2
Hey. Don't change the subject. ;) Does 1.5 make sense or no?

DaveC426913
Gold Member
That would not change the fact that the exponents are 1! that changes the proportionality constant. Pick a different system of units (or base your units on this) and that can be true.
bpu: dushyanth is correct on this one. He is asking about m1.5, which is an exponential change, not a proportional change. You're describing F=1.5ma