Dimensionless Equations: Understanding the Role of Units in Physical Equations

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In summary, the problem is that when we substitute 1kg for 1000g in an equation, the equations always works correctly, but the result is wrong. The reason is that the algebraic symbols are considered to be divided by the unit, even though they are actually multiplied. This issue is resolved by understanding that all physical equations must ultimately be numerical, and that numbers remain numbers even when they are expressed in different units.
  • #1
puzzler7
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I have a problem that might sound simple, but has been bugging me for months. In a physical equation, the units are regarded as multipliers - so to take a very simple example in SI:

1) F[N]=M[kg]a[m/s2]

And, of course, [N] is equivalent to [kg][m/s2], so all is well.

Here's my problem: let's say I want to adjust the equation, so that my mass measurements are in grams [g] rather than [kg].

Direct substitution for 1kg = 1000g into equation 1) gives:

2) F(N)=1000M[g]a[m/s2]

Which is clearly incorrect.

(a mass of 1g accelerated at 1 m/s2 would compute a force of 1000N - wrong - The equation actually needs to be divided by 1000 on the RHS.)

The logic looks perfect - but the result is wrong.

The problem is resolved in *all* equations by regarding the algebraic symbols to be *divided* by the unit - so why do we consider them to be multiplied?

What's my problem!?
 
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  • #2
Hi puzzler. Numerically the mass in kg is 1/1000-th of the mass in grams.

So the correct substitution is F[N] = M[grams]/1000 a[m/s^2]
 
  • #3
uart said:
Hi puzzler. Numerically the mass in kg is 1/1000-th of the mass in grams.

But this doesn't mean M(kg) = M(g)/1000 Because that means 1000kg = 1g

Do you see the problem? In this verbal equation, *in* must stand for divide by - not multiply.

If we apply the rule that *in* stands for divide, the verbal equation now works:

M/kg = (1/000) M/g

This is consistent. But it brings us back to the original point - numbers in physical equations are divided by their units - not multiplied.

Maybe it looks like this: M(kg)/[kg]

The parentheses give the expected unit: the brackets the divisor. This makes the whole equation dimensionless.

Now I can substitute [1kg] = [1000g], and everything will work as expected.

Can anyone expand - it's a worry!

(here's a thought: M of kg = 1000 x M of g. But M in Kg = (1/000) M in g. Note the difference between 'of' multiply, and 'in' divide'. I've been doing physics for 20 years - and I'm suddenly puzzled!)
 
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  • #4
I think I've solved my own problem: but I would appreciate all comments and criticism:

It seems that, maybe, all physical equations must, ultimately, be numerical, and therefore - ultimately dimensionless.

I suggest my mistake is in thinking that the M, in F=Ma, is fixed - as the unit quantities move around it.

(note that: F=Ma is just a simple example - the rules are expected to apply to all equations)


Heres a solution: If M is not fixed, we can have M1, M2 - two versions:

if M1(kg) = M2(g)

Now we can write:

1000M1(g) = M2(g)

Hence M1=M2/1000

If we substitute this into the equation we get the desired result (see above)

The conclusion *must be* that all physical equations - even if they appear to have units - are ultimately dimensionless.

We make them dimensionless by correct choice of units.

If we change those units - we must allow the dimensionless equation to adapt to the units (not vice versa)

Ultimately, even in physical equations, numbers = numbers.

Please feel free to argue and discuss.
 
  • #5


First of all, it's great that you have recognized this issue and are seeking a deeper understanding of it. This is a common confusion for many students and even experienced scientists. Let me try to explain the role of units in physical equations.

Dimensionless equations, or equations without units, are often used in physics because they allow us to simplify complex equations and make them easier to solve. However, these equations are not always accurate and can lead to incorrect results if not used properly.

In your example, when you substituted 1kg with 1000g, you essentially changed the units of the mass without adjusting the units of the other variables. This is why the result was incorrect. In order for the equation to remain accurate, all units must be consistent and balanced on both sides of the equation.

So, why do we consider units to be multiplied rather than divided? This is because units represent physical quantities and they behave like numbers in algebraic equations. Just like how numbers are multiplied and divided in equations, units are also multiplied and divided to maintain balance and consistency.

In your example, the correct way to adjust the equation would be to divide both sides by 1000, which would give you:

F[N]/1000 = M[g]a[m/s2]

This ensures that the units are consistent and the equation remains accurate.

In summary, units in physical equations are not just multipliers, but they are also important indicators of the physical quantities involved. It is crucial to understand their role and use them correctly in equations to avoid incorrect results. I hope this explanation helps to clear your confusion. Keep exploring and questioning, that's what science is all about!
 

1. What are dimensionless equations?

Dimensionless equations are mathematical expressions that do not contain any units. They are used to describe relationships between physical quantities without being affected by the units used to measure those quantities.

2. Why are dimensionless equations important in science?

Dimensionless equations are important because they allow us to compare and analyze physical phenomena without the influence of different units. This helps to simplify complex equations and make them easier to understand and apply in different contexts.

3. How do units affect physical equations?

Units affect physical equations by determining the scale and magnitude of the quantities involved. Units also dictate the relationships between different physical quantities, which is why converting between units is essential in scientific calculations.

4. Can dimensionless equations be used for all physical phenomena?

No, dimensionless equations may not be applicable to all physical phenomena. Certain phenomena may be better described and understood using equations with units, especially when the scale and magnitude of the quantities involved are important.

5. How can understanding dimensionless equations help in problem-solving?

Understanding dimensionless equations can help in problem-solving by providing a deeper understanding of the underlying relationships between physical quantities and how they interact with each other. This can help in making more accurate predictions and finding solutions to complex problems in science and engineering.

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