Dimensional anaylsis and gravitational law

[Shing Ernst]

Pretend that we do not know gravitational law at all, and want to investigate the gravitational law by dimensional analysis:

Let's suppose the gravitational force are proportional to both masses, distance, hence:

$$F \propto m_1^am_2^br_{12}^c$$

But obviously, there is no way to equal the dimensions, since the right side has no dimension of time at all. Making a constant G fitting the dimensions kind of sounds like a cheat to me here. It left me wonder if dimensional analysis fails.

Hence I would like to pose: how do we obtain the gravitational law by dimensional analysis? If impossible, then when, and how dimensional analysis fails?

 

[vanhees71]

I don't understand your problem. $G$ also has a dimension. The correct law is that
$$|\vec{F}| =\frac{G m_1 m_2}{r^2}.$$
The dimensions are
$$\text{N}=\text{kg} \, \text{m}/\text{s}^2=[G] \text{kg}^2/\text{m}^2 \; \Rightarrow \; [G]=\text{m}^3/(\text{kg} \, \text{s}^2).$$
The value is $6.67408(31) \cdot 10^{-11}\, \frac{\text{m}^3}{\text{kg} \, \text{s}^2}$.

 

[A.T.]

how do we obtain the gravitational law by dimensional analysis? If impossible, then when, and how dimensional analysis fails?

Dimensional analysis is not the tool to obtain new physical laws.

 

[BvU]

You mean something like this can make you happy ?

 

[Shing Ernst]

Not exactly...
I mean if we know absolutely nothing about Newton's gravitational law, and we want to find gravitational law by dimensional analysis.
however, this seems impossible to me (as I wrote in my 1st post)

 

[Ibix]

You don't do it by dimensional analysis. You do it by observing the motions of the planets and dropping cannonballs off towers (allegedly).

 

[Shing Ernst]

I don't understand your problem. $G$ also has a dimension. The correct law is that
$$|\vec{F}| =\frac{G m_1 m_2}{r^2}.$$
The dimensions are
$$\text{N}=\text{kg} \, \text{m}/\text{s}^2=[G] \text{kg}^2/\text{m}^2 \; \Rightarrow \; [G]=\text{m}^3/(\text{kg} \, \text{s}^2).$$
The value is $6.67408(31) \cdot 10^{-11}\, \frac{\text{m}^3}{\text{kg} \, \text{s}^2}$.

this is obvious in hindsight. but imagine we live in a time before Newton, and want to figure it out by dimensional analysis - we know nothing about G. While in dimensional analysis, we usually assume no dimensions for the proportional constant.

 

[Shing Ernst]

Dimensional analysis is not the tool to obtain new physical laws.

Would you mind elaborating a bit more?

 

[Shing Ernst]

I actually asked this question here, but got duplicate. however, the other site's answer doesn't satisfy me at all.

 

[Stephen Tashi]

It left me wonder if dimensional analysis fails.

Are there any examples where dimensional analysis succeeds ? - without making some restrictive assumptions about the equation that is to be deduced.

 

[A.T.]

Would you mind elaborating a bit more?

See post #6.

 

[lychette]

I don't understand your problem. $G$ also has a dimension. The correct law is that
$$|\vec{F}| =\frac{G m_1 m_2}{r^2}.$$
The dimensions are
$$\text{N}=\text{kg} \, \text{m}/\text{s}^2=[G] \text{kg}^2/\text{m}^2 \; \Rightarrow \; [G]=\text{m}^3/(\text{kg} \, \text{s}^2).$$
The value is $6.67408(31) \cdot 10^{-11}\, \frac{\text{m}^3}{\text{kg} \, \text{s}^2}$.

I think these are UNITS...not dimensions!

 

[vanhees71]

In a fixed system of units as the here used SI there's a one-to-one correspondence between units and dimensions.

 

[BvU]

The link seems no longer work..

No but a simple search for cantwell dimensional analysis fixes that easily: here

 

[lychette]

In a fixed system of units as the here used SI there's a one-to-one correspondence between units and dimensions.

There may be a one to one correspondence but they are different physics concepts. Why do we use dimensions...M,L, T and C?
If an exam question asks for dimensional analysis, using units would lose marks !

 

[hackhard]

how do we obtain the gravitational law by dimensional analysis? If impossible, then when, and how dimensional analysis fails?

always , how can you derive relations with dimensional analysis.
it is not logical at all

 

[Dadface]

I think dimensional analysis can be helpful. If you suspect there is a relationship between certain variables you can use observations, general knowledge and sometimes even complete guesses to predict what all of the variables may be and how they are related. You can then carry out a dimensional analysis and possibly come up with equations that balance.You can then test the equations experimentally and the experiments can yield the values of any dimensionless constants.

Dimensional analysis has its limitations and it may not work in many if not most cases. But it doesn't take long to carry out and it can give some clues on how to proceed with other methods.