# I Dimensional anaylsis and gravitational law

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1. Nov 8, 2016

### 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?

Last edited: Nov 8, 2016
2. Nov 8, 2016

### 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 $6.67408(31) \cdot 10^{-11}\, \frac{\text{m}^3}{\text{kg} \, \text{s}^2}$.

3. Nov 8, 2016

### A.T.

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

4. Nov 8, 2016

### BvU

You mean something like this can make you happy ?

5. Nov 8, 2016

### 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)

6. Nov 8, 2016

### 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).

7. Nov 8, 2016

### Shing Ernst

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.

8. Nov 8, 2016

### Shing Ernst

Would you mind elaborating a bit more?

9. Nov 8, 2016

### Shing Ernst

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

10. Nov 8, 2016

### Stephen Tashi

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

11. Nov 8, 2016

### A.T.

See post #6.

12. Nov 8, 2016

### lychette

I think these are UNITS....not dimensions!

13. Nov 8, 2016

### vanhees71

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

14. Nov 8, 2016

### BvU

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

15. Nov 8, 2016

### lychette

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 !!

16. Nov 26, 2016

### hackhard

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

17. Nov 26, 2016