Friday 30 August 2019

forces - Gravitation law paradox for very close objects?


We all know that gravitation force between two small (not heavenly) bodies is negligible. We give a reason that their mass is VERY small. But according to inverse square law, as $r\to 0$, then $F\to \infty$. But in real life we observe that even if we bring two objects very close, no such force is seen.


Why is this so?



Answer



The inverse-square law holds for spherically symmetric objects, but in that case the main problem is that $r$ is the distance between their centers. So "very close" spheres are still quite a bit apart--$r$ would be at least the sum of their radii.


For two spheres of equal density and size just touching each other, the magnitude of the gravitational force between them is $$F = G\frac{M^2}{(2r)^2} = \frac{4}{9}G\pi^2\rho^2r^4\text{,}$$ which definitely does not go to infinity as $r\to 0$ unless the density $\rho$ is increased, but ordinary matter has densities of only up to $\rho \sim 20\,\mathrm{g/cm^3}$ or so.



Tests of Newton's law for small spheres began with the Cavendish experiment, and this paper has a collection of references to more modern $1/r^2$ tests.


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...