TITLE: Rotating metal sphere
QUESTION [2 upvotes]: My knowledge of physics is not that great, but I am always willing to learn. 
I have the following question: 
If we have a sphere made out of metal with a radius of 1 meter, and it rotates in an isolated system where there is vacuum and no external gravity applicable, will the molecules fall apart at some certain speed of rotation because the centrifugal force it is greater than the forces keeping the metal structure bonds and if there is a way to calculate it, how would you do it, and what formulas/theorems should be used?

REPLY [1 votes]: When an object is spinning, there are indeed stresses developing that keep the object from ripping apart. But every material has a finite strength (the yield stress), and when you exceed that it will deform plastically (it will not go back to its old form after you remove the stress), and may even break (when you reach the ultimate tensile strength).
If you have a simple ring with radius r and cross sectional area a, you can think of the stresses on this ring with the following diagram (from http://www.tectonic-forces.org/images/fig08.gif):

For a segment subtending a small angle $\delta \phi$, the mass is $m = \rho \cdot a \cdot r \cdot \delta \phi$ and the force $F$ required to provide the centripetal force is calculated from $F\sin(\delta\phi)=m\omega^2 r$. Using the above expression for mass, and noting that for small angles, $\sin\phi = \phi$, we find for the stress
$$\sigma = \frac{F}{a} = r^2\omega^2\rho$$
Substituting values for steel: 
ultimate tensile strength = $5\cdot 10^8 ~\rm{N/m^2}$
density = $8\cdot 10^3~\rm{ kg/m^3}$
r = $1~\rm{ m}$
This gives us an estimate for the speed of rotation $\omega$:
$$\omega = \sqrt{\frac{\sigma}{r^2\rho}} = 250 ~\rm{rad/sec}\approx 2400~\rm{ rpm}$$
Note that this calculation was done for a ring, not a sphere - and in fact the shape of a sphere probably makes the above a lower limit on the speed it can sustain, as most of the sphere has a perpendicular distance less than $r$ from the axis of rotation, and so will act to "help support" the material that is furthest away ("on the equator" of the sphere). Another way to look at it - the curvature of a sphere provides "supporting force" in two perpendicular directions, so probably this gives exactly a factor 2x in strength ($\sqrt{2}$ increase in angular velocity that can be sustained).
You can find more equations and derivations at http://www.ewp.rpi.edu/hartford/~sarric/SMS/Readings/32669_04.pdf - but this should give you a starting point.
{"set_name": "stack_exchange", "score": 2, "question_id": 141312}