Is there a limit to how fast a physical object can spin?

Light speed is the current limit.

The integrity of a spinning object will be lost when centrifugal forces on its mass reaches a critical point.

I was watching a YouTube video just the other day, about black holes and neutron stars and the other crazy things that happen to stars when they run out of fuel. I was shocked to hear that some neutron stars spin at a billion rounds per second. I couldn’t resist doing some math to figure out the maximum size of something spinning that fast. It’s not hard to figure out. Have a look:

Imagine you’re standing on the surface of a spherical object (star, black hole, whatever) that’s spinning at a billion revolutions per second. Light speed is roughly 300 million meters per second. You can’t move any faster than that, so the maximum size of the object you’re standing on is the size that would make you move at light speed. The math is very straightforward. First, how many meters are you moving on every revolution?

L (light speed) = 3×10^8 meters per second
U (revolutions) = 1×10^9 revolutions per second

To get meters per revolution, that is, how far you would move with each revolution, or in other words, the circumference of the neutron star, just take L / U. In this case, you would be moving 3×10^-1 meters per revolution, that is, 300 millimeters. About 12 inches, roughly the length of your forearm. Get a piece of string about that length, and make a circle with it. That’s the maximum size of an object spinning at a billion revolutions per second. How far across is that circle? That is, what is its diameter ( d )? From school, we know that circumference ( c ) is πd, and we have our circumference, L / U. So if we want the diameter of your circle, we take circumference divided by π. That is, d = (L / U) / π. If you want the radius ( r ), just take half, that is, r = (L / U) / 2π.

The radius of your string-circle is L = ^3×108 m/s divided by U = ^1×109 m/rev divided by 2π.

(3×10^{8 / 1×10}9 ) / 2π = about 50mm, or about 2 inches. Your 12-inch string gives you a circle that’s about 4 inches across.

To figure out the general case, that is, the maximum size based on how fast it’s spinning, just do this:

r = (3×10^8 / U) / 2π

Plug in U for the number of revolutions per second, and r will be the maximum radius of your object. But it’s probably more intuitively grasped if you calculate the length of your string, so you can arrange it in a circle. For that, go back to the easy case above: L / U is the length of your string, in meters.

Can you imagine? A thing out there in space about the size of a smallish cantaloupe, spinning at a billion times per second, spewing out all kinds of radiation ripping anything to shreds that comes anywhere near it, weighing…well, let’s figure that out too. The volume of a sphere is 4πr^3 / 3. We can plug in our radius, 50 millimeters, and we find the volume is about 520,000 cubic millimeters, which is a silly and meaningless number. It’s about 32 cubic inches. About 32 little boxes, each 1 inch on a side.

Neutron stars are crazy. Their density is about 5000kg per cubic meter. We’ll use our silly number again, and convert it to cubic meters, that is, about 5.2×10^-4.

5×10^{3 / 5.2×10}-4 = 9615384.62kg, which is about 21 billion pounds, or about 10 million tons, or about the weight of ten Nimitz class aircraft carriers. The size of a cantaloupe. Each of the 32 little 1-inch cubes would weigh 312,000 tons, that is, three of them would weigh as much as a Nimitz.

That is, if I’ve done the math right.

Would you get dizzy though?

Precisely balanced, no. Turbochargers of Diesel engines can hit around 100,000 RPM in a hard pull. They can fly apart at that point because of the balance problem.

I heard that if you kick a scotsman in the sporran hard enough his testicles will spin indefinitely.

@SaganRitual Wow! That is great! I wish I could give you more than 1 GA! You worked out size, mass, density from that one bit of info!

High powered rifle bullets spin at 150,000 to 250,000 rpm. The spin is determined by the muzzle velocity and the twist rate of the gun barrel. If you are pushing the bullet muzzle velocity near 3000 ft/sec through a barrel with a twist rate of 1 turn per 16 inches you will occasionally spin it so quickly the bullet will self destruct and fly apart as it exits the barrel. It is possible to shoot at a target only 10 meters away and not hit the paper since the bullet fragments and sprays in all directions.

@SaganRitual – I think your stellar mass calculations came out far too low.

’ ’... a teaspoon of neutron star material would weigh around a billion tonnes… ’ ’

If a bullet was driven at such a high velocity out of a very twisted rifled barrel, it’s mass/speed might render it unable to catch the spin of the rifling, and thus render it just a swiftly speeding slug with next to no spin at all by ripping the rifling grooves right off the slug itself. ?
Perhaps, perhaps…

@Zaku Ack! You’re totally right. I read 5×10^{3 kg/m3 (which is Earth’s density), instead of 10}17 kg/m3 for neutron stars! So, here’s the correction:

I had the volume figures right (I hope!):

The volume of a sphere is 4πr^3 / 3. We can plug in our radius, 50 millimeters, and we find the volume is about 520,000 cubic millimeters, which is a silly and meaningless number. It’s about 32 cubic inches. About 32 little boxes, each 1 inch on a side.

Now the part that wasn’t right, the density of a neutron star:

Neutron stars are crazy. Their density is about 10^{17kg per cubic meter. We’ll use our silly number again, and convert it to cubic meters, that is, about 5.2×10}-4 cubic meters.

10^{17 / 5.2×10}-4 = about 1.9×10^20, which is a ridiculously larger number, as @Zaku pointed out. That is, 192,307,692,307,692,307,692 kilograms, or 424,000,000,000,000,000,000 pounds, or 212,000,000,000,000,000 tons, or:

212,000,000,000 Nimitz class aircraft carriers. That’s still a hard number to imagine: 212 billion aircraft carriers! How much does each of our 32 little boxes weigh? 6,625,000,000,000,000 tons, or 6,625,000,000—six billion aircraft carriers.

Still a huge number. How’s this: that’s enough to give a Nimitz class carrier to 85% of the people alive in the world right now. I don’t know where we’re going to put them all.

A little one-inch cube of neutron star material weighs enough for most of the people in the world to have a tiny piece that weighs as much as a million ton aircraft carrier. (If I’ve done the math right, and got the right figures!)

@kritiper It definitely spins and disintegrates. Ideally you tune twist rate and muzzle velocity to keep the stresses just below the bullet’s design parameters. The manufacturer supplies the max stress info.
Bullet RPM Calculator
“If you spin your bullets too fast, this heats up the jackets and also increases the centrifugal force acting on the jacket, pulling it outward. The combination of heat, friction, and centrifugal force can cause jacket failure and bullet “blow-ups” if you spin your bullets too fast.”

Video

@SaganRitual, The math is beyond my skill-set. I assume your calculations are based upon common gravity. Would Dark Matter gravity be applicable?

For a number of years, as I understand it, dragsters couldn’t go beyond 300 MPH because, even though the tire beads were reinforced with steel wire, the tires would come off the rims. Racers used screws and bolts to fix the problem, if only somewhat.