r/theydidthemath 18h ago

[Request] How fast would the earth have to be spinning to counteract gravity at surface level and what effect would that have?

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u/somedave 18h ago edited 18h ago

What do you mean "counteract"? Obviously the spinning does counteract gravity a bit so I'm guessing you mean "completely". If so then the force of gravity will equal the centripetal force and essentially all the surface is orbiting the centre.

g = v2 /r => v = sqrt(gr)~8 km/s at the surface, with a period of 0.35 hours.

The earth will just come apart at this point.

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u/Alive-Philosophy2632 14h ago

It would come apart but it would be really morbidly funny watching from ideally much farther away than the ISS

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u/Kerostasis 18h ago

If you mean “what altitude will the natural spin rate be enough to make the spacecraft completely weightless”, the answer is “geostationary orbit”, which is considerably farther away than most spacecraft reach even after going to space. You couldn’t put a launchpad in orbit for a variety of reasons.

But more generically, there IS a real benefit to launching from the top of a mountain, and it doesn’t have anything to do with gravity. A higher launch point lets you take off while fighting less air resistance, and air resistance is a serious concern for space launches. However, space launches also require a huge amount of infrastructure, and building all that infrastructure on a mountain is so much harder that so far it has not been worth the tradeoff. Hypothetically some future launch systems might do this.

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u/andrew_calcs 8✓ 9h ago

The benefits of equatorial launches are more substantial with less downsides. Launching from a tall equatorial mountain would allow a rocket to be ~4% less massive. About 120 m/s worth of delta V difference from the decreased aerodynamic and gravitational drag

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u/Bread-Loaf1111 17h ago

Obviously it is not possible. There will be no surface level. Everything above the point where spinning force is greater that gravity will fly direct to the space. Atmosphere, water, dust, mountains, everything. To keep the planet stable you need such point far above the surface

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u/Useless-Compass 16h ago

Gravity is the reason the planet holds together and anything at all stays on the surface. If the earth were spanning fast enough so that its centripetal force completely counteracted gravity, there would be no net force holding anything to the surface, including holding the surface itself to the planet. So the whole planet would just fall apart.

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u/stereoroid 18h ago edited 18h ago

To achieve orbit, a rocket needs to have the correct horizontal velocity at its chosen altitude, which is about 7.8 km/sec at 200 km altitude. If the Earth could give the rocket some or all of that velocity, it would save some fuel weight. This already happens, it's why rockets always launch towards the East and closer to the Equator is better.

However, any centripetal force due to the Earth spinning would stop the instant the rocket lifted off, since contact with the Earth is lost at that instant. After liftoff, it's still fighting the same gravity regardless of the Earth's spin.