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I have imagined the Earth as a perfect sphere with uniform mass density and I put an object somewhere between the equator and the north pole at rest with respect to earth. And also in my imaginary world, there is no air resistance.

When I draw the free body diagram of the object including the pseudo force, I cannot see how these three forces will cancel.

Would we need force of friction in order to be able to stand still in my imaginary world?

After @Fiatlux's answer, I've realized that I may even have discovered for myself the fundamental reason behind Earth's real shape accidentally.

I have another thing that's bugging me now. Let say we create a perfect frictionless horizontal plane and we put an object there. Is there a chance it might start moving depending on where we conduct the experiment or the Earth's shape and density distribution forbid this to happen anywhere on Earth?

Emilio Pisanty
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physicsguy19
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    Related: https://physics.stackexchange.com/q/8074/2451 , https://physics.stackexchange.com/q/69562/2451 , https://physics.stackexchange.com/q/9751/2451 , https://physics.stackexchange.com/q/44931/2451 and links therein. – Qmechanic Sep 18 '18 at 12:55
  • This is why I love StackExchange! – Kevin Sep 18 '18 at 16:45

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Indeed, you need friction. Otherwise this object will slide to equator just as you expect.

In fact, all the objects will slide there to form equatorial bulge and eventually the sphere will be turned into new shape with surface orthogonal to $\vec{g}+\omega^2 \vec{r}$ at each point. That is exactly what happens to planets in reality while they are still liquid.

FiatLux
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    You can actually see this effect on Earth's oceans - there is an equatorial bulge created by the Earth's rotation. – probably_someone Sep 18 '18 at 15:58
  • @probably_someone is this not also the coriolis effect, that causes weather? – AncientSwordRage Sep 18 '18 at 16:50
  • @Pureferret Technically, no, but both effects are caused by the rotation. The coriolis effect describes motion relative to the surface of a rotating object; this answer describes the formation of the surface of the rotating object. – Sam Sep 18 '18 at 17:39
  • @Pureferret: Indeed, it is all related. For example: the same effect which causes the oceans to bulge at the equator also makes the atmosphere taller at the equator. Which means that there is less air pressure at the poles (this is the "polar vortex"). But the reason that there is a bulge at all is because the earth is spinning, which means that it is spinning beneath those regions of low pressure, which makes them from our perspective appear to spin in the opposite direction, and we call that observed rotation the Coriolis effect. – Eric Lippert Sep 18 '18 at 18:47
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    The relationship between the Coriolis effect and the equatorial bulge is that while material is moving toward the Equator, its path across the surface will curve instead of "falling" toward the Equator in a straight line. – Beanluc Sep 18 '18 at 20:06