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Let's imagine a person standing somewhere on Earth, but not on the equator, i.e. somewhere with a positive net value of latitude. Since the Earth spins around its axis and the person spins along, the sum of forces acting on them should give us the centripetal force: enter image description here

The only forces acting on the person are the gravitational force $F_g$ and the force exerted on the person by the ground, $N$. The gravitational force is directed into the centre of Earth: enter image description here

That means, that the other force $N$ should look like this: enter image description here

So now, when we "zoom" and look at our person, the situation looks like that: enter image description here

Which is rather peculiar, since the reaction force of the ground is usually a normal force, perpendicular to the ground. Is my deduction correct, and if yes, isn't the fact that $N$ is not perpendicular to the ground something we should consider while solving problems (for example problems related to friction, which is proportional to the normal force)?

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neverneve
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  • Related: http://physics.stackexchange.com/q/8074/2451 , http://physics.stackexchange.com/q/9751/2451 , http://physics.stackexchange.com/q/69562/2451 , and links therein. – Qmechanic Feb 23 '14 at 20:16
  • Isn't Normal force defined to be perpendicular to the surface? – mehfoos Feb 23 '14 at 21:26
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    You're assuming that the Earth is perfectly spherical. That's not true. – Peter Shor Feb 24 '14 at 02:45
  • Yes, the normal force by definition is perpendicular, which leaves the question, which garyp tried to answer - from where does the centripetal acceleration come. And Peter, you're right that the shape of the Earth might be adding something into equation, but the question remains the same even for a perfectly spherical object which just started rotating (so it hasn't had the chance to change its shape). The centripetal acc. has to be caused by something. – neverneve Feb 24 '14 at 14:44

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Interesting question. Normal forces are normal: perpendicular to the surface. So where does the centripetal acceleration come from? The only place I can see it coming from is friction between the earth and my shoes.

A simple calculation shows that the centripetal acceleration at mid latitudes is about 20 mm/sec${}^2$. Consider a hanging plumb bob. The tension in the string must play two roles: balancing the gravitational force, and applying the centripetal force. So the plumb bob does not point to the center of the (ideal, rigid, solid, spherical) earth, but away from the center by about 0.002 radians (about 0.1 degrees) at mid latitudes, and negligible for most purposes.

One thing disturbs me about this analysis. The force of friction on my shoes would have to be about (75 kg)*(20 mm/sec${}^2)$ or 1.5 N. I would think that that magnitude of force would be noticeable in certain situations. The force on a 0.1 kg hockey puck or air table puck would be about 0.002 N, and perhaps not noticeable.

garyp
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  • Well, 1.5N acting on you is as if you were,moving on a surface with the coefficient of friction equal to $2 \cdot 10^{-3}$, so maybe it really is negligible. – neverneve Feb 24 '14 at 14:54