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I am trying to figure out if I am right about this(picture at the bottom):

Let's say we have mass M hanging from a string above the surface of the earth, creating θ degrees between it and the north pole.

The way I see it, all forces applied are : The gravitational force towards the center of the earth which has a component acting as the centripetal force towards where the axis of rotation of the earth passing at that latitude (Mgsinθ).
Then we have the suspension force T (please correct me if this is not the way it's called in English) which has a component that together with the centrifugal fictitious force balance out the centrifugal force. The other component of T balances out (Mgcosθ).

My problem with this is that if Tcosθ = Mgcosθ ==> T=Mg Which means that F(centrifugal)+Tsinθ>Mgsinθ. So what happens to F(centrifugal)? What am I missing?enter image description here

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For a spherical etc Earth the force diagram for the mass on the end of the string $m$ looks like this with $\vec F_{\rm cp} = \vec F_{\rm g}- \vec T $ where $\vec F_{\rm g}$ is the gravitational attraction on the mass due to the Earth, $\vec F_{\rm cp}$ is the force causing the centripetal acceleration of the mass $mr\omega^2$ and $-\vec T$ is the tension in the string.

You can think of $\vec F_{\rm cp}$ and $\vec T$ as the two componets of the force $\vec F_{\rm g}$.

enter image description here

You will see from this diagram that the string direction is not directly towards the centre of the Earth.

If you want to make it a statics problem in the rotating frame of the Earth then all you need to do is add a force $ \vec F _{\rm cf} =-\vec F_{\rm cp} $ to the diagram where $\vec F_{\rm cf}$ is the centrifugal force and then $\vec F_{\rm g} - \vec T + \vec F_{\rm cf}=0$.

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If $\theta_1$ is the angle from the axis of Earth’s rotation to your mass and $\theta_2$ is the angle from the string to the axis, then in your diagram $\theta_1 = \theta_2$ and $T = mg$. In this scenario, there would be no net force on the string and it would not rotate with the Earth. So, for a suspended string like this, the actual forces and angles obey the relationships $T < mg$ and $\theta_2 < \theta_1$ (The string is deflected from the normal to Earth’s surface). This allows $T\cos\theta_2 = mg\cos\theta_1$ and $mg\sin\theta_1 = F_{\text{centrifugal}}+ T\sin\theta_2$ to both be true simultaneously.

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