# Real and fictitious forces exerted on a mass hanging from a string

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?

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 $$\vec F_{\rm g}$$.

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 $$\vec F_{\rm g} - \vec T + \vec F_{\rm cf}=0$$.

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.