Can Tidal Force Unlock Unlimited Energy Potential?

[appmathstudent]

Homework Statement

The origin of the Cartesian coordinates is at the Earth’s center. The moon is on the
z-axis, a fixed distance R away (center-to-center distance). The tidal force exerted by
the moon on a particle at the Earth’s surface (point x, y,z) is given by... **See picture on attempted solution**

Relevant Equations

Fx,Fy,Fz

 

[etotheipi]

You are absolutely correct, that the potential is only determined up to an additive constant. Given any function $\phi$ such that $\mathbf{F} = - \nabla \phi$, which in this case you may obtain by simply summing the negatives of the integrals of the components of the force [because of the separability, i.e. each force component only depends on its corresponding co-ordinate], $\phi' = \phi + c$ is also a valid solution.

In this case, the authors chose that the point of zero potential is $\mathbf{x}_0 = (0,0,0)$.

 

[appmathstudent]

You are absolutely correct, that the potential is only determined up to an additive constant. Given any function $\phi$ such that $\mathbf{F} = - \nabla \phi$, which in this case you may obtain by simply summing the negatives of the integrals of the components of the force [because here, each component only depends on its corresponding co-ordinate], $\phi' = \phi + c$ is also a valid solution.

In this case, the authors chose that the point of zero potential is $\mathbf{x}_0 = (0,0,0)$.

Thank you very much for the explanation !

 

[etotheipi]

Thank you very much for the explanation !

No problem!