Differential gravitational force causing tides model?

[21joanna12]

Homework Statement

Homework Equations

The Attempt at a Solution

For a), I initially tried to consider the component of the gravitational force of the moon acting normally to the Earth's surface. This would be F=F_0 cos(theta) where theta is the angle between a horizontal line going through A and B and the point on the Earth's surface in question (giving zero at the poles and maximum at the equator). I know that , due to the effect of the Earth actually undergoing circular motion and 'free-falling' due to the moon's gravitational pull, the force at each point on the side of the Earth facing the moon will be (approximately) mirrored by the opposite side of the Earth since r<<x. But now I am stuck. I wanted to deal with forces rather than equipotentials, as it seems like the question wants you to just deal with the forces. I was considering the fact that the effect is actually spherically symmetrical when looking onto the face of the Earth from the moon, but II don't know what to do with this to find the heights? I could find the total volume of water on the Earth by doing surface area of Earth x 750, but I still don't see how II could use this?Thank you in advance :)

 

[mfb]

as it seems like the question wants you to just deal with the forces

I don't see why. Potentials are fine.

 

[21joanna12]

I don't see why. Potentials are fine.

Help! I cannot figure out how to get the equation with the equipotential. Would I have to consider the Earth's rotation? And how to I create a potential due to the moon?

 

[mfb]

And how to I create a potential due to the moon?

In the same way you do for earth, with Newton's law of gravity.

Would I have to consider the Earth's rotation?

Not the 24h-rotation, but the motion around the center of mass of the earth/moon system is relevant to get the second tidal wave at the opposite side.

 

[paranormal]

I would approach this question by first acknowledging that the tides are caused by the gravitational force of the moon and sun acting on the Earth. This force is not uniform across the Earth, as it varies depending on the distance between the Earth and the moon/sun. This is known as the differential gravitational force.

To model this, we can use the concept of tidal forces, which refers to the difference in gravitational force between two points on the Earth's surface. This difference creates a gradient in the gravitational force, which results in the bulging of the Earth's oceans towards the moon/sun.

To understand this more clearly, we can use the analogy of a rubber band stretching. The side of the Earth facing the moon/sun experiences a stronger gravitational force compared to the opposite side, causing it to bulge outwards. At the same time, the opposite side experiences a weaker force, causing it to bulge inwards. This creates two tidal bulges on opposite sides of the Earth.

To determine the heights of these bulges, we can use the concept of equipotential surfaces, which represent points on the Earth's surface with the same gravitational potential energy. The heights of these bulges will correspond to the points on the equipotential surfaces that are closest to the moon/sun.

In summary, the differential gravitational force model for tides takes into account the varying gravitational force across the Earth's surface and uses the concept of tidal forces and equipotential surfaces to explain the bulging of the Earth's oceans.