Solve IPhO Basic Q: Angular Momentum Conservation of Moon-Earth System

[quantum13]

Homework Statement

http://www.jyu.fi/tdk/kastdk/olympiads/2009/Theo1_Question_Final.pdf

This is a problem on some gravity phenomenon. I can't even get past the first part (though to be honest I'm probably not qualified even to look at these)

The moon - Earth system is changing. Because the tidal bulge axis is not aligned with the moon-earth axis, a torque shifts angular momentum from the Earth's rotational into the moon's translation.

Determine the conservation of momentum equation for the initial and final states of the system. In the final state, the moon's rotational velocity and the Earth's rotational velocity will be equal; the moon will no longer move away from the Earth afterwards.

QUOTE: "Neglecting the contribution of the Earth´s rotation to the final total
angular momentum, write down the equation that expresses the angular
momentum conservation for this problem."

Homework Equations

L = Iw

The Attempt at a Solution

Initial angular momentum:

I(earth)w(earth,initial) + I(moon,initial)w(moon,initial) = L

Final angular momentum:

I(earth)w(final) + I(moon,final)w(final) = L

I(earth)w(earth,initial) + I(moon,initial)w(moon,initial) =
I(earth)w(final) + I(moon,final)w(final)
BUT their solution on http://www.jyu.fi/tdk/kastdk/olympiads/2009/Theo1_Answer.pdfI(earth)w(earth,initial) + I(moon,initial)w(moon,initial) = L = I(moon,final)w(final)

I don't understand why they just blew off the angular momentum of the Earth's rotation from the equation. It's obviously not w=0 that's for sure.

Thanks for your help.

 

[anathema]

Hello,

Thank you for bringing this problem to my attention. I understand your confusion and I will do my best to explain the solution provided in the answer document.

Firstly, let's review the concept of angular momentum conservation. Angular momentum is a measure of an object's rotational motion and is calculated by multiplying its moment of inertia (I) by its angular velocity (ω). In a closed system, the total angular momentum remains constant unless acted upon by an external torque.

In this problem, we are dealing with a closed system consisting of the Earth and the moon. Initially, the Earth is rotating with an angular velocity (ω) and the moon is also rotating with an angular velocity (ω). The total angular momentum of the system is the sum of the individual angular momenta of the Earth and the moon.

Now, let's move on to the final state where the moon's rotational velocity (ω) and the Earth's rotational velocity (ω) are equal. This means that the moon is no longer moving away from the Earth and is in a state of synchronous rotation. In this state, the moon's rotational velocity is equal to its orbital velocity, which is much greater than its initial rotational velocity. This increase in the moon's velocity results in a decrease in its moment of inertia (I).

Now, let's look at the equation for angular momentum conservation in the final state provided in the answer document:

I(earth)w(earth,initial) + I(moon,initial)w(moon,initial) = I(moon,final)w(final)

As we can see, the angular momentum of the Earth's rotation is not included in this equation because the Earth's rotation is no longer contributing to the total angular momentum of the system in the final state. This is because the Earth's rotational velocity (ω) remains constant throughout the process and does not change, while the moon's rotational velocity (ω) increases significantly. Therefore, the contribution of the Earth's rotation to the final total angular momentum is negligible and can be neglected.

I hope this explanation helps to clarify the solution provided in the answer document. If you have any further questions, please do not hesitate to ask. Keep up the good work in your studies!


(PhD, Physics)