How Does Changing Earth's Mass and Size Affect Its Rotation Period?

[mrshappy0]

Homework Statement

Find the hours it takes for the Earth to rotate in one day if the mass is reduced to 2/3 of its original mass and it is shrunk to 3/4 its original size.

Homework Equations

L=Iω

The Attempt at a Solution

To start this problem I assumed that angular momentum is conserved from the original Earth to the final earth. This means the angular velocity must change in order for the angular momentum to remain unchanged. So used this equation: I_earthω_earth=I_kω_k and got that I_earthω_earth/I_k=ω_k. I plugged in the data and got that the new speed is twice the speed of the original earth. This would mean the Earth rotates about 12 hrs a day. This SEEMS wrong but I am not sure.

Note: k is the final earth.

 

[sophiecentaur]

Um.
I would be inclined to do this in two steps. First assume that the mass changed as a result of density, for a start - giving an identical 'size' of sphere as it started off with. That gives a new I. (2/3 of the original) But I can't see how this is relevant to the final answer, actually, because you could have two, spheres rotating around a common axis and each sphere could reduce in size, giving the same answer.
Then conserve angular momentum for the new reduced radius to find the new angular velocity. I is proportional to radius squared so new I is 9/16 of original. this gives a day length of 24X9/16 = 18hours

Buit the question is SLOPPY because what does "size" mean? Volume or radius?

 

[mrshappy0]

OKay, well I didn't copy the exact questions...Here: Suppose the Earth were to suddenly shrink to 3/4 of its initial radius and 2/3 of its initial mass. What would the duration of one day be?

 

[sophiecentaur]

Haha. Sloppy student not sloppy question.
I guess the 9/16 is what you're after then.

 

[asteriatic]

Your approach and calculations are correct. It may seem counterintuitive that reducing the mass and size of the Earth would result in a faster rotation, but this is due to the conservation of angular momentum. As the Earth's size and mass decrease, its moment of inertia (I) also decreases, causing the angular velocity (ω) to increase in order to maintain a constant angular momentum (L=Iω). Therefore, the Earth would rotate in approximately 12 hours instead of the 24 hours it currently takes. This is a very interesting concept and highlights the interconnectedness of different physical quantities in our universe. Great job on your solution!