Total Angular Momentum of the Earth

AI Thread Summary
To determine the period of Earth's rotation that results in zero total angular momentum, the discussion emphasizes the need to equate the rotational and translational angular momentum. The equations Ltot = Ltrans + Lrot and Lrot = Iw are central to the problem, with the angular velocity defined as w = 2pi/T. Participants highlight the importance of correctly identifying variables, particularly the radius used in different contexts. The solution involves solving for the period T while considering Earth's orbital velocity and gravitational forces. The conversation reflects a collaborative effort to clarify the concepts and equations involved in the problem.
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Homework Statement


How long should the day be so that the total angular momentum of the Earth
(including its rotation about its own axis and its (approximately) circular orbit around the
sun) is zero (Note: the magnitude of the angular velocity is 2pi/T where T is the period of rotation?)

Homework Equations


Ltot = Ltrans + Lrot
Lrot = Iw^2
w = 2pi/T

The Attempt at a Solution


Ltot = 0

I am really clueless. I don't know where to start..
 
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v3r said:

Homework Statement


How long should the day be so that the total angular momentum of the Earth
(including its rotation about its own axis and its (approximately) circular orbit around the
sun) is zero (Note: the magnitude of the angular velocity is 2pi/T where T is the period of rotation?)

Homework Equations


Ltot = Ltrans + Lrot
Lrot = Iw^2
w = 2pi/T

The Attempt at a Solution


Ltot = 0

I am really clueless. I don't know where to start..

you're started. double check your equation
Lrot = Iw^2

There shouldn't be a square there. For the Earth's orbital angular momentum, just treat the Earth like a point mass at a distance r.

L=mvr
 
So it should be
Ltot = Ltrans + Lrot
Ltot = 0
Ltrans = Lrot
mvr = Iw
mvr = 2pi/T * I
mvr = 2pi/T * mr^2
v/r = 2pi/T

I'm still confused.
 
v3r said:
So it should be
Ltot = Ltrans + Lrot
Ltot = 0
Ltrans = Lrot
mvr = Iw
mvr = 2pi/T * I
mvr = 2pi/T * mr^2
v/r = 2pi/T

I'm still confused.

You're getting there. Remember, you're looking for the period of one day on Earth under your new conditions. what variable do you want to solve for?

Also, be carefull with your variables. You've used "r" to stand in for two different things.

What is the r in Ltrans = mvr?
What is the r in I = mr^2?
 
r in I would be the perpendicular distance which would be the radius of the earth.
r in Ltrans would be the distance to the center of mass which would be the distance of Earth from sun?

I want to solve for v in order to get the time by dividing it by the radius?
 
You're almost there. Now use another variable to re-name one of your r's. Eventually, you want to solve for period T. However, you have the Earth's orbital velocity v to get rid of. For this, use univorm circular motion, and universal gravitation.

F=m(v^2) / rF=GmM / (r^2)
 

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