Total Angular Momentum of the Earth

[v3r]

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..

 

[flatmaster]

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

 

[v3r]

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.

 

[flatmaster]

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?

 

[v3r]

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?

 

[flatmaster]

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)