Total Angular Momentum of the Earth

In summary: Solve for v in the first equation and plug it into the second equation to eliminate v.In summary, to find the length of a day in which the total angular momentum of the Earth is zero, we can use the equation Ltot = Ltrans + Lrot and set it equal to 0. Then, using the equations Lrot = Iw^2 and w = 2pi/T, we can solve for the period T by setting Ltrans equal to Lrot and using the equations mvr = Iw and F=m(v^2) / r. Finally, we can plug in the value for v from the first equation into the second equation and solve for T.
  • #1
v3r
5
0

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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  • #2
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
 
  • #3
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.
 
  • #4
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?
 
  • #5
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?
 
  • #6
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)
 

1. What is the total angular momentum of the Earth?

The total angular momentum of the Earth is the measure of the combined rotational motion of the Earth around its own axis and its orbit around the Sun. It is a vector quantity and is conserved in the absence of any external torques.

2. How is the total angular momentum of the Earth calculated?

The total angular momentum of the Earth can be calculated by multiplying the Earth's moment of inertia (a measure of its resistance to rotational motion) by its angular velocity (how fast it rotates around its axis) and then adding the angular momentum of its orbit around the Sun.

3. What factors affect the total angular momentum of the Earth?

The total angular momentum of the Earth is affected by changes in its rotation rate, changes in its orbital distance from the Sun, and changes in its shape or mass distribution. Other external factors, such as the gravitational pull of other planets, can also have a small effect on the Earth's total angular momentum.

4. Is the total angular momentum of the Earth constant?

In the absence of any external torques, the total angular momentum of the Earth remains constant due to the principle of conservation of angular momentum. However, small changes in the Earth's rotation rate, orbit, and shape can cause slight variations in its total angular momentum.

5. Why is the total angular momentum of the Earth important?

The total angular momentum of the Earth is important for understanding and predicting the Earth's rotation and orbit. It also plays a role in the Earth's climate, ocean currents, and other geophysical processes. Furthermore, the study of the Earth's total angular momentum helps us understand the formation and evolution of our planet.

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