Astrophysical background: The celestial equator and celestial pole are the line

Question

Astrophysical background: The celestial equator and celestial pole are the line and points respectively in the sky above the Earth's equator and north and south poles. The ecliptic is the plane of Earth's orbit around the Sun. (a) Model the Earth as a uniform sphere. Calculate the angular momentum of the Earth due to its spinning motion about its axis to three significant figures.

(b) Calculate the angular momentum of the Earth due to its orbital motion about the Sun to three significant figures.

Magnitude Direction ---Select--- toward the north celestial pole toward the south celestial pole toward the north ecliptic pole toward the south ecliptic pole

Explanation / Answer

a) Moment of inertia of earth about its axis is I = 2/5MR2, as the earth is considered to be a solid sphere.

M = mass of earth = 5.972 × 1024 kg

R = radius of earth = 6.371X106m

So I = (2/5)(5.972 × 1024 kg)(6.371X106m)2 = 96.96X1036kg-m2

angular velocity of earth = = 2/T

where T = time period of rotation or simply the lenght of a day = 24X3600s = 86400s

So = 2/T = 6.28/(86400s)

So the angular momentum is L = I = (96.96X1036kg-m2)[6.28/(86400s)] = 7.047X1033kg-m2/s

The direction of andular momentum is toward the north celestial pole as the earth rotates west to east.

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b) The angular momentum of the Earth due to its orbital motion is L = MvRs, where v is the speed of the earth in its revolution around the sun.

The earth revolves around the sun in 365 days and covers a distance of 2Rs, where Rs is the distance of earth from sun.

So speed v = 2Rs/365days = 6.28X(149.6X109m)/[365X24X3600s] = 2.979X104m/s

So angular momentum due to its orbital motion is L = MvR = (5.972 × 1024 kg)(2.979X104m/s)(149.6X109m)

or L = 2.66X1040kg-m2/s toward the north ecliptic pole

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c) The orbital angular momentum is larger.

orbital angular momentum is = 2.66X1040kg-m2/s

rotational angular momentum is = 7.047X1033kg-m2/s

So orbital angular momentum is larger by a factor of (2.66X1040kg-m2/s)/(7.047X1033kg-m2/s)

or orbital angular momentum is larger by a factor of 3.77X106.

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This concludes the answers. Check the answer and let me know if it's correct. If you need anymore clarification or correction I will be happy to oblige....

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