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PHYS_2014_Homework_9_soln

# PHYS_2014_Homework_9_soln - PHYS2014 Benton Fall 2010 OSU...

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PHYS2014 Fall 2010 Benton OSU Physics Dept. 9-1 Physics 2014: General Physics I Solutions to Homework Assignment 9 Due: Friday, 12 November 2010 at the beginning of your Recitation session. 1. While the Sun contains nearly all the mass of the Solar System, how much of the total angular momentum of the Solar System is due to the Sun’s rotation on its axis? Find the fractions of the Solar System’s total angular momentum and total mass accounted for by the rotation of the Sun. Use a simplified model that assumes the Sun is a sphere of uniform density that rotates about its axis once every 27 days. This model also makes the following simplifying assumptions: i) only the four outer gas giants—Jupiter, Saturn, Uranus, and Neptune—together with the Sun are important in determining the total angular momentum and the total mass of the Solar System, ii) these four outer planets orbit the Sun in circular orbits, and iii) that the rotation of these planets on their own axes can be neglected. Consult Appendix E of your textbook for values on the masses, mean distances, orbital periods, etc. of the planets and the Sun that you’ll need to work this problem. Solution: Let’s do the easy part first. The total mass of the solar system is obviously going to be the mass of the Sun plus the mass of the planets. We are told we can neglect all the planets except Jupiter, Saturn, Uranus, and Neptune. 30 27 26 25 25 30 30 30 1.99 10 kg 1.90 10 kg 5.68 10 kg 8.68 10 kg 1.02 10 kg 2.01 10 kg 1.99 10 kg 0.99 or 99% 2.01 10 kg SS Sun Jup Sat Ur Nep Sun SS M M M M M M M M = + + + + × + × + × + × + × = × × = = × 99% of the mass of the Solar System is contained in the Sun, so only 1% resides in the planets. We solve the problem for angular momentum in much the same way. We need to find the angular momentum of the Sun rotating on its axis and the angular momenta of the planets orbiting the Sun. The sum of these will be the total angular momentum of the Solar System. SS Sun Jup Sat Ur Nep L L L L L L = + + + + First we find the angular momentum of the Sun rotating on its axis. ( ) 2 6 2 30 8 6 42 2 2 5 2 2 rad 2.69 10 s s 27 d 86,400 d rad 2 1.99 10 kg 6.96 10 m 2.69 10 5 s 1.04 10 kg m Sun Sun Sun Sun Sun Sun Sun Sun L I M r T L L ω ω π π ω = = = = = × = × × × = ×

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PHYS2014 Fall 2010 Benton OSU Physics Dept. 9-2 Next we have to find the angular momenta of each of the gas giant planets. Because we can neglect their rotation on their own axes and because they are so small compared to their radial distances from the Sun, we can treat them as if the mass of each planet was concentrated at a single point. In that case, the moment of inertia for each planet will just be 2 p orb I M r = . Jupiter: ( ) 8 2 2 27 11 8 43 2 2 2 rad 1.67 10 s d s 11.9yr 365.25 86,400 yr d rad 1.90 10 kg 7.78 10 m 1.67 10 s 1.92 10 kg m Jup Jup Jup Jup Jup Jorb Jup Jup Jup T L I M r L L π π ω ω ω = = = × = = = × × × = × Saturn: ( ) 9 2 2 26 12 9 42 2 2 2 rad 6.75 10 s d s 29.5yr 365.25 86,400 yr d rad 5.68 10 kg 1.43 10 m 6.75 10 s 7.84 10 kg m Sat Sat Sat Sat Sat Sorb Sat Sat Sat T L I M r L L π π ω ω ω = = = × = = = × × × = × Uranus: ( ) 9 2 2 25 12 9 42 2 2 2 rad 2.37 10 s d s 84.1yr 365.25 86,400
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PHYS_2014_Homework_9_soln - PHYS2014 Benton Fall 2010 OSU...

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