Homework 33

# Homework 33 - homework 33 BAUTISTA ALDO Due 4:00 am Version...

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homework 33 – BAUTISTA, ALDO – Due: Apr 21 2006, 4:00 am 1 Version number encoded for clicker entry: V1:1, V2:4, V3:2, V4:4, V5:3. Question 1 Part 1 of 2. 10 points. Two satellites A and B orbit the Earth in the same plane. Their masses and radii have the relationships m B = 6 m A and r B = 3 r A . r B r A A B What is the ratio of the orbital speeds v B v A ? 1. v B v A = 1 3 correct 2. v B v A = 3 3. v B v A = 1 3 4. v B v A = 1 2 5. v B v A = 9 6. v B v A = 1 2 7. v B v A = 2 8. v B v A = 3 9. v B v A = 2 10. v B v A = 1 9 Explanation: Basic Concepts: Force of gravity between two masses m 1 and m 2 at a distance r F g = G m 1 m 2 r 2 , where G is the gravitational constant. Circular motion a r = v 2 r , for the radial (centripetal) acceleration. Note: v is the tangential speed. Solution: Since the force of gravity is re- sponsible for holding a satellite in its orbit, the orbital centripetal force is equal to the force of gravity F r = m v 2 r = G M m r 2 , where again M is the mass of the Earth, m is the mass of the satellite, and r is the radius of the orbit (from the Earth’s center). Thus the tangential speed v of an orbit at distance r is, v = r G M r . Note: The speed is independent of m . Thus the ratio v B v A = r G M r B r G M r A = r r A r B . And since r B = 3 r A , we have v B v A = r r A 3 r A = 1 3 . Question 2 Part 2 of 2. 10 points. Let the distance of the satellite A from the center of the Earth be r A = 10 R , where R is the radius of the Earth. Denote the gravitational acceleration at the surface of the Earth by g . The gravitational acceleration due to the Earth at satellite A is given by 1. g A = g 121

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homework 33 – BAUTISTA, ALDO – Due: Apr 21 2006, 4:00 am 2 2. g A = g 81 3. g A = g 4. g A = g 11 5. g A = g 10 6. g A = g 9 7. g A = g 100 correct 8. g A = g 3 9. g A = g 10 10. g A = g 11 Explanation: Since g = G M R 2 , then at r A , g A = G M r 2 A = G M (10 R ) 2 = g 100 . Question 3 Part 1 of 1. 10 points. Given: k = 4 π 2 G M s , where M s is the mass of the Sun. Suppose that the gravitational force law between two massive objects is F g = G m 1 m 2 r 2+ , where is a small number. Which of the following would be the rela- tionship between the period T and radius r of a planet in circular orbit? 1. T 2 = k r 3 / 2. T 2 = k r 3 3. T 2 = k r 3+ correct 4. T 2 = k r 2 - 3 5. T 2 = k r 2+3 6. T 2 = k r 3 - 7. T 2 = k r 3 8. T 2 = k r 3 - 2 9. T 2 = k r 3+2 10. T 2 = k r 3+2 / Explanation: Kepler’s third law changes from its normal form if gravity is not quite an inverse square law. Let M p be the mass of a planet and M s be the mass of the Sun. r will be the radius of the orbit. M p v 2 r = G M s M p r 2+ .
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