Studio 15 Gravity SOLUTIONS

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University of Michigan *

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Course

140

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Physics

Date

Jan 9, 2024

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pdf

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4

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Studio 15: Gravity Physics 140 - University of Michigan, Department of Physics Name: Problem 1 Back to our in-class discussion on the mass of the Milky Way. Clearly there is a discrepancy between the mass of the Milky Way estimated via counting stars and that estimated by using the velocity of objects in the outer rim of the galaxy. Assume the Milky Way can be modeled as a disk of radius 50 kpc (kiloparsecs) and uniformly distributed mass M (which we do not know). Find the shape of the velocity distribution as a function of radial position both inside and outside the Milky Way. Which of the two curves below does this batter match, A or B? When you finish this problem, follow this LINK to learn which one is correct. This will also resolve the discrepancy between mass calculations from class. https://cdms.phy.queensu.ca/Public_Docs/DM_Intro.html Solution: M inside = πr 2 · M πR 2 v inside circ = q GM R 2 · r r v outside circ = q GM r 1 / r So our calculations say curve A. But experimentally it’s curve B. This is evidence of Dark Matter!!!
Problem 2 Geosynchronous satellites are positioned at a specific height above the earth’s surface such that they always hover over the same spot (so for example, a satellite could be always above Ann Arbor). Calculate what this height must be. Assume m E = 5 . 97 × 10 24 kg and R E = 6 . 38 × 10 6 m . (Answer: 35,900 km). Solution: -
Problem 3 Answer the following: a. What is the gravitational acceleration of Mars at its surface? Solution: g mars = 3 . 7 m/s 2 b. What is the escape velocity of Mars? Solution: v escape = 5 km/s c. What is harder to escape from: Earth (if you start on the Earth’s surface) or Sun (if you start at the Earth’s orbit)? M Earth = 6 . 0 × 10 24 kg M Sun = 2 . 0 × 10 30 kg R Earth = 6 . 4 × 10 6 m R Earth s orbit = 1 . 5 × 10 11 m Solution: The sun from Earth’s orbit. v escape,sun = 42 . 2 km/s
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Problem 4 Many stars belong to systems of two or more stars held together by their mutual gravitational attraction. The figure below shows a three star system at an instant when the stars are at the vertices of a 45 right triangle. Find the total gravitational force exerted on the small star by the two large ones. Solution: -