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hw1ans - Phys 341 Homework#1 Solutions 1 To start let's use...

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Phys 341: Homework #1 Solutions 1. To start, let’s use dimensional analysis to estimate the speed of particles moving in a system of mass M and radius R . The things we have to work with are gravity G [ M - 1 L 3 T - 2 ] mass M [ M ] radius R [ L ] The combination with dimensions of velocity is v GM R 1 / 2 = [ L T - 1 ] (a) Consider a globular cluster containing about 10 6 stars in a sphere of radius 10 pc. Use dimensional analysis to estimate the speed with which the stars move inside the cluster. If we assume the stars are similar to the sun we have M = 10 6 M and v (6 . 67 × 10 - 8 cm 3 g - 1 s - 2 ) × (10 6 × 1 . 99 × 10 33 g) 10 × 3 . 086 × 10 18 cm 1 / 2 2 . 1 × 10 6 cm s - 1 21 km s Notice how I wrote out the units, and carefully converted from M to g, and pc to cm. Always be careful with units and conversions! (b) Now repeat the analysis for a galaxy containing 10 11 stars in a radius of 10 kpc. v (6 . 67 × 10 - 8 cm 3 g - 1 s - 2 ) × (10 11 × 1 . 99 × 10 33 g) 10 4 × 3 . 086 × 10 18 cm 1 / 2 210 km s - 1 (c) Finally, estimate the speed of galaxies moving in a cluster of galaxies with mass M 10 14 M and radius 1 Mpc.
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