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solutions-ps01 - Physics 341 Problem Set#1 Solutions 1(a...

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Physics 341: Problem Set #1 Solutions 1. (a) Use dimensional analysis to derive a relationship between the total mass of a gravitationally bound system M , its typical size R , and the typical speed v of its components. We can start by writing the dimensions for the quantities involved, as well as Newton’s gravitational constant (because gravity is holding the system together). gravity G [ M - 1 L 3 T - 2 ] mass M [ M ] size R [ L ] speed v [ L T - 1 ] If we solve for the speed, some trial and error quickly yields v GM R 1 / 2 Or similarly, we could have solved for the mass or the size M v 2 R G R GM v 2 (b) A globular cluster has about 10 6 stars in a sphere of radius 10 pc. Using your analysis above, estimate the speed with which the stars move inside the cluster, in units of km s - 1 . You will need to make an assumption about the average star; be sure to state any assumptions explicitly. We need to know the mass of the cluster, so we’ll have to make an assumption. If we assume the stars are similar to the Sun on average then for the total mass of the cluster we have M = 10 6 M
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