v 9What's New X Assignment 4 X E Assignment 4 1 / 6 118% + Problem 1. Recall that the moment of inertia of an object D of density function p(x, y) about a point (xo, yo) is defined as follows: I(20. 30) = {(x -20)2+ (9 -30)2} P(x, y) dA. Note that the moment of inertia is always non-negative. a) Show that the moment of inertia about the center of mass (x, y) is I (1. 0) = (2 2 + 32 ) p(x, y ) dA - (12 + y? ) M, where M is the total mass in D. b) Evidently, when the mass is entirely concentrated at the center of mass, we have I(z, y) = 0. Based on this fact, we can interpret I(z, y) as the dispersion of mass about the center of mass. Consider the density function Pt ( 2 , y ) = - Art At , (x, y) E RR2 where t > 0 is a parameter. Determine the dispersion of mass about the center of mass in terms of t. Hint: The improper integral in R2 is defined by integrating over a disk of radius R when R- co. Remark. The figures below show the mass concentration for t= 0.1 and t= 1. As we observe, the density spreads out in time. When t- 0, all mass is concentrated at the origin, the center of mass. This is called a singularity. = 1 O 0.5 -0.5 -0.5 -1 0 0 Q Search ENG 8:18 PM US 2023-11-02