Question

Problem 2. (a) Let Di be a disc of radius 1 centred around the point (1,3), and suppose there is a lamina occupying D1. Assume that the mass density of the lamina at a point (x, y) E Di is given by p(, y) exp-K1 (y 3], where K is a positive constant. What is the total mass and what is the centre of mass of the lamina? Hint. You may use Formula (1) from Problem 1 if you want to.] (b) (Bonus) Assume now that a second lamina occupies a bigger region B CUD where C is another closed and bounded domain, and where C and Di overlap only on their boundary. Assume also that the distance of any point in B from the point (1,3) is at most 100, and that the density function of the second lamina is given by the same function p in part (a). Show that we can choose K sufficiently large so that the centre of mass of the lamina B is again within D Remark. This question suggests that we can redistribute mass in many possible ways so that we force the centre of mass of an object to be as close to a given point as we want. Hint. Note that by our assumptions C is contained in the disc D100 which has centre the point (1,3) and radius 100. Note also that, since C and D do not overlap except on their boundary C is enclosed in the region C ((a,y) : 1 <(+(v-3)2 100). Estimate then the contribution of the density at points in C to the total mass by bounding it from above by the corresponding contribution of density from the entire C, and show that, if we choose K sufficiently large, this is negligible, and cannot force the centre of mass to he ontside D

Answer 1

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