Materials needed: Cardboard, scissors, compass, ruler, and your brain. Total cos
ID: 2868396 • Letter: M
Question
Materials needed: Cardboard, scissors, compass, ruler, and your brain. Total cost: Less than $ 10.00 (excluding cost of college education). This is an artsy-craftsy type project designed to amuse and amaze even the most metallic-music hardened of you out there. We will develop the concept of the center of mass of a plane region through a coordinated experimental and theoretical investigation. Because genuinely plane regions are unavailable in the real three-dimensional world, we will content ourselves with laminae which are regions with area but negligible thickness. Experiment Take your cardboard and cut out three laminae: A triangle with sides 3 4 and 5 respectively A semicircle radius 3 A horseshoe, i.e. a half-ring, whose inner and outer edges are composed of circular arcs of radii 3 and 5 respectively. For each lamina, find several balancing lines by balancing your figure on the edge of the ruler (the sharper the better). Mark each on your figure. What do you notice about how these balancing lines intersect? Mark this intersection; it is called the center of mass of the figure. The figure should balance on a point placed at the center of mass, since each line through it is a balancing line. Theory: For purposes of describing the laminae mathematically, we set up an (x.y) coordinate system for each. The first moment of a planar mass destribution about a line l is the integral over the region of the (area) density sigma(P) times the distance of P from the line. For example, the first moment about the line x = k is Mk = (x-k)sigma(P)dA. In order for a line to be a balancing line, the first moment about this line must be zero. Now we finally come to the work in this project: Prove that any line through the center of mass is a balancing line. Hint: Make your line the x axis of a new coordinate system rotted by an angle Theta with respect to your original axis. Place the origin of both coordinate systems at the center of mass. Now express the moment about the new x axis in terms of the moments about the old x and y axes. Can you now conclude that all balancing lines intersect at the center of mass?Explanation / Answer
Any triangle has 3 medians. Every median of a triangle passes through the centroid of the triangle. Centroid of a triangle is the center of mass of the triangle of uniform density. The triangle balances on the point of intersection of the medians. Therefore any line through the center of mass is a balancing line.