6.042/18.] Mathematics for Computer Science March 15.2005 Srini devadas and Eric Lehman Lecture notes Sums, Approximations, and Asymptotics II 1 Block Stacking How far can a stack of identical blocks overhang the end of a table without toppling over? Can a block be suspended entirely beyond the tables edge? Table Physics imposes some constraints on the arrangement of the blocks. In particular, the stack falls off the desk if its center of mass lies beyond the desk's edge. Moreover, the center of mass of the top k blocks must lie above the(k+ 1)-st block; otherwise, the top k would fall over In order to find the best configuration of blocks satisfying these constraints, we'll need a fact about centers of mass Fact 1. If two objects have masses m1 and m2 and centers-of-mass at positions a1 and 22, center of mass of the two objects together is at position 31m1+2m2 m1+m2 Define the offset of a particular configuration of blocks to be the horizonal distar from its center of mass to its rightmost edge offset center of mass Table6.042/18.062J Mathematics for Computer Science March 15, 2005 Srini Devadas and Eric Lehman Lecture Notes Sums, Approximations, and Asymptotics II 1 Block Stacking How far can a stack of identical blocks overhang the end of a table without toppling over? Can a block be suspended entirely beyond the table’s edge? Table Physics imposes some constraints on the arrangement of the blocks. In particular, the stack falls off the desk if its center of mass lies beyond the desk’s edge. Moreover, the center of mass of the top k blocks must lie above the (k + 1)­st block; otherwise, the top k would fall over. In order to find the best configuration of blocks satisfying these constraints, we’ll need a fact about centers of mass. Fact 1. If two objects have masses m1 and m2 and centers­of­mass at positions z1 and z2, then the center of mass of the two objects together is at position: z1m1 + z2m2 m1 + m2 Define the offset of a particular configuration of blocks to be the horizonal distance from its center of mass to its rightmost edge. offset ? center of mass - s ? Table