The center of mass of a solid bounded by the planes z = 1 –- y, z = 0 and y = Vx where density 8(x,y,z) = y is :=1-y

Nixon is a tile worker, he hates physics and only likes cutting and shaping his tiles. Her boss has given him a tile. It has a width of 50 cm. The tile must fit around a corner of a fireplace so a piece is sawed out. The extracted piece is a smaller square and its side length is only 25.0 cm. Because of the cutout segment, the center of mass of the ceramic tile shifts a bit from point A to point B. Nixon needs help to find the distance between point A and B the ceramic tile her boss gave him. 50cm 25cm 8 S

The height varies from h to zero according to this function: y(x) = h ( – 1)´ . The constants h and e replace 1.00 m and 3.00 m. There is also a thickness t and a density p. You need two integrals, the total mass and the center of mass. Possibly surprisingly, you don't actually need the numbers t, h, and p. Ax y(x) X The column at x has a mass Am = (density * volume) = y(x) p t Ax. You add all the Am values to get %3D the total mass M. The sum becomes an integral: М — pt y(x) dx For the center of mass, you add each column's x Am, and divide by M: pt Xc х у(x) dx Calculate xc. The only quantity you'll need is e = 5 m.

An outlaw cuts loose a wagon with two boxes of gold, of total mass 300 kg, when the wagon is at rest 50 m up a 6.0° slope. The outlaw plans to have the wagon roll down the slope and across the level ground, and then fall into a canyon where his accomplices wait. But in a tree 40 m from the canyon’s cliff wait the Lone Ranger (mass 75.0 kg) and Tonto (mass 60.0 kg). They drop vertically into the wagon as it passes beneath them (Fig.). (a) If they require 5.0 s to grab the gold and jump out, will they make it before the wagon goes over the cliff? The wagon rolls with negligible friction. (b) When the two heroes drop into the wagon, is the kinetic energy of the system of heroes plus wagon conserved? If not, does it increase or decrease, and by how much?