In two dimensions I would no center it at the origin, I use ##(0,0)## as one of the corners of the square, then I would find the ##x## component by ##X_{cm}=\frac{\int\sigma x dA}{\int\sigma dA}## and the ##y## component by ##Y_{cm}=\frac{\int\sigma y dA}{\int\sigma dA}##. So ##X_{cm}=\frac{\int_{0}^L Lxdx }{\int_{0}^L Ldx}=\frac{L\int_{0}^L xdx}{L\int_{0}^L dx}=\frac{\frac{x^2}{2}}{L}=\frac{L}{2}##, the same would be for ##y##, so ##Y_{cm}=\frac{\int_{0}^L Lydy }{\int_{0}^L Ldy}=\frac{L\int_{0}^L ydy}{L\int_{0}^L dy}=\frac{\frac{y^2}{2}}{L}=\frac{L}{2}##, so the center of mass would be on the point ##\left(\frac{L}{2},\frac{L}{2}\right)##. How should I do this with three dimensions?andrewkirk said:How would you do it in two dimensions? If you think about the technique you use in that case, it should become apparent how you might extend that to three dimensions.
So I would end up with ##L/2## again? Would be something like ##Z_{cm}=\frac{\int_{0}^L Lzdz }{\int_{0}^L Ldz}=\frac{L\int_{0}^L ydz}{L\int_{0}^L dz}=\frac{\frac{z^2}{2}}{L}=\frac{L}{2}##? Or would something change?andrewkirk said:Just do the same calculation for the z dimension as you did for the x and y dimensions, using (0,0,0) as one of the corners of the cube, the cube lying entirely in the first octant of the space (all three coordinates positive) and with each edge of the cube parallel to one of the three axes.
Yes, you'd end up with that. Nothing would change.Davidllerenav said:So I would end up with ##L/2## again? Would be something like ##Z_{cm}=\frac{\int_{0}^L Lzdz }{\int_{0}^L Ldz}=\frac{L\int_{0}^L ydz}{L\int_{0}^L dz}=\frac{\frac{z^2}{2}}{L}=\frac{L}{2}##? Or would something change?
Ok. Then the aswer would be just to write thos three calculations and that's it?andrewkirk said:Yes, you'd end up with that. Nothing would change.
The center of mass of a cube can be found by identifying the midpoint of each side and then connecting these points to form the diagonals of the cube. The intersection of these diagonals is the center of mass.
The formula for finding the center of mass of a cube is (a/2, a/2, a/2), where a is the length of one side of the cube. This means that the center of mass is located at the midpoint of each side of the cube.
No, the center of mass of a cube will always be located within the boundaries of the cube. This is because the center of mass is determined by the distribution of mass within an object.
The center of mass of a cube is directly influenced by the distribution of mass within the cube. If the mass is evenly distributed, the center of mass will be at the geometric center of the cube. However, if the mass is unevenly distributed, the center of mass will shift towards the side with the higher amount of mass.
Yes, determining the center of mass of an object is crucial in physics as it helps in understanding the motion and stability of an object. It also plays a role in calculating the moment of inertia and torque of a cube, which are important concepts in physics.