The motion of a solid object can be analyzed by thinking of the mass as concentrated at a single point, the center of mass. If the object has density
at the point and occupies a region W, then the coordinates of the center of mass are given by
where is the total mass of the body.
Consider a solid is bounded below by the square ,, and above by the surface the density of the solid be 1 g/cm$^3$, with x,y,z measured in cm. Find each of the following:
The mass of the solid
I am having troubles with starting and setting the integral
Best Answer
The mass of the solid is $\int_W \rho \mathrm{d}V$ which in this case is equal to $\rho \int_W \mathrm{d}V$ as the solid has a uniform mass density. To find the mass consider a small rectangular element of sides $\mathrm{d}x$ and $\mathrm{d}y$ on the $xy$ plane. The volume of the cuboidal rod of solid with this element as the base is $z\mathrm{d}x\mathrm{d}y$. So the total volume of the solid will be $$\int_0^5\int_0^4(x+y+3) \mathrm{d}x \mathrm{d}y = 150$$ and therefore the mass = $1gm/cm^3\cdot150 cm^3 = 150 gm$
For the coordinates of center of mass the integrals are similar $$ \overline{x} = \frac{1}{m}\int_0^5\int_0^4\rho x(x+y+3) \mathrm{d}x \mathrm{d}y$$ $$ \overline{y} = \frac{1}{m}\int_0^5\int_0^4\rho y(x+y+3) \mathrm{d}x \mathrm{d}y$$ $$ \overline{z} = \frac{1}{m}\int_0^5\int_0^4\rho \frac{(x+y+3)^2}{2} \mathrm{d}x \mathrm{d}y$$