find the center of mass of the upper half of the ball $x^2+y^2+z^2 \le 16$ for $z\ge 0$
can someone help me set up the integrals?
Its symmetry on the x and y axis right so we only have to find z-axis.
find the center of mass of the upper half of the ball $x^2+y^2+z^2 \le 16$ for $z\ge 0$
can someone help me set up the integrals?
Its symmetry on the x and y axis right so we only have to find z-axis.
Imagine the half-ball is a ham, and take a thin horizontal slice going from height $z$ to height $z+dz$. The cross section at height $z$ is a disk. Since $x^2+y^2=16-z^2$, the disk has radius $\sqrt{16-z^2}$. This disk has area $\pi(16-z^2)$. So the volume of the slice is about $\pi(16-z^2)\,dz$.
This slice is at height $z$ above the $x$-$y$ plane, so it has moment approximately equal to $(z)(\pi)(16-z^2)\,dz$ about the $x$-$y$ plane.
"Add up," $z=0$ to $z=4$. We get that the moment is $\int_0^4\pi(16z-z^3)\,dz.$ Evaluate this integral, divide by the volume of the half-ball.