Answer to Question #85858 in Classical Mechanics for Rinku Rai

Question

A solid in the shape of a hemisphere with a radius of 2 units, has its base in the xy-plane
and the centre of the base at the origin. If the density of the solid is given by the function
ρ(x, y,z) = xyz, determine the mass of the hemisphere.

Answer 1

Mass of the hemisphere could be calculated as:

$$M=\int\rho\left(x,y,z\right)dV$$

In polar coordinates:

$$x=rcos\varphi sin\theta,\ y=rsin\varphi sin\theta,\ \ z=rcos\theta$$

This gives us:

$$M=\int\rho\left(x,y,z\right)dV=\int_{0}^{2}\int_{0}^{2\pi}\int_{0}^{\pi/2}\left(rcos\varphi sin\theta\right)\left(rsin\varphi sin\theta\right)\left(rcos\theta\right)r^2sin\theta d\theta d\varphi dr=$$

$$=\int_{0}^{2}{r^5dr\int_{0}^{2\pi}{cos\varphi sin\varphi d\varphi\int_{0}^{\pi/2}{sin^3\theta cos\theta}}}d\theta=(\frac{64}{6})(0)(0.25)=0$$

So, the total mass of the hemisphere is zero