a) M = ∫∫∫ ρ(x, y, z) dV
b) The z-coordinate of the center of mass is given by:
z = (1/M) ∫∫∫ zρ(x, y, z) dV
c) The general expression for the moment of inertia about the z-axis is:
I_z = ∫∫∫ ρ(x, y, z) (x² + y²) dV
Consider the solid described in Exercise 21, where the density is defined as ρ(x, y, z) = √x² + y².
(a) To determine the mass of the solid, we need to integrate the density over the entire volume of the solid. The bounds of integration will depend on the specific shape of the solid, but the general expression for the mass is:
M = ∫∫∫ ρ(x, y, z) dV
(b) To determine the center of mass of the solid, we need to calculate the average coordinates of the mass distribution. The x-coordinate of the center of mass is given by:
x = (1/M) ∫∫∫ xρ(x, y, z) dV
Similarly, the y-coordinate of the center of mass is given by:
y = (1/M) ∫∫∫ yρ(x, y, z) dV
and the z-coordinate of the center of mass is given by:
z = (1/M) ∫∫∫ zρ(x, y, z) dV
(c) To determine the moment of inertia about the z-axis, we need to calculate the integral of the square of the distance from each point in the solid to the z-axis, multiplied by the density.
The general expression for the moment of inertia about the z-axis is:
I_z = ∫∫∫ ρ(x, y, z) (x² + y²) dV