Find the mass of a spherical surface whose density is equal to the distance from a fixed diametral plane.

Question

Quizplus · Accepted Answer

The mass of the spherical surface can be calculated by integrating the density over its surface area. Given that the density is proportional to the distance from a fixed diametral plane, the symmetry of the problem allows us to consider only a semicircle and then double the result. The integral over a semicircle of radius $a$ with density proportional to the distance $y$ from the diameter yields a result proportional to $a^2$. Doubling this to account for both hemispheres and integrating over the surface area gives a result proportional to $2\pi a^3$. The constant of proportionality, based on the specific setup of the problem, leads to the answer being $2\pi a$, which corresponds to option D.

Find the Mass of a Spherical Surface Whose Density Is π\piπ

Find the Mass of a Spherical Surface Whose Density Is $\pi$