Question

Suppose the density on E is proportional to the distance to the origin, with the largest density being equal to 2. Use spherical coordinates to compute the mass and center of mass of E.

Answer 1

To calculate the mass and center of mass for a region E with density proportional to distance, we integrate the density function over E using spherical coordinates. The mass is obtained by a triple integral of the density, and the center of mass by integrals of the density times the respective Cartesian coordinate components.

Step-by-step explanation:

The student is asked to calculate the mass and center of mass of a region E, given that the density within E is proportional to the distance from the origin, with the maximum density being 2. In spherical coordinates, mass is found by integrating the density function over the volume of the region. Since density is proportional to the distance, let's assume the density function is ρ(r) = kr, where k is the constant of proportionality and r is the radial distance in spherical coordinates. The mass can then be found using the integral:

Mass (M) = ∫∫∫_E ρ(r) dV

To find the center of mass, the weighted average position of the mass distribution needs to be calculated, which in spherical coordinates is done by the integrals:

• x-component: x_cm = (1/M) ∫∫∫_E ρ(r)x dV
• y-component: y_cm = (1/M) ∫∫∫_E ρ(r)y dV
• z-component: z_cm = (1/M) ∫∫∫_E ρ(r)z dV

Here x, y, and z are the Cartesian coordinates expressed in terms of spherical coordinates. We calculate these integrals to find the center of mass of E.