Find average density of the sphere of radius 𝑎 whose density at a distance𝑟 from the centre of the sphere is 𝜌 = 𝜌0 [1 + 𝑘 𝑟3𝑎3

To find the average density of the sphere, we need to integrate the density function over the entire volume of the sphere and then divide by the total volume.

Step 1: Determine the volume of the sphere. The volume of a sphere with radius 𝑎 is given by the formula V = (4/3)π𝑎^3.

Step 2: Set up the integral for the average density. The average density (𝜌_avg) is given by the equation: 𝜌_avg = (1/V) ∫𝜌 dV

Step 3: Express the density function in terms of 𝑟. Given that 𝜌 = 𝜌0 [1 + 𝑘(𝑟/𝑎)^3], we can substitute this expression into the integral: 𝜌_avg = (1/V) ∫[𝜌0 (1 + 𝑘(𝑟/𝑎)^3)] dV

Step 4: Convert the integral to spherical coordinates. Since we are dealing with a sphere, it is more convenient to use spherical coordinates. The volume element in spherical coordinates is given by dV = 𝑟^2 sin(𝜃) d𝑟 d𝜃 d𝜙, where 𝜃 ranges from 0 to 𝜋 and 𝜙 ranges from 0 to 2𝜋.

Step 5: Evaluate the integral. Substitute the expression for dV and the limits of integration into the integral: 𝜌_avg = (1/V) ∫[𝜌0 (1 + 𝑘(𝑟/𝑎)^3)] 𝑟^2 sin(𝜃) d𝑟 d𝜃 d𝜙

Integrate with respect to 𝑟 from 0 to 𝑎, 𝜃 from 0 to 𝜋, and 𝜙 from 0 to 2𝜋: 𝜌_avg = (1/V) ∫[𝜌0 (1 + 𝑘(𝑟/𝑎)^3)] 𝑟^2 sin(𝜃) d𝑟 d𝜃 d𝜙 = (1/V) ∫[𝜌0 (1 + 𝑘(𝑟/𝑎)^3)] 𝑟^2 sin(𝜃) d𝑟 ∫𝜃 d𝜃 ∫𝜙 d𝜙

Evaluate the integrals: 𝜌_avg = (1/V) ∫[𝜌0 (1 + 𝑘(𝑟/𝑎)^3)] 𝑟^2 sin(𝜃) d𝑟 ∫𝜃 d𝜃 ∫𝜙 d𝜙 = (1/V) ∫[𝜌0 (1 + 𝑘(𝑟/𝑎)^3)] 𝑟^2 sin(𝜃) d𝑟 [-cos(𝜃)] [2𝜋]

Simplify the expression: 𝜌_avg = (1/V) ∫[𝜌0 (1 + 𝑘(𝑟/𝑎)^3)] 𝑟^2 sin(𝜃) d𝑟 [-cos(𝜃)] [2𝜋] = (2𝜋/V) ∫[𝜌0 (1 + 𝑘(𝑟/𝑎)^3)] 𝑟^2 sin(𝜃) d𝑟 [-cos(𝜃)]

Step 6: Evaluate the remaining integral. Integrate with respect to 𝑟 from 0 to 𝑎: 𝜌_avg = (2𝜋/V) ∫[𝜌0 (1 + 𝑘(𝑟/𝑎)^3)] 𝑟^2 sin(𝜃) d𝑟 [-cos(𝜃)] = (2𝜋/V) ∫[𝜌0 (1 + 𝑘(𝑟/𝑎)^3)] 𝑟^2 sin(𝜃) d𝑟 [-cos(𝜃)] evaluated from 0 to 𝑎

Evaluate the integral at the upper limit 𝑎 and subtract the value at the lower limit 0: 𝜌_avg = (2𝜋/V) [𝜌0 (1 + 𝑘(𝑎/𝑎)^3)] 𝑎^2 sin(𝜃) [-cos(𝜃)] evaluated from 0 to 𝑎

Simplify the expression: 𝜌_avg = (2𝜋/V) [𝜌0 (1 + 𝑘)] 𝑎^2 sin(𝜃) [-cos(𝜃)] evaluated from 0 to 𝑎

Step 7: Calculate the average density. Substitute the values into the expression: 𝜌_avg = (2𝜋/V) [𝜌0 (1 + 𝑘)] 𝑎^2 sin(𝜃) [-cos(𝜃)] evaluated from 0 to 𝑎 = (2𝜋/V) [𝜌0 (1 + 𝑘)] 𝑎^2 sin(𝜃) [-cos(𝜃)] evaluated at 𝑎 minus evaluated at 0

Evaluate the expression at 𝑎: 𝜌_avg = (2𝜋/V) [𝜌0 (1 + 𝑘)] 𝑎^2 sin(𝜃) [-cos(𝜃)] evaluated at 𝑎 = (2𝜋/V) [𝜌0 (1 + 𝑘)] 𝑎^2 sin(𝜃) [-cos(𝜃)] evaluated at 𝑎

Evaluate the expression at 0: 𝜌_avg = (2𝜋/V) [𝜌0 (1 + 𝑘)] 𝑎^2 sin(𝜃) [-cos(𝜃)] evaluated at 𝑎 = (2𝜋/V) [𝜌0 (1 + 𝑘)] 𝑎^2 sin(𝜃) [-cos(𝜃)] evaluated at 𝑎 - (2𝜋/V) [𝜌0 (1 + 𝑘)] 0^2 sin(𝜃) [-cos(𝜃)] evaluated at 0

Simplify the expression: 𝜌_avg = (2𝜋/V) [𝜌0 (1 + 𝑘)] 𝑎^2 sin(𝜃) [-cos(𝜃)] evaluated at 𝑎 - (2𝜋/V) [𝜌0 (1 + 𝑘)] 0^2 sin(𝜃) [-cos(𝜃)] evaluated at 0 = (2𝜋/V) [𝜌0 (1 + 𝑘)] 𝑎^2 sin(𝜃) [-cos(𝜃)] evaluated at 𝑎 - (2𝜋/V) [𝜌0 (1 + 𝑘)] 0^2 sin