A flat circular disc, of radius R, can be modelled as a thin disc of negligible thickness. It has a surface mass density function given by f(r,φ) = k(1 - r2/R2), where k is the surface density at the centre and r is the distance from the centre of the disc.Using area integral in plane polar coordinates, calculate the total mass of the disc, in kg, when R = 0.21 m and k = 27.47 kg m-2. Give your answer to 3 decimal places. Take π = 3.142.

Question

A flat circular disc, of radius R, can be modelled as a thin disc of negligible thickness. It has a surface mass density function given by f(r,φ) = k(1 - r2/R2), where k is the surface density at the centre and r is the distance from the centre of the disc.Using area integral in plane polar coordinates, calculate the total mass of the disc, in kg, when R = 0.21 m and k = 27.47 kg m-2. Give your answer to 3 decimal places. Take π = 3.142.
Answer

Steve · Answer

Consider the disc as a set of concentric rings, each with area 2πr dr. The mass of each ring is thus 2πr f(r) dr

Note that φ does not affect the density on any ring.

The mass is thus

∫[0,0.21] 2kπr(1-r^2/0.441) dr
= 3.61