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

Answer 1

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

Answer 2

To calculate the total mass of the disc, we need to integrate the surface mass density function over the entire area of the disc.

The surface mass density function is given by f(r, φ) = k(1 - r^2/R^2), where k is the surface density at the center, r is the distance from the center of the disc, and R is the radius of the disc.

To calculate the total mass, we need to integrate this function over the entire area of the disc. Since the function is given in polar coordinates, we will use an area integral in polar coordinates.

The area element in polar coordinates is given by dA = r * dr * dφ, where r is the radial distance, and φ is the angle.

To calculate the limits of integration, we need to consider the range of r and φ. Since the disc has a radius of R, the radial distance, r, ranges from 0 to R. The angle, φ, ranges from 0 to 2π since we want to cover the entire circular surface of the disc.

The mass of the disc is given by the integral:

m = ∫∫f(r, φ) * dA

Substituting the expression for f(r, φ) and dA, we have:

m = ∫[0,2π]∫[0,R] k(1 - r^2/R^2) * r * dr * dφ

Expanding the integral, we have:

m = k * ∫[0,2π]∫[0,R] (r - r^3/R^2) * dr * dφ

Now, we can simplify the integral and evaluate it:

m = k * ∫[0,2π] (1/2 * R^2 - 1/4 * R^2) * dφ [applying the limits of integration for the inner integral]

m = k * ∫[0,2π] (1/4 * R^2) * dφ

m = k * 1/4 * R^2 * [φ] evaluated from 0 to 2π [integrating with respect to φ]

m = k * 1/4 * R^2 * 2π

m = k * 1/2 * π * R^2

Now, we can substitute the given values for k and R:

m = 27.47 * 1/2 * 3.142 * (0.21)^2

Calculating this expression will give the total mass of the disc in kg.