Calculation of Centre of Mass of Non-Uniform Semicircular Disc


In physics, the centre of mass is the point where the entire mass of an object or system is concentrated. It is the point where the net force acts on the object. The centre of mass of a system can be calculated using the mass distribution of the system.


A semicircular disc of non-uniform mass having sigma(mass per unit area) =2r where r is the radius of the disc.


To calculate the centre of mass of a semicircular disc, we divide the disc into small elements, each having a mass dm. The centre of mass of each element is located at its midpoint. We assume that the semicircular disc is lying on a coordinate plane with its flat side on the plane.


Divide the semicircular disc into small elements of thickness dx as shown in the diagram below.



Calculate the mass of each element dm. The mass per unit area of the disc is given as sigma = 2r. Therefore, the mass of each element is dm = sigma * area of the element. The area of the element can be calculated as dA = 2x * dx.

Therefore, dm = sigma * dA = 2r * 2x * dx = 4rx dx


Calculate the x-coordinate of the centre of mass of each element. The x-coordinate of each element is located at its midpoint, which is x/2.


Calculate the total mass of the semicircular disc by integrating dm over the entire length of the disc.

M = integral(dm) from 0 to r

M = integral(4rx dx) from 0 to r

M = 2r^2


Calculate the x-coordinate of the centre of mass of the semicircular disc using the formula:

xcm = (1/M) * integral(x dm) from 0 to r

xcm = (1/2r^2) * integral(x * 4rx dx) from 0 to r

xcm = (1/2r^2) * 4r * integral(x^2 dx) from 0 to r

xcm = (1/2r^2) * 4r * (r^3/3)

xcm = (2/3) * r

Therefore, the centre of mass of the semicircular disc is located at a distance of 2/3 times the radius of the disc from the centre of the flat side of the disc.