Centre of Mass of Semicircular Ring

The centre of mass is a unique position of an object or a system of objects where the entire mass of the system is concentrated. The motion of this unique point is identical to the motion of a single-particle whose mass is equal to the sum of all individual particles of the system. The method to find the centre of mass of a semicircular ring is discussed in this page.

How to Find Centre of Mass of Semicircular Ring

Centre of mass of circular ring

We are taking a half ring, which has a uniformly distributed mass with a radius R. The centre of mass of the ring will lie on the vertical line passing through the centre of the ring. The vertical line is considered as the y-axis. The horizontal line seen in the figure is the x-axis.

We are considering an elemental mass dM on the ring. The element is taken at an angle θ from the y-axis, it has an angular width of dθ and a linear width of Rdθ.

The mass of element dM = (M/π)dθ

The y-coordinate of dM is, y = Rcosθ

The y-coordinate of the centre of mass, yc = (1/M)ഽydM

yc=1Mπ2+π2Mπdθ(Rcosθ)y_{c}=\frac{1}{M}\int_{-\frac{\pi }{2}}^{+\frac{\pi }{2}}\frac{M}{\pi }d\theta (Rcos\theta ) yc=Rππ2+π2cosθdθy_{c}=\frac{R}{\pi }\int_{-\frac{\pi }{2}}^{+\frac{\pi }{2}}cos\theta d\theta

Integrating the above equation, we get

y = (R/π)[1+1] = 2R/π

The centre of mass of the ring is given by 2R/π, where R is the radius of the semicircular ring