2.2 Semicircular ring

  Centre of Mass and Conservation of Linear Momentum

Consider a uniform semicircular ring of radius $R$ and mass $M$, whose centre is at the origin of a cartesian co-ordinate (for simplicity).

Length of the semicircular ring $(L) = \pi R\quad ...(i)$

Mass per unit length $\left( {{\lambda _L}} \right) = \frac{M}{{\pi R}}\quad ...(ii)$

Consider an infinitesimal small mass $dm$ at point $P$ on the ring.

Co-ordinate of point $P$ is $(R\sin \theta ,\;R\cos \theta )$

So, the position vector of the centre of mass $(COM)$ of point $P$ is,

Note: The small section at point $P$ is infinitesimally small, so, the centre of mass of that section is assumed to be at point $P$ only.

$${\overrightarrow r _P} = R\sin \theta \widehat i + R\cos \theta \widehat j\quad ...(iii)$$ Also, $$\begin{equation} \begin{aligned} dm = {\lambda _L}.\;dx \\ dm = {\lambda _L}Rd\theta \quad ...(iv) \\\end{aligned} \end{equation} $$

Therefore, the centre of mass $(COM)$ of the semicircular ring is given by,

$${\overrightarrow r _{COM}} = \frac{{\int {{{\overrightarrow r }_P}dm} }}{{\int {dm} }} = \frac{{\int {\left( {R\sin \theta \widehat i + R\cos \theta \widehat j} \right)dm} }}{{\int {dm} }}\quad ...(v)$$ From equation $(iv)$ & $(v)$ we get, $$\begin{equation} \begin{aligned} {\overrightarrow r _{COM}} = \frac{{\int {\left( {R\sin \theta \widehat i + R\cos \theta \widehat j} \right)} {\lambda _L}Rd\theta }}{{\int {{\lambda _L}Rd\theta } }} \\ {\overrightarrow r _{COM}} = \frac{{\int {\left( {R\sin \theta \widehat i + R\cos \theta \widehat j} \right)} d\theta }}{{\int {d\theta } }} \\\end{aligned} \end{equation} $$

Integrating for the semicircular ring from point $C$ to $A$ along point $B$,

At point $C$, $\theta = 0$ and at point $A$, $\theta = \pi $, so the limit becomes, $$\begin{equation} \begin{aligned} {\overrightarrow r _{COM}} = \frac{{\int\limits_0^\pi {\left( {R\sin \theta \widehat i + R\cos \theta \widehat j} \right)d\theta } }}{{\int\limits_0^\pi {d\theta } }} \\ {\overrightarrow r _{COM}} = \frac{{R\left[ { - \cos \theta \widehat i + \sin \theta \widehat j} \right]_0^\pi }}{{\left[ \theta \right]_0^\pi }} \\ {\overrightarrow r _{COM}} = \frac{{2R}}{\pi }\widehat j \\\end{aligned} \end{equation} $$

So, the centre of mass $(COM)$ of the uniform semicircular ring is $\left( {0,\frac{{2R}}{\pi }} \right)$