Question

A plate in the form of a semicircle of radius $R$ has a mass per unit area of $kr$ where $k$ is a constant and $r$ is the distance form the centre of a straight edge. By dividing plate into semicircular rings find the centre of mass of the plate from the centre of its straight edge.

• A. $\frac{4R}{3\pi }$
• B. $\frac{2R}{\pi }$
• C. $\frac{3R}{2\pi }$
• D. $\frac{3R}{4\pi }$

Answer 1

The correct option is C $\frac{3R}{2\pi }$

$\begin{array}{l}{y_{CM}} = \frac{{\int_{}^{} {dm \times {y_{cm}}} }}{{\int_{}^{} {dm} }}\\\frac{{dm}}{{dA}} = kr\\dA = 2\pi {r^2}dr\\
\end{array}$
$\begin{array}{c}=\frac{{\int }^{}k2\pi r^{2}dr\times \frac{2r}{\pi }}{{\int }^{}k2\pi r^{2}dr}\\ =\frac{2}{\pi }\frac{R^4/4}{R^3/3}\end{array}$
$=\frac{3\pi }{2R}$