Question

Locate the COM of a uniform hemisherical solid with respect to 0 as shown in figure. (Radius = r)

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

This is pretty simple as we already know that COM of hemispherical shell lies at $\frac{R}{2}$ on symmetrical axis.
We can choose hemispherical shell as our element and by integration we can form the complete given solid hemisphere, as shown in diagram.

$dV=2\pi x^{2}dx$
Let p be the volume density
$\to \rho =\frac{M}{V}$
$\Rightarrow dm=2\pi x^{2}dx\rho$
We know that the COM has to lie on the symmetrical axis {i.e., y-axis} and also COM of uniform hemispherical shell is at half its Radius.
$Y_{COM}=\frac{\underset{0}{\overset{\gamma }{\int }}2\pi x^{2}dx\frac{x}{2}\rho }{\rho \frac{4}{6}\pi r^3}$ = $\frac{3r}{8}$