Centre Of Mass Of Solid Hemisphere - Practice Questions & MCQ

Have a look at the figure of solid Hemisphere

Since it is symmetrical about y-axis
So we can say that its $x_{c m}=0$ and $z_{c m}=0$
Now we will calculate its $y_{c m}$ which is given by

$
y_{c m}=\frac{\int y \cdot d m}{\int d m}
$

So Take a small elemental hollow hemisphere of mass $d m$ of radius $r$ as shown in figure.

Now have a look on elemental hollow hemisphere of mass dm of radius r

Since our element mass is hollow hemisphere so its C.O.M is at (r/2)
Now $\quad d m=\rho d v=\rho\left(2 \pi r^2\right) d r$

Where,

$
\begin{aligned}
\rho & =\frac{M}{\frac{2}{3} \pi R^3} \\
y_{c m} & =\frac{\int \frac{r}{2} d m}{M}=\frac{\int_0^R \frac{r}{2} * \frac{3 M}{2 \pi R^3} * 2 \pi r^2 d r}{M}=\frac{3}{2 R^3} * \int_0^R r^3 d r=\frac{3 R}{8}
\end{aligned}
$

So $y_{c m}=\frac{3 R}{8}$ from base