Centre Of Mass Of Solid Hemisphere MCQ - Practice Questions with Answers

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 $y_{\mathrm{cm}}$ which is given by

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

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

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

Since our element mass is hollow hemisphere 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,

$$
\rho=\frac{M}{\frac{2}{3} \pi R^3}
$$

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

so $y_{c m}=\frac{3 R}{8}_{\text {from base }}$