Question

The moment of inertial about the center of mass of the unified body after the collision is

• A. $\frac{48}{5}m{r}^2$
• B. $\frac{48}{3}m{r}^2$
• C. $\frac{24}{5}m{r}^2$
• D. $\frac{24}{3}m{r}^2$

Answer 1

The correct option is A $\frac{48}{5}m{r}^2$

As $OC=x=r$ (solved earlier)

$impliesC{O}^{\prime }=3r-r=2r$

Moment of bigger sphere about point O, $I_O=\frac{2}{5}\left(2m\right)(2r{)}^2=\frac{16}{5}m{r}^2$

MOI of bigger sphere about C, $I_C^{\prime }=I_O+\left(2m\right)C{O}^2$

$I_C^{\prime }=\frac{16}{5}m{r}^2+\left(2m\right)r^2=\frac{26}{5}m{r}^2$

Moment of smaller sphere about point O', $I_O^{\prime }=\frac{2}{5}\left(m\right)(r{)}^2=\frac{2}{5}m{r}^2$

MOI of smaller sphere about C, $I_C^{\prime \prime }=I_O^{\prime }+\left(m\right)C{O}^{\prime 2}$

$I_C^{\prime \prime }=\frac{2}{5}m{r}^2+\left(m\right)4r^2=\frac{22}{5}m{r}^2$

MOI of the system about C, $I=I_C^{\prime }+I{"}_C=\frac{48}{5}m{r}^2$