Parallel Axis Theorem

Solution:

Let's label spherical snowballs as 1, 2 and 3 as shown in the picture.

First we can find mass of each snowball using their radius and density of snow:

$m_1=\rho V_1=\left(300\ \frac{kg}{m^3}\right)\left(\frac{4}{3}\pi\right)\left(0.3\right)^3=10.8\ \pi\ kg$

$m_2=86.4\pi\ kg\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m_3=291.6\pi\ kg$

Now we need to find the center mass position. We can guess that due to symmetry it should be on the line passing through center of all snowballs. We can find its height respect to any arbitrary point. I chose the center of lowest snowball (the third one in the picture).

So, the position of the snowman center of mass respect to the center of the lowest snowball is:

$X_{cm}=\dfrac{m_1x_1+m_2x_2+m_3x_3}{m_1+m_2+m_3}=\dfrac{\left(10.8\right)\left(2.4\ m\right)+\left(86.4\right)\left(1.5\right)+\left(291.6\right)\left(0\right)}{\left(10.8+86.4+291.6\right)}$

So, the snowman center of mass is 0.4m above the center of the lowest snowball or 1.3m above the ground.

Since the snowman is rotating about its center of mass, we need to find its moment of inertia respect to this point. We can use the formula given to us to find the moment of inertia of each snowball about its own center of mass and then use parallel axis theorem to calculate its moment of inertia about the snowman's center of mass.

$I_{cm1}=\frac{2}{5}m_1r_1^2=\frac{2}{5}\left(10.8\pi\right)\left(0.3\right)^2=0.3888\pi\ kg.m^2$

$I_{cm2}=12.4416\pi\ kg.m^2\ \ \ \ \ \ \ \ I_{cm3}=94.4784\pi\ \ kg.m^2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

We can use the radius of snowballs to find the distance between their center of mass and the snowman's center of mass (axis of rotation)

$I_1=I_{cm1}+m_1 h_1^2=0.3888\pi+\left(10.8\pi\right)\left(2.0\right)^2=136.9\ kg\ m^2$

$I_2=I_{cm2}+m_2 h_2^2=12.4416\pi+\left(86.4\pi\right)\left(1.1\right)^2=367.5\ kg\ .\ m^2$

$I_3=I_{cm3}+m_3 h_3^2=94.4784\pi+\left(291.6\pi\right)\left(0.4\right)^2=443.4\ kg.\ m^2$

Finally, we need to add above moments of inertia to find the snowman's moment of inertia