Solutions to Some Random Questions

Question 1:

The height of the center of mass is a mass-weighted sum of the coordinates of the individual centers of mass.

$y_{CM} = \frac{y_1 M_1 + y_2 M_2 + y_3 M_3}{M_1 + M_2 + M_3}$

Values of these quantities:

$y_1 = 7R \hspace{1cm} M_1 = \frac{4}{3} \pi R^3 \rho_1 \\ y_2 = 4R \hspace{1cm} M_2 = \frac{32}{3} \pi R^3 \rho_2 \\ y_3 = R \hspace{1cm} M_3 = 8 R^3 \rho_3$

Plugging in:

$y_{CM} = \frac{7R \times \frac{4}{3} \pi R^3 \rho_1 + 4R \times \frac{32}{3} \pi R^3 \rho_2 + R \times 8 R^3 \rho_3}{\frac{4}{3} \pi R^3 \rho_1 + \frac{32}{3} \pi R^3 \rho_2 + 8 R^3 \rho_3 } = 4R \\ \frac{28}{3} \pi R^4 \rho_1 + \frac{128}{3} \pi R^4 \rho_2 + 8 R^4 \rho_3 = \frac{16}{3} \pi R^4 \rho_1 + \frac{128}{3} \pi R^4 \rho_2 + 32 R^4 \rho_3 \\ \frac{28}{3} \pi R^4 \rho_1 + 8 R^4 \rho_3 = \frac{16}{3} \pi R^4 \rho_1 + 32 R^4 \rho_3 \\ 4 \pi R^4 \rho_1 = 24 R^4 \rho_3 \\ \frac{\rho_1}{\rho_3} = \boxed{\frac{6}{\pi}}$

Question 2:

The height of the center of mass is a length-weighted sum of the coordinates of the individual centers of mass of the straight and curved parts. We can use length instead of mass because of the uniform mass density per unit length.

$y_{CM} = \frac{y_1 L_1 + y_2 L_2 }{L_1 + L_2 }$

Straight Part:

$L_1 = \pi R \hspace{1cm} y_1 = 0$

Curved Part:

$L_2 = \pi R \hspace{1cm} y_2 = \frac{2R}{\pi}$

Plugging in:

$y_{CM} = \frac{0 \pi R + \frac{2R}{\pi} \pi R }{\pi R + \pi R} = \boxed{\frac{R}{\pi}}$