How to find the center of mass for a system of multiple solid spheres?

Question

Assuming spheres have continuous and uniform mass distribution ($\rho$). How can we compute the center of mass (com), e.g. for a system with 3 solid spheres: $(x_1, y_1, z_1, R_1)$, $(x_2, y_2, z_2, R_2)$, $(x_3, y_3, z_3, R_3)$ mathematically?

Answer 1

Let calculate the mass of each sphere

$$M_i=\frac43 \pi \rho R_i^3$$

and then

$$x_C=\frac{\sum x_iM_i}{\sum M_i} \quad y_C=\frac{\sum y_iM_i}{\sum M_i} \quad z_C=\frac{\sum z_iM_i}{\sum M_i}$$

Note that since the mass is uniform we can substitute $M_i$ with $V_i$.

EDIT

In integral form, for the x component we have

$$x_C=\frac{\int\int\int_V x\,dm}{\int\int\int_V \,dm}=\frac{\int\int\int_{V_1} x\,dm+\int\int\int_{V_2} x\,dm+\int\int\int_{V_3} x\,dm}{\sum M_i}$$

note also that

$$\int\int\int_{V_i} x\,dm=x_i M_i$$

thus

$$x_C=\frac{x_1M_1+x_2M_2+x_3M3}{\sum M_i}=\frac{\sum x_iM_i}{\sum M_i}$$