Two uniform solid spheres of the same material having radii R and...

Let the density of the material be $\rho$.

Mass of the smaller sphere, $M_1 = \dfrac{4}{3}\pi R^3 \rho = M$

Mass of the larger sphere, $M_2 = \dfrac{4}{3}\pi (2R)^3 \rho = 8M$

The distance between the centers of the two spheres is $d = R + 2R = 3R$.

Taking the center of the smaller sphere as the origin $(0,0)$, the coordinates of the centers are $x_1 = 0$ and $x_2 = 3R$.

The position of the center of mass is given by:

$x_{cm} = \dfrac{M_1 x_1 + M_2 x_2}{M_1 + M_2}$

$x_{cm} = \dfrac{M(0) + 8M(3R)}{M + 8M}$

$x_{cm} = \dfrac{24MR}{9M} = \dfrac{8R}{3}$

Answer: $\dfrac{8R}{3}$