Two uniform solid spheres of the same material having radii R and 2R are placed such that they are in contact with each other. What is the distance of the center of mass of this two-sphere system from the center of the smaller sphere?

Step-by-step solution

Let the density of the material be . Mass of the smaller sphere, M₁ = 4 3 R^3 = M Mass of the larger sphere, M₂ = 4 3 (2R)^3 = 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 x₁ = 0 and x₂ = 3R . The position of the center of mass is given by: x_ cm = M₁ x₁ + M₂ x₂ M₁ + M₂ x_ cm = M(0) + 8M(3R) M + 8M x_ cm = 24MR 9M = 8R 3 Answer: 8R 3