A disc of radius R is cut out from a larger disc of radius 2R such that the edge of the hole touches the edge of the larger disc. Determine the location of the centre of mass of the residual disc.

Step-by-step solution

Let the centre of the larger disc be at the origin (0,0) . Radius of the larger disc is 2R . Its mass is proportional to its area, M₁ = (2R)^2 = 4 R^2 , where is the surface mass density. The hole has a radius R and touches the edge of the larger disc. The distance of its centre from the origin is 2R - R = R . Let its centre be at (R, 0) . Mass of the cut-out disc is M₂ = R^2 . The x -coordinate of the centre of mass of the residual disc is given by: x_ cm = M₁ x₁ - M₂ x₂ M₁ - M₂ Substituting the values: x_ cm = 4