The y – coordinate of centre of mass of a thin and uniform semicircular ring of radius r shown in figure is

Step-by-step solution

The wire is uniform. So, the mass per unit length of the wire μ = M π r . Let us assume a small element on the semicircular ring as shown in the figure. Mass of the element d m = M π r r d θ = M π d θ The y -coordinate of the centre of mass y cm = 1 M ∫ y d m ⇒   y cm = 1 M ∫ 0 π r sin θ M π d θ ⇒   y cm = r π 1 + 1 ⇒   y cm = 2 r π