Question of Physics

Question

A half uniform annulus of inner and outer radii R₁ and R₂ has a nonuniform surface mass density given by σ = σ₀ r where σ₀ is a positive constant and r is the radial distance from the origin (radial coordinate in the cylindrical coordinates). Find

i) the mass of the half annulus, and

ii) the position of the center of mass, i.e., (x_cm, y_cm) the center of mass in terms of R₁ and R₂.

Answer 1

Hello Hasan I have done the first part

If you follow this one , I can go to the second part

Pl let me know

Dr Gulshan S

σ = σ₀ r

dA = pi r dr

dm =σ₀ r pi r dr

dm = σ₀ pi r² dr

M = ∫ σ₀ pi r² dr

Limits of integration are from R1 to R2

Integrating

M = σ₀ pi ∫ r² dr

M = (σ₀ pi) (R₂ ³ - R₁ ³ )/3