Answer:
(0, πR/4)
Explanation:
The linear mass density (mass per length) is λ = λ₀ sin θ.
A short segment of arc length is ds = R dθ.
The mass of this short length is:
dm = λ ds
dm = (λ₀ sin θ) (R dθ)
dm = R λ₀ sin θ dθ
The x coordinate of the center of mass is:
X = ∫ x dm / ∫ dm
X = ∫₀ᵖ (R cos θ) (R λ₀ sin θ dθ) / ∫₀ᵖ R λ₀ sin θ dθ
X = R ∫₀ᵖ sin θ cos θ dθ / ∫₀ᵖ sin θ dθ
X = R ∫₀ᵖ ½ sin 2θ dθ / ∫₀ᵖ sin θ dθ
X = ¼R ∫₀ᵖ 2 sin 2θ dθ / ∫₀ᵖ sin θ dθ
X = ¼R (-cos 2θ)|₀ᵖ / (-cos θ)|₀ᵖ
X = ¼R (-cos 2π − (-cos 0)) / (-cos π − (-cos 0))
X = ¼R (-1 + 1) / (1 + 1)
X = 0
The y coordinate of the center of mass is:
Y = ∫ y dm / ∫ dm
Y = ∫₀ᵖ (R sin θ) (R λ₀ sin θ dθ) / ∫₀ᵖ R λ₀ sin θ dθ
Y = R ∫₀ᵖ sin² θ dθ / ∫₀ᵖ sin θ dθ
Y = R ∫₀ᵖ ½ (1 − cos 2θ) dθ / ∫₀ᵖ sin θ dθ
Y = ½R ∫₀ᵖ (1 − cos 2θ) dθ / ∫₀ᵖ sin θ dθ
Y = ½R (θ − ½ sin 2θ)|₀ᵖ / (-cos θ)|₀ᵖ
Y = ½R [(π − ½ sin 2π) − (0 − ½ sin 0)] / (-cos π − (-cos 0))
Y = ½R (π − 0) / (1 + 1)
Y = ¼πR
