Find the Y-coordinate of the centroid of that piece of the arc of the cycloid for which 0 ≤ θ ≤ π/2?
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Any help would be much appreciated!
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x = (t - sin(t)) ; y = (1 - cos(t)) ..... a unit cycloid (r = 1)
A = ∫ y dx = ∫ (1 - cos(t))² dt [0,π/2]
= 3π/4 - 2
Cy A = 1/2 ∫ y² dx = 1/2 ∫ (1 - cos(t))³ dt [0,π/2]
= 5π/4 - 11/3
Cy = (5π/4 - 11/3) / (3π/4 - 2)
= (44 - 15π) / (24 - 9π)
Answer: Cy = (44 - 15π) / (24 - 9π) ≈ 0.7308483