If an ellipse is rotated about X-axis an ellipsoid is formed, then the coordinates of the centre of mass of a uniform semi-ellipsoid formed by rotating a quarter ellipse given by the equation a 2 x 2

Question

If an ellipse is rotated about X-axis an ellipsoid is formed, then the coordinates of the centre of mass of a uniform semi-ellipsoid formed by rotating a quarter ellipse given by the equation a 2 x 2 ​ + b 2 y 2 ​ = 1 as shown in diagram is :-

TestHub · Accepted Answer

X c m ​ = ∫ d m ∫ x d m ​ with disk slices of radius y : d m = ρ π y 2 d x , b 2 y 2 ​ = 1 − a 2 x 2 ​ . ∫ d m = ρ π b 2 ∫ 0 a ​ ( 1 − a 2 x 2 ​ ) d x = 3 2 ​ ρ π b 2 a . ∫ x d m = ρ π b 2 ∫ 0 a ​ ( x − a 2 x 3 ​ ) d x = 4 1 ​ ρ π b 2 a 2 . X c m ​ = 2 ρ π b 2 a /3 ρ π b 2 a 2 /4 ​ = 8 3 a ​ - (A)

Physics - Center of Mass Question with Solution | TestHub

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