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Find the coordinate of the centroid for the shaded part of the lamina with respect to given reference axis.[Figure: origin O with X-axis to the right and Y-axis upward. Points: B(3,0) on the X-axis, A(0,3) on the Y-axis, C(−3,0) on the negative X-axi…

Q.2 (A) Find the coordinate of the centroid for the shaded part of the lamina with respect to given reference axis.

[Figure: origin O with X-axis to the right and Y-axis upward. Points: B(3,0) on the X-axis, A(0,3) on the Y-axis, C(−3,0) on the negative X-axis, D(−3,3). Arc AB is a quarter circle centred at O with R = 3 cm. Arc CA is a quarter circle centred at D with R = 3 cm (concave towards D). The shaded region is bounded by OB, the arc BA, the arc AC and the line CO, i.e. the quarter circle OAB together with the part of square OCDA lying outside the quarter circle centred at D. Dimensions in cm.]

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Split the shaded area (all dimensions in cm, \(r = 3\)): (1) square OCDA (+), (2) quarter circle centred at O in the first quadrant (+), (3) quarter circle centred at D(−3,3) (−).

Quarter circle: \(A = \dfrac{\pi r^2}{4} = 7.069\ \text{cm}^2\), centroid distance from its centre \(= \dfrac{4r}{3\pi} = 1.273\) cm.

PartA (cm²)x (cm)y (cm)AxAy
1 Square 3×39.000−1.5001.500−13.50013.500
2 Quarter circle (centre O)7.0691.2731.2739.0009.000
3 Quarter circle (centre D), removed−7.069−3 + 1.273 = −1.7273 − 1.273 = 1.727+12.206−12.206
Σ9.0007.70610.294

\(\bar x = \dfrac{\sum Ax}{\sum A} = \dfrac{7.706}{9} = 0.856\ \text{cm}\), \(\bar y = \dfrac{\sum Ay}{\sum A} = \dfrac{10.294}{9} = 1.144\ \text{cm}\)

Answer: centroid G = (0.856 cm, 1.144 cm) w.r.t. O.

Doubtly AI · 25 Oct 2026 AI-verified · Oct 2026 Report
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