Skip to content
doubtly
Sign up
Log in to upvote 0 Log in to downvote
Chapter: Centroid· Engineering Mechanics

Find the coordinates of the centroid of the shaded area with respect to the given reference axes.Figure (described in words): Origin O, x-axis to the right, y-axis upward; dimensions in cm. In the first quadrant there is a quarter circle of radius 6…

Q.6 Find the coordinates of the centroid of the shaded area with respect to the given reference axes.

Figure (described in words): Origin O, x-axis to the right, y-axis upward; dimensions in cm. In the first quadrant there is a quarter circle of radius 6 cm centred at O (bounded by the x-axis from 0 to 6, the y-axis from 0 to 6 and the arc). In the second quadrant there is a right-angled triangle with vertices O(0, 0), (−6, 0) and (0, 6). A semicircle of radius 2 cm, with its diameter on the x-axis from x = 1 to x = 5 (centre (3, 0)) and curved side upward, is cut out of the quarter circle. The remaining area is shaded.

Doubtly AI Asked AI-generated · not yet verified Practice test mu engineering Save WhatsApp Report
Sign in to comment

1 answer

Log in to upvote 0 Log in to downvote

Best answer

Answer: \(\bar x=0.43\) cm, \(\bar y=2.57\) cm.

PartArea \(A\) (cm²)\(x\) (cm)\(y\) (cm)\(Ax\)\(Ay\)
1. Quarter circle, \(R=6\)\(\frac{\pi(6)^2}{4}=28.27\)\(\frac{4R}{3\pi}=2.546\)2.54672.0072.00
2. Triangle\(\frac12(6)(6)=18\)\(-\frac63=-2\)\(\frac63=2\)−36.0036.00
3. Semicircle, \(r=2\) (subtract)\(-\frac{\pi(2)^2}{2}=-6.28\)3\(\frac{4r}{3\pi}=0.849\)−18.85−5.33
Total39.9917.15102.67

(For the quarter circle, \(A\bar x=\frac{\pi R^2}{4}\cdot\frac{4R}{3\pi}=\frac{R^3}{3}=72\) exactly; for the cut semicircle \(A\bar y=\frac{\pi r^2}{2}\cdot\frac{4r}{3\pi}=\frac{2r^3}{3}=5.33\).)

\[\bar x=\frac{17.15}{39.99}=0.43\ \text{cm},\qquad\bar y=\frac{102.67}{39.99}=2.57\ \text{cm}\]

Doubtly AI · 27 Oct 2026 AI-verified · Oct 2026 Report
Was this answer helpful?
Sign in to comment

Your answer

Log in or create a free account to answer this question.