For the shaded area in Problem F9-1, bounded by y=x^3, y=1 m, and the y-axis, determine the x-coordinate of the centroid (x_bar).
Explanation
To solve this, one must identify the correct area described in Problem F9-1 on page 461, which is bounded by y=x^3, the y-axis, and the line y=1. Then, the centroid formula x_bar = [integral of x_tilde*dA]/[integral of dA] must be applied. Using a horizontal differential area element (dA = x*dy) is often easier for this orientation. Here dA = y^(1/3) dy and the centroid of this strip is at x_tilde = x/2 = y^(1/3)/2. Integrating these from y=0 to y=1 gives the correct x_bar. It seems my calculation gave 0.4 m. Let me re-check. Integral of x_tilde * dA = Integral( (y^(1/3))/2 * y^(1/3) dy ) = 1/2 Integral( y^(2/3) dy) from 0 to 1 = 1/2 * [y^(5/3) / (5/3)] = 1/2 * 3/5 = 3/10. Area = Integral(y^(1/3) dy) from 0 to 1 = [y^(4/3) / (4/3)] = 3/4. x_bar = (3/10) / (3/4) = 12/30 = 0.4. It appears I have made an error in my initial answer choices. Let me correct the choices to reflect the right answer. The correct answer should be 0.4 m. I will change my choices. Oh, I see the issue. F9-1 has y=x^3. But F9-2 is y=x^3. The image for F9-1 and F9-2 are different. F9-1 is the area between curve and y=1. F9-2 is area between curve and x-axis. I will ask about F9-2. 'For the shaded area in Problem F9-2, bounded by y=x^3, the x-axis, and the line x=1m, what is the y-coordinate of the centroid?' Answer should be 0.286m. I will use that. Let me rephrase the question to match the correct problem number and calculation. I will ask for the y-coordinate of F9-2.
Other questions
Under what condition does the center of gravity of a body coincide with its centroid?
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For a triangular area with base 'b' and height 'h', where is the centroid located with respect to the base?
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What is the x-coordinate of the centroid for the area bounded by the curve y=x^2, the line x=1 m, and the x-axis?
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What is the formula for the x-coordinate of the centroid of a line segment described by a curve y = f(x)?
For the composite wire shown in Example 9.9, which consists of a semicircular arc and two straight segments, what is the calculated x-coordinate of the centroid?
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A straight rod of length L has a mass per unit length given by m = m0*(1 + x^2/L^2). Where is its center of mass, x_bar, measured from x=0?
A straight rod of length L has a mass per unit length given by m = m0*(1 + x^2/L^2). Where is its center of mass, x_bar?
A homogeneous solid is formed by revolving the shaded area from Problem F9-5 (z^2 = y/4, from z=0 to z=0.5m) about the y-axis. What is the y-coordinate of the centroid?
A homogeneous solid is formed by revolving the shaded area from Problem F9-6 (a triangle with vertices at (0,0), (1.5, 2), and (1.5, 0)) about the z-axis. What is the z-coordinate of the centroid?
What is the formula for the z-coordinate of the center of gravity, z_bar, for a general body?
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A wire is bent into a U-shape defined by y=x^2 from x=-2 ft to x=2 ft. The wire is symmetric about the y-axis. What is the x-coordinate of the centroid?
A uniform parabolic-shaped rod is defined by the equation y^2 = 4x, extending from x=0, y=0 to x=4 m, y=4 m. The mass per unit length is 2 kg/m. What is the total mass of the rod?
Determine the area of the shaded region bounded by the parabola y^2 = 4x and the vertical line x = 4 m, as depicted in Problem 9-8.
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The z-coordinate of the centroid of a semicircular wire of radius 'r' and angle alpha (as shown in Problem 9-7), when the wire is in the y-z plane and symmetric about the z-axis, is given by z_bar = r*sin(alpha)/alpha. What is the x_bar coordinate?
What is the x-coordinate of the centroid for the area bounded by the curves y^2=x and y=x^2, as shown in Problem 9-26/27?
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Determine the y-coordinate of the centroid for the T-shaped beam's cross-sectional area shown in F9-8.
Determine the y-coordinate of the centroid for the T-shaped beam's cross-sectional area shown in F9-8, which consists of a 300mm by 50mm flange on top of a 300mm by 50mm web.
An L-shaped cross-sectional area is shown in F9-10. It consists of a 4in by 0.5in vertical rectangle and a 2.5in by 0.5in horizontal rectangle. What is the x-coordinate of the centroid?
A homogeneous solid block has the shape shown in F9-11. It is a composite solid made of a rectangular block (A) and a triangular prism (B). Block A is 2ft x 4ft x 3ft. Block B has a triangular base (width 2ft, height 3ft) and length 4ft. What is the x-coordinate of the center of mass?
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A wire is bent in the shape shown in Problem 9-44, consisting of three straight segments. The top horizontal segment is 100mm, the vertical segment is 150mm, and the bottom horizontal segment is 50mm. What is the x-coordinate of the centroid?
An area is shaped like a C-channel, as in Problem 9-53/54. If the flanges are 12in wide and 2in thick, and the web is 2in thick and has a total height of 4in between the flanges, what is the y-coordinate of the centroid, measured from the bottom surface?
What is the key idea behind the 'Procedure for Analysis' for finding the centroid of a shape by single integration?
A semi-circular area of radius 'a' has its centroid at y_bar = 4a/(3*pi). If this area is revolved 360 degrees (2*pi radians) about its base (the x-axis), what is the volume of the resulting sphere?
Determine the x-coordinate of the centroid of the shaded area shown in Problem 9-9. The area is bounded by the curve y^2=x^3, the line x=1m, and the x-axis.
For the composite body in Example 9.10, the area of the triangular part (1) is 4.5, the area of the square part (2) is 9, and the area of the hole (3) is -2. The respective x-coordinates of their centroids are 1, -1.5, and -2.5. What is the overall x-coordinate?
When can a single integration be used to find the centroid of an area defined by y=f(x)?
What is the centroid (x_bar, y_bar) of the area of a quarter circle of radius R in the first quadrant?
The center of mass is derived from the center of gravity equations by substituting dW with what expression?
A plate is made in the shape of the shaded area in Problem 9-17, bounded by y = h - (h/a^2)*x^2 and the x-axis. The shape is symmetric about the y-axis. What can be concluded about the x-coordinate of its centroid?
For the cardioid r = a(1 - cos(theta)) shown in Problem 9-19, what method would be most appropriate for finding the centroid?
To determine the center of gravity of the water tank shown on the chapter's title page (and in Problem 9-81), what is the most practical first step?
The composite body equations for the centroid (e.g., x_bar = Sigma(x_tilde*A)/Sigma(A)) are analogous to the integral equations (e.g., x_bar = integral(x_tilde*dA)/integral(dA)). What does the summation operator (Sigma) in the composite equations replace from the integral equations?