The centroid, also known as the center of mass, is a crucial concept in geometry. It represents the average position of a figure’s points, providing insight into its balance and distribution. In the case of a parabolic arc, a curve defined by a quadratic equation, finding the centroid is essential for comprehending its physical and mathematical properties.
To embark on this journey, we must first lay the mathematical groundwork. A parabolic arc can be described by the equation y = ax^2 + bx + c, where a, b, and c are constants. By employing integral calculus, we dissect the parabolic arc into infinitesimal vertical strips, each of which contributes a tiny bit of mass to the overall system. Integrating these infinitesimal contributions allows us to determine the total mass and the x-coordinate of the centroid.
However, the ‘y’ coordinate of the centroid requires a different approach. We employ the concept of moments, which measures the tendency of a mass to rotate about a point. By calculating the moments of the infinitesimal strips around the x-axis, we can determine the ‘y’ coordinate of the centroid. Once both coordinates are known, the centroid can be pinpointed, offering valuable information about the parabolic arc’s distribution and behavior.
How to Find the Centroid of a Parabolic Arc
A parabolic arc is a portion of a parabola, which is a U-shaped curve that opens either upward or downward and is symmetrical about its axis of symmetry. The centroid of a figure is the geometric center of its area. To find the centroid of a parabolic arc, we need to use the integral calculus.
Centroid of Parabolic Arc
Consider a parabolic arc y = ax2 + bx + c, where a ≠ 0. Without loss of generality, we let x = 0 at the vertex. So, the coordinates of the endpoints are (-h, 0) and (h, 0), where h = -b/2a.
The formula for the centroid of the parabolic arc is given by:
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(x̄, ȳ) = (0, 3h/8)
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Proof
We first find the area of the parabolic arc using integration:
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A = ∫[-h, h] (ax2 + bx + c) dx = 2ah3/3
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Next, we find the x-coordinate of the centroid using the formula:
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x̄ = (1/A) ∫[-h, h] x(ax2 + bx + c) dx = 0
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This shows that the x-coordinate of the centroid is 0. This is expected since the parabolic arc is symmetric about the y-axis.
Finally, we find the y-coordinate of the centroid using the formula:
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ȳ = (1/A) ∫[-h, h] (1/2)(ax2 + bx + c)2 dx = 3h/8
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Therefore, the centroid of the parabolic arc is (0, 3h/8).
