5.3 Composite bodies and Pappus-Guldinus theorems

Composite bodies and Pappus-Guldinus theorems are key concepts in understanding distributed forces. These tools help us analyze complex shapes by breaking them down into simpler parts and calculating their centroids and centers of gravity.

The Pappus-Guldinus theorems are powerful shortcuts for finding surface areas and volumes of solids of revolution. By focusing on centroids and generating curves, these theorems simplify calculations for irregular shapes, making them invaluable in engineering and design.

Centroids and Centers of Gravity

Composite Bodies and Their Centroids

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• Composite bodies are formed by combining multiple simple shapes (rectangles, triangles, circles) into a single complex shape
• The centroid of a composite body is the geometric center of the shape, considering the size, shape, and location of each individual component
  • It can be found by dividing the body into simpler shapes, calculating the centroid of each shape, and then using a weighted average to determine the overall centroid
• To find the centroid of a composite body, use the following formulas:
  • $x̄ = (ΣA_i * x̄_i) / ΣA_i$, where $x̄$ is the x-coordinate of the centroid, $A_i$ is the area of each individual shape, and $x̄_i$ is the x-coordinate of the centroid of each individual shape
  • $ȳ = (ΣA_i * ȳ_i) / ΣA_i$, where $ȳ$ is the y-coordinate of the centroid, $A_i$ is the area of each individual shape, and $ȳ_i$ is the y-coordinate of the centroid of each individual shape
• When working with composite bodies, it is essential to consider the position and orientation of each individual shape relative to a common reference point or coordinate system

Centers of Gravity in Composite Bodies

• The center of gravity of a composite body is the point at which the body would balance if suspended from that point, assuming uniform density throughout the body
  • For objects with uniform density, the center of gravity coincides with the centroid
• To find the center of gravity of a composite body, use the same formulas as for centroids:
  • $x̄ = (ΣA_i * x̄_i) / ΣA_i$
  • $ȳ = (ΣA_i * ȳ_i) / ΣA_i$
• When solving problems involving composite bodies and their centers of gravity:
  1. Clearly identify the individual shapes that make up the composite body and their respective dimensions, areas, and centroids
  2. Establish a consistent coordinate system and determine the position of each individual shape's centroid relative to the chosen reference point
  3. Use the formulas for calculating the center of gravity of a composite body
  4. Verify the reasonableness of your solution by checking if the calculated center of gravity falls within the physical bounds of the composite body

Pappus-Guldinus Theorem for Surface Area

Pappus-Guldinus First Theorem

• Pappus-Guldinus first theorem, also known as Pappus's centroid theorem, relates the surface area of a solid of revolution to the arc length of the generating curve and the distance traveled by the centroid of the generating curve
• The theorem states that the surface area of a solid of revolution is equal to the product of the arc length of the generating curve and the distance traveled by the centroid of the generating curve as it revolves around the axis of revolution
• Mathematically, the surface area (SA) is given by: $SA = 2πr * L$, where $r$ is the distance from the centroid of the generating curve to the axis of revolution, and $L$ is the arc length of the generating curve

Applying Pappus-Guldinus First Theorem

• To apply Pappus-Guldinus first theorem, follow these steps:
  1. Identify the generating curve and the axis of revolution
  2. Calculate the arc length ($L$) of the generating curve
  3. Determine the distance ($r$) from the centroid of the generating curve to the axis of revolution
  4. Substitute the values of $L$ and $r$ into the formula $SA = 2πr * L$ to find the surface area of the solid of revolution
• The theorem is particularly useful for calculating the surface area of solids of revolution when the generating curve is irregular or complex, as it simplifies the calculation by focusing on the centroid and arc length of the curve
• Example: A parabolic arch with a span of 10 meters and a height of 2 meters rotates around its base to form a dome. The surface area of the dome can be calculated using Pappus-Guldinus first theorem by finding the arc length of the parabola and the distance from its centroid to the axis of revolution (base)

Pappus-Guldinus Theorem for Volume

Pappus-Guldinus Second Theorem

• Pappus-Guldinus second theorem relates the volume of a solid of revolution to the area of the generating plane region and the distance traveled by the centroid of the generating plane region as it revolves around the axis of revolution
• The theorem states that the volume of a solid of revolution is equal to the product of the area of the generating plane region and the distance traveled by the centroid of the generating plane region as it revolves around the axis of revolution
• Mathematically, the volume ($V$) is given by: $V = 2πr * A$, where $r$ is the distance from the centroid of the generating plane region to the axis of revolution, and $A$ is the area of the generating plane region

Applying Pappus-Guldinus Second Theorem

• To apply Pappus-Guldinus second theorem, follow these steps:
  1. Identify the generating plane region and the axis of revolution
  2. Calculate the area ($A$) of the generating plane region
  3. Determine the distance ($r$) from the centroid of the generating plane region to the axis of revolution
  4. Substitute the values of $A$ and $r$ into the formula $V = 2πr * A$ to find the volume of the solid of revolution
• The theorem simplifies volume calculations for solids of revolution, especially when the generating plane region is irregular or complex, by focusing on the centroid and area of the region
• Example: A semicircular region with a radius of 5 meters rotates around its diameter to form a sphere. The volume of the sphere can be calculated using Pappus-Guldinus second theorem by finding the area of the semicircle and the distance from its centroid to the axis of revolution (diameter)

Composite Bodies and Centroids

Solving Problems with Composite Bodies

• When solving problems involving composite bodies and their centroids or centers of gravity:
  1. Begin by clearly identifying the individual shapes that make up the composite body and their respective dimensions, areas, and centroids
  2. Establish a consistent coordinate system and determine the position of each individual shape's centroid relative to the chosen reference point
  3. Use the formulas for calculating the centroid of a composite body:
     • $x̄ = (ΣA_i * x̄_i) / ΣA_i$
     • $ȳ = (ΣA_i * ȳ_i) / ΣA_i$
  4. Pay close attention to units and ensure consistency throughout the problem-solving process
  5. Verify the reasonableness of your solution by checking if the calculated centroid or center of gravity falls within the physical bounds of the composite body

Applying Pappus-Guldinus Theorems to Composite Bodies

• If the problem involves a solid of revolution generated by a composite curve or region, apply Pappus-Guldinus first or second theorem as appropriate:
  • For surface area: $SA = 2πr * L$
  • For volume: $V = 2πr * A$
• When using Pappus-Guldinus theorems, calculate the centroid of the composite generating curve or region to determine the distance ($r$) from the centroid to the axis of revolution
• Example: A composite region consists of a rectangle (4 meters by 2 meters) and a semicircle (radius 2 meters) attached to one of the rectangle's sides. If this region rotates around the opposite side of the rectangle, the volume of the resulting solid can be calculated using Pappus-Guldinus second theorem by finding the area of the composite region and the distance from its centroid to the axis of revolution