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17Calculus Physics - Moments, Center of Mass and Centroids - Pappus-Guldinus Theorems

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Pappus-Guldinus Theorems

Pappus-Guldinus Theorem 1

Here is a great video explaining the first theorem of Pappus-Guldinus.

Michel vanBiezen - Mechanical Engineering: Pappus-Guldinus Theorem 1 Explained [3min-3sec]

video by Michel vanBiezen

Pappus-Guldinus Theorem 2

Here is a great video explaining the second theorem of Pappus-Guldinus.

Michel vanBiezen - Mechanical Engineering: Pappus-Guldinus Theorem 2 Explained [2min-49sec]

video by Michel vanBiezen

Pappus-Guldinus Theorems Side-By-Side

Here is a great video explaining the theorems of Pappus-Guldinus side-by-side with an example. Here is the example. He has the same figure, a semi-circle, shown below. For the first part of the example, the semi-circle with radius \(R\) is a line which is rotated about the x-axis and he calculates the resulting surface area. For the second part of the example, the semi-circle defines an area which is also rotated about the x-axis. This results in a volume which he calculates. At the end, he compares the results.

Line of Rotation

\( \bar{y} = 2R/\pi \)

Area of Rotation

\( \bar{y} = 4R/(3\pi) \)

Michel vanBiezen - Mechanical Engineering: using Pappus-Guldinus [3min-42sec]

video by Michel vanBiezen

Okay, you should have all the information you need to solve these problems.

How to Read and Do Proofs: An Introduction to Mathematical Thought Processes

Practice

Unless otherwise instructed, solve these problems using the theorems on this page and give your answers in exact simplified form.

Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the straight line from \((0,0)\) to \((6,3)\) about the x-axis.

Problem Statement

Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the straight line from \((0,0)\) to \((6,3)\) about the x-axis.

Final Answer

\( A = 9\pi\sqrt{5} \approx 63.2 \)
We were not given units in the question, so none are required in the answer or you could say the answer is in square units.

Problem Statement

Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the straight line from \((0,0)\) to \((6,3)\) about the x-axis.

Solution

Michel vanBiezen - 3586 video solution

video by Michel vanBiezen

Final Answer

\( A = 9\pi\sqrt{5} \approx 63.2 \)
We were not given units in the question, so none are required in the answer or you could say the answer is in square units.

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Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the quarter circle with radius \(R\) shown in the figure about the x-axis.

Problem Statement

Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the quarter circle with radius \(R\) shown in the figure about the x-axis.

Final Answer

\( A = 2\pi R^2 \) units2

Problem Statement

Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the quarter circle with radius \(R\) shown in the figure about the x-axis.

Solution

Michel vanBiezen - 3587 video solution

video by Michel vanBiezen

Final Answer

\( A = 2\pi R^2 \) units2

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Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the quarter circle with radius \(R\) shown in the figure about the x-axis.

Problem Statement

Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the quarter circle with radius \(R\) shown in the figure about the x-axis.

Hint

From the previous problem, the center of mass is at the point \( (2R/\pi, 2R/\pi ) \)

Problem Statement

Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the quarter circle with radius \(R\) shown in the figure about the x-axis.

Final Answer

\( A = \pi R^2( \pi - 2 ) \) units2

Problem Statement

Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the quarter circle with radius \(R\) shown in the figure about the x-axis.

Hint

From the previous problem, the center of mass is at the point \( (2R/\pi, 2R/\pi ) \)

Solution

Michel vanBiezen - 3588 video solution

video by Michel vanBiezen

Final Answer

\( A = \pi R^2( \pi - 2 ) \) units2

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Use the theorems of Pappus-Guldinus to calculate the area produced when revolving circle with radius \(R = 2\) shown in the figure about the x-axis.   Units are in centimeters.

Problem Statement

Use the theorems of Pappus-Guldinus to calculate the area produced when revolving circle with radius \(R = 2\) shown in the figure about the x-axis.   Units are in centimeters.

Final Answer

\( A = 40\pi^2 \) cm2

Problem Statement

Use the theorems of Pappus-Guldinus to calculate the area produced when revolving circle with radius \(R = 2\) shown in the figure about the x-axis.   Units are in centimeters.

Solution

Michel vanBiezen - 3589 video solution

video by Michel vanBiezen

Final Answer

\( A = 40\pi^2 \) cm2

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Use the theorems of Pappus-Guldinus to calculate the area produced when revolving semi-circular line segment with radius \(R = 2\) shown in the figure about the x-axis.   Units are in centimeters.

Problem Statement

Use the theorems of Pappus-Guldinus to calculate the area produced when revolving semi-circular line segment with radius \(R = 2\) shown in the figure about the x-axis.   Units are in centimeters.

Solution

Michel vanBiezen - 3590 video solution

video by Michel vanBiezen

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Use the theorems of Pappus-Guldinus to calculate the area produced by revolving the straight line segment from the point \( (4,2) \) to the point \( (12,10) \) about the x-axis.

Problem Statement

Use the theorems of Pappus-Guldinus to calculate the area produced by revolving the straight line segment from the point \( (4,2) \) to the point \( (12,10) \) about the x-axis.

Solution

Michel vanBiezen - 3591 video solution

video by Michel vanBiezen

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Use the theorems of Pappus-Guldinus to calculate the volume produced by the circle with radius \(R = 3\)cm with center \( 10 \)cm above the x-axis when revolved about the x-axis.

Problem Statement

Use the theorems of Pappus-Guldinus to calculate the volume produced by the circle with radius \(R = 3\)cm with center \( 10 \)cm above the x-axis when revolved about the x-axis.

Solution

Michel vanBiezen - 3592 video solution

video by Michel vanBiezen

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Use the theorems of Pappus-Guldinus to calculate the volume produced when revolving the object shown in the figure about the x-axis.

Problem Statement

Use the theorems of Pappus-Guldinus to calculate the volume produced when revolving the object shown in the figure about the x-axis.

Solution

3593 video solution

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Use the theorems of Pappus-Guldinus to calculate the volume produced when revolving the object shown in the figure about the x-axis.   The curved line is the equation \( y = 3x^2 \)

Problem Statement

Use the theorems of Pappus-Guldinus to calculate the volume produced when revolving the object shown in the figure about the x-axis.   The curved line is the equation \( y = 3x^2 \)

Solution

Michel vanBiezen - 3594 video solution

video by Michel vanBiezen

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Unless otherwise instructed, solve these problems using the theorems on this page and give your answers in exact simplified form.

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