Explanation Draw the free body diagram of the ring formed by rotating the square about the vertical axis as in Figure 1. Express the surface area of the ring using the theorem of Pappus and Guldinus. A = θ × ∑ r ¯ L (I) Here, the surface area of revolution is A , angle of revolution measured in radians is θ , perpendicular distance from the axis of revolution to the centroid of the generating curve is r ¯ , and length of the generating curve is L . Since the angle of revolution is measured in radians, θ = 2 π rad Substitute 2 π rad for θ in Equation (I). A = 2 π × ∑ r ¯ L (II) Express the volume of the water in the tank using the theorem of Pappus and Guldinus. V = θ × r ¯ A (III) Here, the generating area is A , perpendicular distance from the axis of revolution to the centroid of the generating area is r ¯ , volume of revolution is V , and angle of revolution measured in radians is r . Since the angle of revolution is measured in radians, θ = 2 π rad Substitute 2 π rad for θ in Equation (III). V = 2 π × r ¯ A (IV) Conclusion: Refer to the Figure (1) and use spreadsheet to calculate the value of L , r ¯ , and r ¯ L for two segments as shown in Table (1). Segment L ( mm ) r ¯ ( mm ) r ¯ L ( mm 2 ) 1 a 2 ( b − a 2 sin 45 ° ) 2 a ( b − a 2 sin

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Author: HIBBELER

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Chapter 9, Problem 103P

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The surface area of the ring (A) .

The volume of the ring (V) .

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Chapter 9, Problem 102P

Chapter 9, Problem 104P

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The surface area of the ring ( A ) . The volume of the ring ( V ) .

Solution Summary: The author illustrates the surface area of the ring using the theorem of Pappus and Guldinus.

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The surface area of the ring (A) .

The volume of the ring (V) .

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Chapter 9 Solutions

The surface area of the ring ( A ) . The volume of the ring ( V ) .

Solution Summary: The author illustrates the surface area of the ring using the theorem of Pappus and Guldinus.

Concept explainers

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The surface area of the ring (A) . The volume of the ring (V) .

Want to see the full answer?

Chapter 9 Solutions

The surface area of the ring (A) .

The volume of the ring (V) .