Explanation Draw the free body diagram of the cup filled with liquid at a height h as in Figure 1. Express the surface area on the inside of the cup using the theorem of Pappus and Guldinus. A = ∑ θ r ¯ L (I) Here, the angle of revolution measured in radians is θ , the perpendicular distance from the axis of revolution to the centroid of the composite part is r ¯ , and the length of the composite part 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) Refer to Figure 1 and calculate the value of x . x = 20 h 50 = 2 h 5 (III) Calculate the total surface area of the cup. A = 2 π ∑ r ¯ L (IV) Conclusion: Refer to Figure (1) and use a 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 20 2 + 50 2 = 53.85 20 1077.033 2 10 5 50 ∑ 1127.03 Refer to Table (1), Substitute 1 , 127.03 mm 2 for ∑ r ¯ L in Equation (II). A = 2 π × 1 , 127.03 mm 2 = 7 , 081.357 mm 2 (V) Refer to Equation (V) and calculate the height h when liquid is poured into the cup so that it contains half the surface area on the inside of the cup. A = 1 2 ( 7 , 081.357 mm 2 ) = 3 , 540.678 mm 2 (VI) Refer to Figure (1) and use a spreadsheet to calculate the value of L , r ¯ , and r ¯ L for two segments as shown in Table 2

13th Edition

ISBN: 9780132915540

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Chapter 9.3, Problem 111P

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The height of the liquid poured into the cup (h) .

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The height of the liquid poured into the cup ( h ) .

Solution Summary: The author explains how to determine the height of the liquid poured into the cup using the theorem of Pappus and Guldinus.

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