Explanation Given: The solid generated by revolving the region bounded by y = 2 x and y = x 2 about the y axis. Formula used: Moment of inertia of a solid: The moment of inertia of a thin plate, with respect to the y axis bounded by y 1 = f 1 ( x ) and y 2 = f 2 ( x ) is given by I y = 2 π k ∫ a b ( y 2 − y 1 ) x 3 d x . The moment of inertia of a thin plate, with respect to the x axis bounded by x 1 = g 1 ( y ) and x 2 = g 2 ( y ) is given by I x = 2 π k ∫ a b ( x 2 − x 1 ) y 3 d x . Calculation: Consider the functions y = 2 x and y = x 2 . In order to find the radius of gyration of the given plate, first draw the graph of the functions y = 2 x and y = x 2 as shown below in Figure 1. From the Figure 1, the horizontal element is marked by dark shaded and area is bounded from 0 to 4. Obtain the moment of inertia of a solid as follows. By using the above formula, I y = 2 π k ∫ 0 2 ( y 2 − y 1 ) x 3 d x = 2 π k ∫ 0 2 ( 2 x − x 2 ) x 3 d x = 2 π k ∫ 0 2 ( 2 x 4 − x 5 ) d x = 2 π k [ 2 x 5 5 − x 6 6 ] 0 2 On further simplification, I x = 2 π k [ ( 2 ( 2 ) 5 5 − ( 2 ) 6 6 ) − ( 4 ( 0 ) 5 5 − ( 0 ) 6 6 ) ] = 2 π k [ ( 2 ( 32 ) 5 − 64 6 ) − ( 0 ) ] = 2 π k ( 384 − 320 30 ) = 2 π k ( 64 30 ) = 64 π k 15 Thus, the moment of inertia of a solid is I y = 64 π k 15

Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

11th Edition

ISBN: 9780137554843

Author: Allyn Washington, Richard Evans

Publisher: PEARSON+

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Question

Chapter 26.5, Problem 20E

To determine

To find: The radius of gyration of the solid with respect to y axis.

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Chapter 26.5, Problem 19E

Chapter 26.5, Problem 21E

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Chapter 26 Solutions

Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

Ch. 26.1 - In Example 4, change the acceleration to a = 4...Ch. 26.1 - Prob. 2PE Ch. 26.1 - In Example 3, change 5.0 s to 2.5 s and then solve...Ch. 26.1 - Prob. 2E Ch. 26.1 - What is the velocity (in ft/s) of a sandbag 1.5 s...Ch. 26.1 - Prob. 4E Ch. 26.1 - A conveyor belt 8.00 m long moves at 0.25 m/s. If...Ch. 26.1 - Prob. 6E Ch. 26.1 - The velocity (in km/h) of a plane flying into an...Ch. 26.1 - A cyclist goes downhill for 15 min with a velocity...

Ch. 26.1 - A car crosses an intersection as a fire engine...Ch. 26.1 - In designing a highway, a civil engineer must...Ch. 26.1 - Prob. 11E Ch. 26.1 - Prob. 12E Ch. 26.1 - A certain Chevrolet Corvette goes from 0 mi/h to...Ch. 26.1 - Prob. 14E Ch. 26.1 - Prob. 15E Ch. 26.1 - Prob. 16E Ch. 26.1 - Prob. 17E Ch. 26.1 - Prob. 18E Ch. 26.1 - Prob. 19E Ch. 26.1 - Prob. 20E Ch. 26.1 - Prob. 21E Ch. 26.1 - Prob. 22E Ch. 26.1 - Prob. 23E Ch. 26.1 - Prob. 24E Ch. 26.1 - Prob. 25E Ch. 26.1 - Prob. 26E Ch. 26.1 - The voltage across a 3.75-μF capacitor in a...Ch. 26.1 - Prob. 28E Ch. 26.1 - Prob. 29E Ch. 26.1 - Prob. 30E Ch. 26.1 - Prob. 31E Ch. 26.1 - Prob. 32E Ch. 26.1 - Prob. 33E Ch. 26.1 - Prob. 34E Ch. 26.1 - Prob. 35E Ch. 26.1 - Prob. 36E Ch. 26.2 - Find the area in the first quadrant bounded by y =...Ch. 26.2 - Prob. 2PE Ch. 26.2 - In Exercises 1 and 2, make the given changes in...Ch. 26.2 - In Exercises 1 and 2, make the given changes in...Ch. 26.2 - Prob. 3E Ch. 26.2 - Prob. 4E Ch. 26.2 - 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