Explanation Given information: The region is bounded by the curve y = 1 − x 3 3 and the coordinate axis. Formula used: A region R with uniform density is said to have moments of inertia I x about the x-axis and I y about the y-axis, where I x = ∫ y 2 d A and I y = ∫ x 2 d A The d A represents the area of an approximating strips. Calculation: The region bounded by the parabola y = 1 − x 3 3 and the coordinate axis. y = 1 − x 3 3 ⇒ x = 1 − y 3 3 I x = ∫ y 2 d A where d A = x d y The y ranges from 0 to 1. I x = ∫ y 2 d A = ∫ 0 1 y 2 ( x d y ) = ∫ 0 1 y 2 ( 1 − y 3 3 ) d y = ∫ 0 1 ( u 3 ) ( − d u 3 ) w h e r e u = 1 − y 3 ⇒ y 2 d y = − d u 3 = − 1 3 ∫ 0 1 u 1 3 d u = − 1 3 ( u 4 3 4 3 ) 0 1

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 6.5, Problem 52PS

To determine

To find: Ix and Iy for the given region.

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Chapter 6.5, Problem 51PS

Chapter 6.5, Problem 53PS

Chapter 6 Solutions

Calculus

Ch. 6.1 - Prob. 1PS Ch. 6.1 - Prob. 2PS Ch. 6.1 - Prob. 3PS Ch. 6.1 - Prob. 4PS Ch. 6.1 - Prob. 5PS Ch. 6.1 - Prob. 6PS Ch. 6.1 - Prob. 7PS Ch. 6.1 - Prob. 8PS Ch. 6.1 - Prob. 9PS Ch. 6.1 - Prob. 10PS

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Refer to Figure 29-9 to determine the values in the table. Allowance is equal to the maximum interference. All dimensions are in millimeters
Tsunami Waves and BreakwatersThis is a continuation of Exercise 16. Breakwaters affect wave height by reducing energy. See Figure 5.30. If a tsunami wave of height H in a channel of width W encounters a breakwater that narrows the channel to a width w, then the height h of the wave beyond the breakwater is given by h=HR0.5, where R is the width ratio R=w/W. a. Suppose a wave of height 8 feet in a channel of width 5000feet encounters a breakwater that narrows the channel to 3000feet. What is the height of the wave beyond the breakwater? b. If a channel width is cut in half by a breakwater, what is the effect on wave height? 16. Height of Tsunami WavesWhen waves generated by tsunamis approach shore, the height of the waves generally increases. Understanding the factors that contribute to this increase can aid in controlling potential damage to areas at risk. Greens law tells how water depth affects the height of a tsunami wave. If a tsunami wave has height H at an ocean depth D, and the wave travels to a location with water depth d, then the new height h of the wave is given by h=HR0.25, where R is the water depth ratio given by R=D/d. a. Calculate the height of a tsunami wave in water 25feet deep if its height is 3feet at its point of origin in water 15,000feet deep. b. If water depth decreases by half, the depth ratio R is doubled. How is the height of the tsunami wave affected?

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