Explanation 1) Concept: Approximate the required work by using the concept of Riemann sum. Then express the work as an integral and evaluate it. 2) Definition: W = lim n → ∞ ⁡ ∑ t = 1 n f x i * ∆ x = ∫ a b f x d x 3) Calculation: Let’s place the rope at the top of the height and hang it with the bucket; x - axis is pointing downward Divide the height of the rope into small parts with length ∆ x . If x i * is a point in the i t h such interval, then all points in the interval are lifted by approximately the same amount, x i * Suppose the part x i * of the rope has just reached level 12 m. Then ( 12 - x i * ) m is hanging downwards. The force acting on x i * is due to gravitational force. The total mass below x i * contributes towards the force acting on x i * . Thus the force acting on x i * is due to the sum mass of rope hanging below x i * , mass of water remaining and mass of bucket. Mass of water goes from 36kg to 0 as bucket moves from height 0 to 12. The rate of leaking is constant. So for every meter 36/12 = 3kg water is leaked out. So when the x i * part of rope has just reached top the mass of water remaining is 36-3 x i * = 3 ( 12 - x i * ) kg. Let’s compute the total mass at x i * . Total mass per meter is the sum of mass of rope, mass of bucket, and mass of water. First mass of rope By the formula, M a s s = l i n e a r d e n s i t y * l e n g t h ( 0.8 k g / m ) ( 12 - x i * m ) = 0.8 12 - x i * = 9.6 - 0.8 x i * k g Now, Mass of water 3 k g / m 12 - x i * m = 36 - 3 x i * k g And mass of the bucket is 10 k g Therefore, T o t a l m a s s = 9

8th Edition

ISBN: 9781285740621

Author: James Stewart

Publisher: Cengage Learning

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Chapter 5.4, Problem 17E

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The amount of work donewhen the bucket finishes draining just as it reaches the 12 m level.

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Chapter 5.4, Problem 16E

Chapter 5.4, Problem 18E

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The amount of work donewhen the bucket finishes draining just as it reaches the 12 m level.

Solution Summary: The author explains how the force acting on x_i* is due to gravitational force.

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To find: The amount of work donewhen the bucket finishes draining just as it reaches the 12 m level.

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Chapter 5 Solutions

To find:

The amount of work donewhen the bucket finishes draining just as it reaches the 12 m level.