Textbook Question
9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.
a. The displacement between t=0 and t=5
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Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 6, Problem 6.7.31a
Winding a chain A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and the chain has a density of 5kg/m.
a. How much work is required to wind the entire chain onto the cylinder using the winch?
Verified step by step guidance1
Identify the physical quantities involved: the length of the chain \(L = 30\) m, the linear density \(\rho = 5\) kg/m, and gravitational acceleration \(g = 9.8\) m/s² (assuming standard gravity).
Express the mass of a small segment of the chain of length \(dx\) at a distance \(x\) from the bottom as \(dm = \rho \\ dx\), where \(x\) measures the length of chain lifted from the bottom upwards.
Determine the work required to lift this small segment \(dm\) by a height \(x\) (since the chain is being wound up from the bottom, each segment must be lifted a distance equal to its position \(x\)). The infinitesimal work is \(dW = (dm)(g)(x) = \rho g x \\ dx\).
Set up the integral for the total work by integrating \(dW\) over the entire length of the chain from \(x=0\) (bottom) to \(x=L=30\) m: \(W = \int_0^{30} \rho g x \\ dx\).
Evaluate the integral to find the total work done in winding the entire chain onto the cylinder.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Work Done by a Variable Force
Work is the integral of force over distance. When the force varies with position, such as lifting different parts of a chain, we calculate work by integrating the force function over the displacement. This approach accounts for the changing weight as the chain is wound up.
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Linear Density and Weight Calculation
Linear density is the mass per unit length of an object, here given as 5 kg/m. Multiplying linear density by gravitational acceleration and length gives the weight of a segment of the chain. This helps determine the force needed to lift varying lengths of the chain.
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Setting up and Evaluating Definite Integrals
To find total work, we set up a definite integral representing the sum of infinitesimal work elements over the chain's length. Each element corresponds to lifting a small segment of the chain a certain distance. Evaluating this integral yields the total work required.
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Related Practice
Textbook Question
Consider a solid whose base is the region in the first quadrant bounded by the curve y=√3−x and the line x=2, and whose cross sections through the solid perpendicular to the x-axis are squares.
a. Find an expression for the area A(x) of a cross section of the solid at a point x in [0, 2].
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Textbook Question
Oil production An oil refinery produces oil at a variable rate given by Q'(t) = <1x3 matrix>, where is measured in days and is measured in barrels.
a. How many barrels are produced in the first 35 days?
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Textbook Question
Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.
a. Suppose P=10, A=20, and r=0. If the initial population is N(0)=10, does the population ever become extinct? Explain.
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Textbook Question
Find the area of the region (see figure) in two ways.
a. Using integration with respect to x.
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Textbook Question
Determine whether the following statements are true and give an explanation or counterexample.
a. The area of the region bounded by y=x and x=y^2 can be found only by integrating with respect to x.
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