Textbook Question
Given the velocity function of an object moving along a line, explain how definite integrals can be used to find the displacement of the object.
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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 6.7.60a
A vertical spring A 10-kg mass is attached to a spring that hangs vertically and is stretched 2 m from the equilibrium position of the spring. Assume a linear spring with F(x) = kx.
a. How much work is required to compress the spring and lift the mass 0.5 m?
Verified step by step guidance1
Identify the forces and displacements involved: The mass stretches the spring 2 m from its equilibrium position, so the spring constant \( k \) can be found using Hooke's Law \( F = kx \), where \( F \) is the weight of the mass and \( x \) is the displacement.
Calculate the spring constant \( k \) by setting the force equal to the weight of the mass: \( F = mg = kx \), where \( m = 10 \) kg, \( g = 9.8 \) m/s\^2, and \( x = 2 \) m. Rearrange to find \( k = \frac{mg}{x} \).
Determine the total displacement for the work calculation: compressing the spring and lifting the mass by 0.5 m means the spring is compressed from its stretched position by 0.5 m, so the new displacement from equilibrium is \( 2 - 0.5 = 1.5 \) m.
Set up the integral for the work done in compressing the spring from 2 m to 1.5 m using the formula for work done by a variable force: \( W = \int_{x_1}^{x_2} F(x) \, dx = \int_{1.5}^{2} kx \, dx \).
Evaluate the integral to find the work done: integrate \( kx \) with respect to \( x \) over the limits from 1.5 to 2, which gives \( W = \left[ \frac{kx^2}{2} \right]_{1.5}^{2} \). This expression represents the work required to compress the spring and lift the mass.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hooke's Law
Hooke's Law states that the force exerted by a linear spring is proportional to its displacement from equilibrium, expressed as F(x) = kx, where k is the spring constant and x is the displacement. This relationship is fundamental for calculating forces and work involving springs.
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Work Done On A Spring (Hooke's Law)
Work Done by a Variable Force
When a force varies with position, the work done is found by integrating the force over the displacement. For a spring, work is calculated as the integral of F(x) = kx from the initial to final position, representing the energy required to compress or stretch the spring.
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05:40Work Done On A Spring (Hooke's Law)
Gravitational Potential Energy
Lifting a mass against gravity increases its gravitational potential energy, calculated as mgh, where m is mass, g is acceleration due to gravity, and h is height. This energy must be included when determining the total work done in lifting the mass.
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05:59Determining Concavity Given a Function
Related Practice
Textbook Question
6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.
Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.
a. What is the radius of a cylindrical shell at a point x in [0, 4]?
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Textbook Question
Depletion of natural resources Suppose r(t) = r0e^−kt, with k>0, is the rate at which a nation extracts oil, where r0=10⁷ barrels/yr is the current rate of extraction. Suppose also that the estimate of the total oil reserve is 2×10⁹ barrels.
a. Find Q(t), the total amount of oil extracted by the nation after t years.
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Textbook Question
Emptying a water trough A water trough has a semicircular cross section with a radius of 0.25 m and a length of 3 m (see figure).
a. How much work is required to pump the water out of the trough (to the level of the top of the trough) when it is full?
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Textbook Question
17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.
a. Determine the position function, for t≥0, using the antiderivative method
v(t) = 6−2t on [0, 5]; s(0)=0
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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.
b. Suppose P=10, A=20, and r=0. If the initial population is N(0)=100, does the population ever become extinct? Explain.
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