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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.7.23d
Chapter 6, Problem 6.7.23d

Compressing and stretching a spring Suppose a force of 30 N is required to stretch and hold a spring 0.2 m from its equilibrium position.
d. How much additional work is required to stretch the spring 0.2m if it has already been stretched 0.2m from its equilibrium position?

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1
Identify the spring constant \( k \) using Hooke's Law, which states that the force \( F \) required to stretch or compress a spring is proportional to the displacement \( x \) from its equilibrium position: \( F = kx \). Given \( F = 30 \) N and \( x = 0.2 \) m, solve for \( k \) by rearranging the formula to \( k = \frac{F}{x} \).
Recall the formula for the work done in stretching or compressing a spring from position \( x = a \) to \( x = b \), which is given by the integral of the force over the displacement: \( W = \int_{a}^{b} kx \, dx \).
Set the limits of integration to represent the additional stretch from \( 0.2 \) m to \( 0.4 \) m, since the spring is already stretched 0.2 m and we want the work to stretch it an additional 0.2 m.
Evaluate the integral \( W = \int_{0.2}^{0.4} kx \, dx \) by finding the antiderivative \( \frac{kx^2}{2} \) and then computing the difference \( \frac{k(0.4)^2}{2} - \frac{k(0.2)^2}{2} \).
Interpret the result as the additional work required to stretch the spring from 0.2 m to 0.4 m beyond its equilibrium position.

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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 needed to stretch or compress a spring is proportional to the displacement from its equilibrium position, expressed as F = kx, where k is the spring constant and x is the displacement. This law helps determine the spring constant from the given force and displacement.
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Work Done On A Spring (Hooke's Law)

Work Done by a Variable Force

The work done in stretching or compressing a spring is calculated by integrating the force over the displacement, since the force varies with position. For a spring, work W = (1/2) k x^2, representing the energy stored in the spring when stretched or compressed.
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Work Done On A Spring (Hooke's Law)

Additional Work for Incremental Stretching

To find the additional work required to stretch a spring further from an already stretched position, calculate the difference in work done between the final and initial displacements. This accounts for the extra energy needed beyond the initial stretch.
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Introduction To Work
Related Practice
Textbook Question

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.


Region R is revolved about the x-axis to form a solid of revolution whose cross sections are washers.


d. Write an integral for the volume of the solid.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


d. A particular marginal cost function has the property that it is positive and decreasing. The cost of increasing production from A units to 2A units is greater than the cost of increasing production from 2A units to 3A units.

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Textbook Question

Where do they meet? Kelly started at noon (t=0) riding a bike from Niwot to Berthoud, a distance of 20 km, with velocity v(t) = 15 / (t + 1)² (decreasing because of fatigue). Sandy started at noon (t=0) riding a bike in the opposite direction from Berthoud to Niwot with velocity u(t) = 20 / (t + 1)² (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours.


d. More generally, if the riders’ speeds are v(t)=A(t+1)² and u(t)=B(t+1)² and the distance between the towns is D, what conditions on A, B, and D must be met to ensure that the riders will pass each other?

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Textbook Question

Piecewise velocity The velocity of a (fast) automobile on a straight highway is given by the function

v(t)={3t if 0≤t<2060 if 20≤t<45240−4t if t≥45v(t)= \(\begin{cases}\)3 t & \(\text\) { if } 0 \(\leq\) t<20 \\ 60 & \(\text\) { if } 20 \(\leq\) t<45 \\ 240-4 t & \(\text\) { if } t \(\geq\) 45\(\end{cases}\)

where is measured in seconds and v has units of m/s. 

d. What is the position of the automobile when t=75?

Textbook Question

Displacement and distance from velocity Consider the graph shown in the figure, which gives the velocity of an object moving along a line. Assume time is measured in hours and distance is measured in miles. The areas of three regions bounded by the velocity curve and the t-axis are also given.

d. What is the displacement of the object over the interval [0,5]?

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample. 


d. Let f(x)=12x^2.. The area of the surface generated when the graph of f on [−4, 4] is revolved about the y-axis is twice the area of the surface generated when the graph of f on [0, 4] is revolved about the y-axis.