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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 6.7.24b
Chapter 6, Problem 6.7.24b

Compressing and stretching a spring Suppose a force of 15 N is required to stretch and hold a spring 0.25 m from its equilibrium position.
b. How much work is required to compress the spring 0.2 m 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 \( F = 15 \) N and \( x = 0.25 \) m, solve for \( k \) by rearranging the formula to \( k = \frac{F}{x} \).
Recall that the work \( W \) done in compressing or stretching a spring from the equilibrium position to a displacement \( x \) is given by the integral of the force over the distance, which simplifies to the formula \( W = \frac{1}{2}kx^2 \).
Substitute the value of \( k \) found in step 1 and the compression distance \( x = 0.2 \) m into the work formula \( W = \frac{1}{2}kx^2 \).
Set up the expression for the work done: \( W = \frac{1}{2} \times k \times (0.2)^2 \).
Evaluate the expression to find the amount of work required to compress the spring 0.2 m from its equilibrium position (do not calculate the final numeric value here, just set up the expression).

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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 given force and displacement values.
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Work Done On A Spring (Hooke's Law)

Spring Constant Calculation

The spring constant k measures the stiffness of a spring and is calculated by dividing the applied force by the displacement (k = F/x). Knowing k allows us to analyze forces and work involved in stretching or compressing the spring at different distances.
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Work Done On A Spring (Hooke's Law)

Work Done on a Spring

The work done to compress or stretch a spring is the energy stored in it, calculated by the integral of force over displacement. For a spring, work W = (1/2)kx², representing the area under the force-displacement curve, which is essential for finding the energy required to compress the spring.
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Work Done On A Spring (Hooke's Law)
Related Practice
Textbook Question

40–43. Population growth


A culture of bacteria in a Petri dish has an initial population of 1500 cells and grows at a rate (in cells/day) of N′(t) = 100e^−0.25t. Assume t is measured in days.


a. What is the population after 20 days? After 40 days?

Textbook Question

Distance traveled and displacement Suppose an object moves along a line with velocity (in ft/s) v(t)=6−2t, for 0≤t≤6, where t is measured in seconds.


b. Find the displacement of the object on the interval 0≤t≤6.

Textbook Question

40–43. Population growth


Starting with an initial value of P(0)=55, the population of a prairie dog community grows at a rate of P′(t)=20−t/5 (prairie dogs/month), for 0≤t≤200, where t is measured in months.


b. Find the population P(t), for 0≤t≤200.

Textbook Question

Find the area of the region (see figure) in two ways.

a. Using integration with respect to x.

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]?

Textbook Question

13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.


b. Find the displacement over the given interval. 


v(t) = 50e^−2t on [0, 4]