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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 6.7.28b
Chapter 6, Problem 6.7.28b

Calculating work for different springs Calculate the work required to stretch the following springs 0.4 m from their equilibrium positions. Assume Hooke’s law is obeyed.
b. A spring that requires 2 J of work to be stretched 0.1 m from its equilibrium position

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1
Recall that the work done to stretch or compress a spring from its equilibrium position is given by the formula \(W = \frac{1}{2} k x^2\), where \(k\) is the spring constant and \(x\) is the displacement from equilibrium.
Use the information given for the spring: it requires 2 J of work to stretch it 0.1 m. Substitute these values into the work formula to find the spring constant \(k\): \(2 = \frac{1}{2} k (0.1)^2\).
Solve the equation for \(k\) by isolating it: multiply both sides by 2 and divide by \((0.1)^2\) to get \(k = \frac{2 \times 2}{(0.1)^2}\).
Now that you have the spring constant \(k\), use it to calculate the work required to stretch the spring 0.4 m by substituting into the work formula: \(W = \frac{1}{2} k (0.4)^2\).
Simplify the expression to find the work done for the 0.4 m stretch, which will give you the answer to the problem.

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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 required 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 linear relationship is fundamental for calculating forces and work in spring problems.
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Work Done On A Spring (Hooke's Law)

Work Done by a Variable Force

When stretching a spring, the force varies with displacement, so work is calculated as the integral of force over distance. For springs, work done W = (1/2)kx², representing the area under the force-displacement curve, which is essential for determining energy stored or work required.
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Work Done On A Spring (Hooke's Law)

Determining the Spring Constant from Work

Given the work done to stretch a spring a certain distance, the spring constant k can be found by rearranging the work formula: k = 2W / x². This allows solving for k when work and displacement are known, enabling calculation of work for other displacements.
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Work Done On A Spring (Hooke's Law)
Related Practice
Textbook Question

Two runners At noon (t=0), Alicia starts running along a long straight road at 4 mi/hr. Her velocity decreases according to the function v(t) = 4 / t + 1 for t≥0. At noon, Boris also starts running along the same road with a 2-mi head start on Alicia; his velocity is given by u(t) = 2 / t + 1, for t≥0. Assume t is measured in hours.


b. When, if ever, does Alicia overtake Boris?

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b. What is the height of a cylindrical shell at a point x in [0, 2]?

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Textbook Question

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.

b. How much work is required to wind the chain onto the cylinder if a 50-kg block is attached to the end of the chain?

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