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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 6 - Applications of IntegrationProblema 6.7.29a
Capitolo 6, Problema 6.7.29a

Calculating work for different springs Calculate the work required to stretch the following springs 1.25 m from their equilibrium positions. Assume Hooke’s law is obeyed.
a. A spring that requires 100 J of work to be stretched 0.5 m from its equilibrium position

Guida verificata passo dopo passo
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 first spring: it requires 100 J of work to stretch it 0.5 m. Substitute \(W = 100\) J and \(x = 0.5\) m into the formula to find the spring constant \(k\) by solving \(100 = \frac{1}{2} k (0.5)^2\).
Rearrange the equation to solve for \(k\): multiply both sides by 2 and divide by \((0.5)^2\) to isolate \(k\).
Once you have the value of \(k\), use it to calculate the work required to stretch the spring 1.25 m by substituting \(x = 1.25\) m into the work formula \(W = \frac{1}{2} k x^2\).
Evaluate the expression to find the work done for the 1.25 m stretch, which will give you the required answer.

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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: W = ∫ F dx. For springs, this results in W = (1/2) k x², representing the energy stored in the spring when stretched or compressed.
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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. Knowing k allows calculation of work for other displacements, enabling solutions to problems involving different stretch lengths.
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Work Done On A Spring (Hooke's Law)
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