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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 6.7.23c
Capitolo 6, Problema 6.7.23c

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.
c. How much work is required to stretch the spring 0.3 m from its equilibrium position?

Guida verificata passo dopo passo
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 that the work \( W \) done in stretching or compressing a spring from the equilibrium position to a displacement \( x \) is given by the integral of the force over the distance, which results in the formula \( W = \frac{1}{2}kx^2 \).
Use the value of \( k \) found in step 1 and substitute \( x = 0.3 \) m into the work formula \( W = \frac{1}{2}kx^2 \) to set up the expression for the work required to stretch the spring 0.3 m.
Write down the expression explicitly: \( W = \frac{1}{2} \times k \times (0.3)^2 \), where \( k \) is the spring constant calculated earlier.
Evaluate the expression to find the amount of work required to stretch the spring 0.3 m, remembering that this value represents the energy stored in the spring due to the stretching.

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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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Percorso guidato
05:40
Work Done On A Spring (Hooke's Law)

Spring Constant

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 quantify how much force is needed for any given stretch or compression of the spring.
Video consigliato:
Percorso guidato
05:40
Work Done On A Spring (Hooke's Law)

Work Done by a Variable Force

Work done in stretching a spring is calculated by integrating the force over the displacement, since the force varies with position. The work done to stretch a spring from 0 to x is W = (1/2)kx², representing the area under the force-displacement curve.
Video consigliato:
Percorso guidato
05:40
Work Done On A Spring (Hooke's Law)
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