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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 6.7.23c
Chapter 6, Problem 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?

Verified step by step guidance
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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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

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.
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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.
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Work Done On A Spring (Hooke's Law)
Related Practice
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. 


c. What is the distance traveled by the automobile in the first 60 s?

Textbook Question

Emptying a water trough A water trough has a semicircular cross section with a radius of 0.25 m and a length of 3 m (see figure).

c. If the radius is doubled, is the required work doubled? Explain.

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Textbook Question

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


c. ∫₀¹(x−x^2) dx=∫₀¹(√y−y) dy

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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 y-axis to form a solid of revolution whose cross sections are washers.


c. What is the area A(y) of a cross section of the solid at a point y in [1, 3]?

Textbook Question

Day hike The velocity (in mi/hr) of a hiker walking along a straight trail is given by v(t) = 3 sin² πt/2, for 0≤t≤4. Assume s(0)=0 and t is measured in hours. 


c. What is the hiker’s position at t=3?

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Textbook Question

Flying into a headwind The velocity (in mi/hr) of an airplane flying into a headwind is given by v(t) = 30(16−t²), for 0≤t≤3. Assume s(0)=0 and t is measured in hours.


c. How far has the airplane traveled at the instant its velocity reaches 400 mi/hr?