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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

Bike race Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in mi/hr). Assume t is measured in hours.

Theo: vT(t)=10, for t≥0

Sasha: vS(t)=15t, for 0≤t≤1, and vS(t)=15, for t>1


c. If the riders ride for 2 hr, who rides farther? Interpret your answer geometrically using the graphs of part (a). 

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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?

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Textbook Question

{Use of Tech} Oscillating motion A mass hanging from a spring is set in motion, and its ensuing velocity is given by v(t) = 2π cos πt, for t≥0. Assume the positive direction is upward and s(0)=0. 


c. At what times does the mass reach its low point the first three times? 

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Textbook Question

Cycling distance A cyclist rides down a long straight road with a velocity (in m/min) given by v(t) = 400−20t, for 0≤t≤10, where t is measured in minutes.


c. How far has the cyclist traveled when her velocity is 250 m/min?

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Textbook Question

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

c. The position at t=5

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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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