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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 6.7.27a
Chapter 6, Problem 6.7.27a

Calculating work for different springs Calculate the work required to stretch the following springs 0.5m from their equilibrium positions. Assume Hooke’s law is obeyed.
a. A spring that requires a force of 50 N to be stretched 0.2 m from its equilibrium position

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Identify the given information: the force required to stretch the spring 0.2 m is 50 N, and we want to find the work done to stretch it 0.5 m.
Recall Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from equilibrium: \(F = k \times x\), where \(k\) is the spring constant and \(x\) is the displacement.
Use the given force and displacement to solve for the spring constant \(k\): rearrange Hooke's Law to \(k = \frac{F}{x}\), then substitute \(F = 50\) N and \(x = 0.2\) m.
Calculate the work done to stretch the spring from 0 to 0.5 m using the formula for work done on a spring: \(W = \frac{1}{2} k x^2\), where \(x\) is the final displacement (0.5 m).
Substitute the value of \(k\) found in step 3 and \(x = 0.5\) m into the work formula to express the work required to stretch the spring 0.5 m.

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

Spring Constant (k)

The spring constant k measures the stiffness of a spring and is calculated by dividing the force applied by the displacement (k = F/x). Knowing k allows you to determine the force required for any displacement and is essential for computing the work done on the spring.
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Work Done On A Spring (Hooke's Law)

Work Done on a Spring

The work done to stretch or compress a spring is the energy stored in it, calculated by the integral of force over displacement. For springs obeying Hooke’s Law, work is W = (1/2)kx², representing the area under the force-displacement curve, which is a triangle.
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Work Done On A Spring (Hooke's Law)
Related Practice
Textbook Question

In the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. Animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. Therefore, animals with a low SAV ratio tend to retain heat, whereas animals with a high SAV ratio (such as children and hummingbirds) lose heat relatively quickly.


a. What is the SAV ratio of a cube with side lengths a?

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.

a. The displacement between t=0 and t=5

Textbook Question

Consider a solid whose base is the region in the first quadrant bounded by the curve y=√3−x and the line x=2, and whose cross sections through the solid perpendicular to the x-axis are squares.


a. Find an expression for the area A(x) of a cross section of the solid at a point x in [0, 2].

Textbook Question

21–30. {Use of Tech} Arc length by calculator


a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 

y = ln x, for 1≤x≤4

Textbook Question

Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.

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

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).

a. How much work is required to pump the water out of the trough (to the level of the top of the trough) when it is full?

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