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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 6.7.25b
Chapter 6, Problem 6.7.25b

Work done by a spring A spring on a horizontal surface can be stretched and held 0.5 m from its equilibrium position with a force of 50 N.
b. How much work is done in compressing the spring 0.5 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 \( x \): \[ F = k \times x \] Given \( F = 50 \) N and \( x = 0.5 \) m, solve for \( k \).
Recall that the work done \( W \) in stretching or compressing a spring from its equilibrium position to a displacement \( x \) is given by the formula for elastic potential energy: \[ W = \frac{1}{2} k x^2 \]
Substitute the value of \( k \) found in step 1 and the displacement \( x = 0.5 \) m into the work formula.
Calculate the expression to find the amount of work done in compressing the spring 0.5 m from its equilibrium position.
Interpret the result as the energy stored in the spring due to compression.

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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 exerted by a spring is proportional to its displacement from the 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 the given force and displacement.
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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 force applied by the displacement (k = F/x). Knowing k allows us to quantify how much force is needed to stretch or compress the spring by a certain amount.
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Work Done On A Spring (Hooke's Law)

Work Done by a Spring

The work done in stretching or compressing a spring is the energy stored in it, calculated by W = (1/2)kx². This formula integrates the variable force over the displacement, reflecting that more work is needed as the spring stretches further.
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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.


b. What is the SAV ratio of a ball with radius a? 

Textbook Question

Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.


c. Suppose P=10, A=50, and r=5. If the initial population is N(0)=10, does the population ever become extinct? Explain.

Textbook Question

Region R is revolved about the line y=1 to form a solid of revolution.


c. Write an integral for the volume of the solid.

Textbook Question

Probe speed A data collection probe is dropped from a stationary balloon, and it falls with a velocity (in m/s) given by v(t) = 9.8t, neglecting air resistance. After 10 s, a chute deploys and the probe immediately slows to a constant speed of 10 m/s, which it maintains until it enters the ocean.


b. How far does the probe fall in the first 30 s after it is released?

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

Textbook Question

Let R be the region bounded by the curve y=cos^−1x and the x-axis on [0, 1]. A solid of revolution is obtained by revolving R about the y-axis (see figures). 


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