Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.7.59c

Chapter 6, Problem 6.7.59c

A nonlinear spring Hooke’s law is applicable to idealized (linear) springs that are not stretched or compressed too far from their equilibrium positions. Consider a nonlinear spring whose restoring force is given by F(x) = 16x−0.1x³, for |x|≤7.
c. How much work is done in compressing the spring from its equilibrium position (x=0) to x=−2?

Verified step by step guidance

Recall that the work done by a variable force \( F(x) \) when moving an object from position \( a \) to \( b \) is given by the integral \( W = \int_a^b F(x) \, dx \).

Identify the force function given: \( F(x) = 16x - 0.1x^3 \), and the limits of compression from \( x=0 \) to \( x=-2 \).

Set up the integral for the work done in compressing the spring: \( W = \int_0^{-2} (16x - 0.1x^3) \, dx \).

Since the upper limit is less than the lower limit, consider reversing the limits and changing the sign of the integral: \( W = - \int_{-2}^0 (16x - 0.1x^3) \, dx \).

Evaluate the integral by finding the antiderivative of \( 16x - 0.1x^3 \), then substitute the limits \( -2 \) and \( 0 \) to find the work done.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Work Done by a Variable Force

Work done by a force that varies with position is calculated by integrating the force function over the displacement interval. For a spring, this means integrating the restoring force from the initial to the final position to find the total work done in compressing or stretching it.

Nonlinear Spring Force

Unlike Hooke’s law for linear springs (F = -kx), a nonlinear spring has a restoring force that depends on higher powers of displacement, such as F(x) = 16x - 0.1x³. This means the force changes in a more complex way as the spring is compressed or stretched, affecting the work calculation.

Limits of Integration and Sign Conventions

When calculating work done from equilibrium to a compressed position, it is important to set the correct limits of integration (from 0 to -2) and consider the direction of force and displacement. The sign of the force and displacement affects whether the work is positive or negative.

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