A non-linear spring obeys the modified Hooke's Law, F = -kxe^x, where F is the force in Newtons,...

Question:

A non-linear spring obeys the modified Hooke's Law, {eq}F = -kxe^x {/eq}, where F is the force in Newtons, k is 2 N/m, and x is horizontal displacement if the right end of the spring in meters measured from the origin.

What is the work done in stretching this spring from the origin to x = 2?

Hooke's Law

Hooke's law in engineering defines the relationship between the force required to deform the spring by one unit and the spring constant. It is mathematically expressed as {eq}\text{F=kx} {/eq}

where F is the force in N

x is the deformation in m

Answer and Explanation: 1

Given Data:

  • Stiffness of the spring is {eq}k=2\ \text{N/m} {/eq}

The work done in stretching the spring is expressed by the relation

{eq}\begin{align} W&=F dx\\[0.3 cm] &= \int_{0}^{2}-kxe^xdx\\ \end{align} {/eq}

We have to solve the integral using integration by parts.


Integration by parts formula {eq}\displaystyle \int u \cdot vdx=u\int vdx-\int u'\left ( \int vdx \right )dx {/eq}


According to the question {eq}u=x, \ v=e^{x} {/eq}

{eq}\begin{align} &=-k\int x e^x \ dx\\[0.3 cm] &=-k \left [x\int e^xdx-\int \left (\frac{\mathrm{d} }{\mathrm{d} x}(x)\int e^xdx \right )dx \right ] & \left [\displaystyle \int u \cdot vdx=u\int vdx-\int u'\left ( \int vdx \right )dx \right ]\\[0.3cm]\\ &=-k\left [xe^x-\int (1)e^xdx \ \right ]& \left [ \frac{\mathrm{d} }{\mathrm{d} x}(x)=1, \int e^xdx=e^x \right ]\\[0.3cm] &=-k\left [xe^x-e^x \right ]_{0}^{2}&\left [\text{Spring is stretched from the origin (0) to 2m} \right ]\\[0.3cm] &=-k\left [xe^x-e^x \right ]_{0}^{2}\\[0.3 cm] &=-k\left [\left (2e^2-e^2 \right )-\left (0e^0-e^0 \right )\right ]\\[0.3 cm] &=-k\left [\left (2e^2-e^2 \right )-\left (0e^0-e^0 \right )\right ]\\[0.3 cm] &=-k\left [ (e^2)-(0-1) \right ]\\[0.3 cm] &=-2\left [ 8.4\right ]\\[0.3 cm] &=\boxed{\color{blue}{-16.8\ \text{N.m}}} \end{align} {/eq}


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Practice Applying Spring Constant Formulas

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Chapter 17 / Lesson 11
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In this lesson, you'll have the chance to practice using the spring constant formula. The lesson includes four problems of medium difficulty involving a variety of real-life applications.


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