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A force of 20 lbs is required to stretch a spring from its natural length of 6 inches to a length...

Question:

A force of 20 lbs is required to stretch a spring from its natural length of 6 inches to a length of 8 inches. Find the work done in stretching the spring.

1. From its natural length to a length of 10 inches

2. From a length of 7 inches to a length of 9 inches

Find the Work Using Hooke's Law:

To solve this problem we will use the following theorem:

Hooke's Law : {eq}F = kx {/eq}

Where {eq}x {/eq} is the distance the spring is stretched from its natural length and {eq}k {/eq} is the spring constant.

Answer and Explanation: 1

We are given:

{eq}F = 20 \, lbs \, \, and \, \, x = (8 - 6) = 2 \, inches \, or \, \frac{2}{12} \, feet \,\, or \,\, \frac{1}{6} \, feet {/eq}

So, using the Hooke's Law we can write:

{eq}20 = k(\frac{1}{6}) \\ \Rightarrow k = 20 \times 6 = 120 {/eq}

If force required to stretch the spring {eq}x {/eq} units then {eq}F(x) = kx {/eq}

Again, we can also write:

{eq}F(x) = 120x {/eq}


Then, the work:

{eq}W = \int_a^b F(x) \, dx {/eq}

1.)

When:

{eq}a = (6 - 6) = 0 \, inches \, or \, 0 \, feet \, \, \, and \, \, b = (10 - 6) = 4 \, inches \, or \, \frac{4}{12} \, feet \,\, or \,\, \frac{1}{3} \, feet {/eq}

So, work done is:

{eq}\begin{align*} W&= \int_{0}^{ \frac{1}{3}} 120 x \, dx \\ &= 120 \int_{0}^{\frac{1}{3}} x \, dx \\ &= \frac{120}{2} \left[ x^{2} \right]_{0}^{\frac{1}{3}} \\ &= \frac{60}{9} \approx 6.67 \end{align*} {/eq}

So, work done is = {eq}6.67 ft-lb {/eq}


2.)

When:

{eq}a = (7 - 7) = 0 \, inches \, or \, 0 \, feet \, \, \, and \, \, b = (9 - 7) = 2 \, inches \, or \, \frac{2}{12} \, feet \,\, or \,\, \frac{1}{6} \, feet {/eq}

So, work done is:

{eq}\begin{align*} W&= \int_{0}^{ \frac{1}{6}} 120 x \, dx \\ &= 120 \int_{0}^{\frac{1}{6}} x \, dx \\ &= \frac{120}{2} \left[ x^{2} \right]_{0}^{\frac{1}{6}} \\ &= \frac{60}{36} \approx 1.67 \end{align*} {/eq}

So, work done is = {eq}1.67 ft-lb {/eq}


Learn more about this topic:

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Hooke's Law & the Spring Constant: Definition & Equation

from

Chapter 4 / Lesson 19
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After watching this video, you will be able to explain what Hooke's Law is and use the equation for Hooke's Law to solve problems. A short quiz will follow.


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