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A spring of negligible mass has a spring constant (K) of 1600 \ N/m. a. How much potential...

Question:

A spring of negligible mass has a spring constant (K) of {eq}1600 \ N/m {/eq}.

a. How much potential energy (PE) will be stored at equilibrium?

b. How much PE will be stored if it is extended {eq}0.15 {/eq} meters away from equilibrium?

c. How far must the spring be compressed to store {eq}3.20 \ J {/eq} of potential energy?

Spring:

The potential energy can be developed in the spring by its deformation. The deformation of the spring depends on the loading condition. Generally, in the field of mechanical engineering, the use of spring is large.

Answer and Explanation: 1


Given Data


  • Spring constant, {eq}K = 1600\;{\rm{N/m}} {/eq}


(a)

The expression for the potential energy stored in the spring is


{eq}U = \dfrac{1}{2}K{x^2} {/eq}


Here {eq}x {/eq} is the displacement of the spring.


At equilibrium condition, the displacement of the spring is zero, so potential energy can be calculated as


{eq}\begin{align*} U &= \dfrac{1}{2}K{x^2}\\ U &= \dfrac{1}{2} \times 1600 \times 0\\ U &= 0\;{\rm{J}} \end{align*} {/eq}


Thus, at equilibrium the potential energy is zero.


(b)

If the spring is extended {eq}0.15\;{\rm{m}} {/eq} away, then the potential energy can be calculated as


{eq}\begin{align*} U &= \dfrac{1}{2}k{x^2}\\ U &= \dfrac{1}{2} \times 1600 \times {0.15^2}\\ U &= 800 \times 0.0225\\ U &= 18\;{\rm{J}} \end{align*} {/eq}


Thus, the potential energy is {eq}18\;{\rm{J}} {/eq}.


(c)

The compression of the spring can be calculated as


{eq}\begin{align*} U &= \dfrac{1}{2}K{x^2}\\ x &= \sqrt {\dfrac{{2U}}{K}} \\ x &= \sqrt {\dfrac{{2 \times 3.20}}{{1600}}} \\ x &= 0.063\;{\rm{m}} \end{align*} {/eq}


On further solving


{eq}x = 6.3\;{\rm{cm}} {/eq}


Thus, to store {eq}3.20\;{\rm{J}} {/eq} of potential energy, the spring must be compressed by {eq}6.3\;{\rm{cm}} {/eq}.


Learn more about this topic:

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Practice Applying Spring Constant Formulas

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Chapter 17 / Lesson 11
3.3K

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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