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In a room that is 2.52 m high, a spring (unstrained length = 0.30 m)hangs from the ceiling. A...

Question:

In a room that is 2.52 m high, a spring (unstrained length = 0.30 m)hangs from the ceiling. A board whose length is 1.96 m is attached to the free end of the spring. The board hangs straight down, so that its 1.96-m length is perpendicular to the floor. The weight of the board (104 N) stretches the spring so that the lower end of the board just extends to, but does not touch, the floor. What is the spring constant of the spring?

Spring:

A mechanical component used in those devices that are subjected to high amplitude of vibrations or oscillations in order to prevent the after-effects of vibrations on the device is known as spring. It is a shock-absorbing component that stores and release energy by its compression and rarefaction.

Answer and Explanation: 1


Given Data

  • The height of the room is: {eq}{h_r} = 2.52\;{\rm{m}} {/eq}.
  • The unstrained length of the spring is: {eq}{L_{us}} = 0.30\;{\rm{m}} {/eq}.
  • The length of the board is: {eq}{L_b} = 1.96\;{\rm{m}} {/eq}.
  • The weight of the board is: {eq}{W_b} = 104\;{\rm{N}} {/eq}.


The equation to calculate the strain of the spring is given by,

{eq}x = {h_r} - {L_{us}} - {L_b}...(I) {/eq}


The expression to calculate the spring constant is given by,

{eq}\begin{align*} {W_b} &= sx\\ s &= \dfrac{{{W_b}}}{x} \end{align*} {/eq}


Substitute equation {eq}(I) {/eq}, and all the values in the above equation.

{eq}\begin{align*} s &= \dfrac{{{W_b}}}{{{h_r} - {L_{us}} - {L_b}}}\\ & = \dfrac{{104}}{{2.52 - 0.3 - 1.96}}\\ & = 400\;{\rm{N/m}} \end{align*} {/eq}


Thus the spring constant of the spring is {eq}400\;{\rm{N/m}} {/eq}.


Learn more about this topic:

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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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