A spring has a natural length of 30 in. A force of 2000 lb stretches the spring to 40 in. How far...

Question:

A spring has a natural length of 30 in. A force of 2000 lb stretches the spring to 40 in. How far beyond its natural length will a 500 lb force stretch the spring?

Spring Force:

The spring force is measured using the spring force constant and the extension or compression (beyond its natural length) in the spring. So, we find the expansion (or compression) in the spring using the following formula.

$${F_{spring}} = kx $$

Where,

  • The spring force constant is {eq}k {/eq}
  • The spring force is {eq}{F_{spring}} {/eq}, and the expansion (or compression) in the spring is {eq}x {/eq}

Answer and Explanation:


Given Data: -

  • The initial length of the spring is: {eq}{L_0} = 30\;{\rm{in}} {/eq}
  • The final length of the spring is: {eq}L = 40\;{\rm{in}} {/eq}
  • The initial spring force is: {eq}{F_1} = 2000\;{\rm{lb}} {/eq}
  • The final spring force is: {eq}{F_2} = 500\;{\rm{lb}} {/eq}


Calculate the expansion in the spring length beyond its natural length.

$$\begin{align*} {x_1} &= L - {L_0}\\[0.3cm] &= 40\;{\rm{in}} - 30\;{\rm{in}}\\[0.3cm] &= 10\;{\rm{in}} \end{align*} $$


Write the expression for the initial spring force applied to the spring.

$${F_1} = k{x_1} $$

Here, the spring force constant is {eq}k. {/eq}


Substitute the known values.

$$\begin{align*} \left( {2000\;{\rm{lb}}} \right) &= k\left( {10\;{\rm{in}}} \right)\\[0.3cm] k &= \dfrac{{2000\;{\rm{lb}}}}{{10\;{\rm{in}}}}\\[0.3cm] k &= 200\;{\rm{lb/in}} \end{align*} $$


Write the expression for the spring force required (final spring force) to expand the spring beyond its natural length.

$${F_2} = k{x_2} $$

Here, {eq}{x_2} {/eq} is the extension in the spring.


Substitute the known values.

$$\begin{align*} 500\;{\rm{lb}} &= \left( {200\;{\rm{lb/in}}} \right){x_2}\\[0.3cm] {x_2} &= \dfrac{{500\;{\rm{lb}}}}{{200\;{\rm{lb/in}}}}\\[0.3cm] {x_2} &= 2.5\;{\rm{in}} \end{align*} $$

Thus, the spring is stretched {eq}\bf{2.5\;{\rm{in}}} {/eq} beyond its natural length while applying {eq}500\;{\rm{lb}} {/eq} of force.


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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