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A 96.1\ \mathrm{g} ball is dropped from a height of 59.1\ \mathrm{cm} above a spring of...

Question:

A {eq}96.1\ \mathrm{g} {/eq} ball is dropped from a height of {eq}59.1\ \mathrm{cm} {/eq} above a spring of negligible mass. The ball compresses the spring to a maximum displacement of {eq}4.75403\ \mathrm{cm} {/eq}.The acceleration of gravity is {eq}9.8\ \mathrm{m/s^2} {/eq}. Calculate the spring force constant {eq}k {/eq}. Answer in units of {eq}\mathrm{N/m} {/eq}.

Spring Force:

The springs force depends on spring stiffness or spring force constant (varies according to the springs) and the spring's displacement. The spring force varies linearly with the displacement (extension or compression) of the springs.

Answer and Explanation: 1


Given Data

  • The mass of the ball is: {eq}m = 96.1\;{\rm{g}} {/eq}
  • The height of the ball is: {eq}h = 59.1\;{\rm{cm}} {/eq}
  • The compression of the spring is: {eq}x = 4.75403\;{\rm{cm}} {/eq}


The expression for the energy absorbed or gain by the spring is,

{eq}{E_{spring}} = \dfrac{1}{2}k{x^2} {/eq}

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


The loss of potential energy of the ball is,

{eq}P{E_{ball}} = mg\left( {h + x} \right) {/eq}


The expression for the energy conservation is,

{eq}\begin{align*} {E_{spring}} &= P{E_{ball}}\\ \dfrac{1}{2}k{x^2} &= mg\left( {h + x} \right) \end{align*} {/eq}


Substitute the known values.

{eq}\begin{align*} \dfrac{1}{2}k{\left( {4.75403\;{\rm{cm}}} \right)^2} &= \left( {96.1\;{\rm{g}}} \right)\left( {9.81\;{\rm{m/}}{{\rm{s}}^2}} \right)\left( {59.1\;{\rm{cm}} + 4.75403\;{\rm{cm}}} \right)\\ \dfrac{1}{2}k{\left( {4.75403\;{\rm{cm}}\left( {\dfrac{{1\;{\rm{m}}}}{{100\;{\rm{cm}}}}} \right)} \right)^2} &= \left( {96.1\;{\rm{g}}} \right)\left( {\dfrac{{1\;{\rm{kg}}}}{{1000\;{\rm{g}}}}} \right)\left( {9.81\;{\rm{m/}}{{\rm{s}}^2}} \right)\left( {63.85403\;{\rm{cm}}\left( {\dfrac{{1\;{\rm{m}}}}{{100\;{\rm{cm}}}}} \right)} \right)\\ k &= 532.70\;{\rm{N/m}} \end{align*} {/eq}


Thus, the spring force constant is {eq}532.70\;{\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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