# Two spheres are mounted on identical horizontal springs and rest on a frictionless table. When...

## Question:

Two spheres are mounted on identical horizontal springs and rest on a frictionless table. When the spheres are uncharged, the spacing between them is {eq}0.05 \ m {/eq}, and the springs are unstrained. When each sphere has a charge of {eq}+1.6 \ \mu C {/eq}, the spacing doubles. Assuming that the spheres have a negligible diameter, determine the spring constant of the springs?

## Spring constant:

Spring force is dependent on the spring constant, as the property of the spring will change on changing the spring's material. The spring constant does not change due to the deformation of the spring because it the property of the spring.

## Answer and Explanation: 1

**Given Data**

- When the spheres are uncharged the distance between the two spheres, {eq}d = 0.05\;{\rm{m}} {/eq}

- When the spheres are charged the distance between the two spheres, {eq}D = 2d = 0.1\;{\rm{m}} {/eq}

- Charge of the each sphere, {eq}q = 1.6\;\mu {\rm{C}} = 1.6 \times {10^{ - 6}}\;{\rm{C}} {/eq}

When the spheres are charged and the distance between the two spheres becomes doubled then the force between them can be calculated as

{eq}\begin{align*} F &= \dfrac{{Kqq}}{{{D^2}}}\\ F &= \dfrac{{K{q^2}}}{{{D^2}}}\\ F &= \dfrac{{9 \times {{10}^9} \times {{\left( {1.6 \times {{10}^{ - 6}}} \right)}^2}}}{{{{0.1}^2}}}\\ F &= 2.304\;{\rm{N}} \end{align*} {/eq}

Now from the expression of the spring force, the spring constant can be calculated as

{eq}F = kx .... (1) {/eq}

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

Here when spheres becomes charged the spacing gets doubled and compression of spring occurs, so displacement of the each spring due to compression is

{eq}\begin{align*} x &= \dfrac{d}{2}\\ x &= \dfrac{{0.05}}{2}\\ x &= 0.025\;{\rm{m}} \end{align*} {/eq}

Now substitute the values in the equation (1)

{eq}\begin{align*} 2.304 &= k\left( {0.025} \right)\\ K &= 92.16\;{\rm{N/m}} \end{align*} {/eq}

Thus, the value of the spring constant is {eq}92.16\;{\rm{N/m}} {/eq}.

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Chapter 17 / Lesson 11In 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.