Question

If 6 J of work is needed to stretch a spring from 10 cm to 12 cm and another 10 J is needed to stretch it from 12 cm to 14 cm, what is the natural length of the spring?

Short answer

The natural length of the spring is obtained as \({\rm{8cm}}\).

Step-by-step solution

Concept Introduction

In mathematics, an integral is a numerical number equal to the area under a function's graph for some interval or a new function whose derivative equals the original function.

Deriving the equations

Let the natural length be \({\rm{l}}\).

Then it can be written –

\(\begin{aligned}{}{\rm{6 = }}\int_{{\rm{10 - l}}}^{{\rm{12 - l}}} {\rm{k}} {\rm{xdx}}\\{\rm{6 = }}\left( {\frac{{\rm{1}}}{{\rm{2}}}{\rm{k}}{{\rm{x}}^{\rm{2}}}} \right)_{{\rm{10 - l}}}^{{\rm{12 - l}}}\\{\rm{6 = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{k}}\left( {{{{\rm{(12 - l)}}}^{\rm{2}}}{\rm{ - (10 - l}}{{\rm{)}}^{\rm{2}}}} \right){\rm{ Equation 1}}\end{aligned}\)

And also, it can further be written –

\(\begin{aligned}{}{\rm{10 = }}\int_{{\rm{12 - l}}}^{{\rm{14 - l}}} {\rm{k}} {\rm{xdx}}\\{\rm{10 = }}\left( {\frac{{\rm{1}}}{{\rm{2}}}{\rm{k}}{{\rm{x}}^{\rm{2}}}} \right)_{{\rm{12 - l}}}^{{\rm{14 - l}}}\\{\rm{10 = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{k}}\left( {{{{\rm{(14 - l)}}}^{\rm{2}}}{\rm{ - (12 - l}}{{\rm{)}}^{\rm{2}}}} \right){\rm{ Equation 2}}\end{aligned}\)

Calculation for work done

Divide Equation \({\rm{2}}\) by Equation \({\rm{1}}\) -

\(\begin{aligned}{}\frac{{{\rm{10}}}}{{\rm{6}}}{\rm{ = }}\frac{{{{{\rm{(14 - l)}}}^{\rm{2}}}{\rm{ - (12 - l}}{{\rm{)}}^{\rm{2}}}}}{{{{{\rm{(12 - l)}}}^{\rm{2}}}{\rm{ - (10 - l}}{{\rm{)}}^{\rm{2}}}}}\\\frac{{\rm{5}}}{{\rm{3}}}{\rm{ = }}\frac{{{\rm{(14 - l - 12 + l)(14 - l + 12 - l)}}}}{{{\rm{(12 - l - 10 + l)(12 - l + 10 - l)}}}}.......({{\rm{a}}^{\rm{2}}}{\rm{ - }}{{\rm{b}}^{\rm{2}}}{\rm{ = (a - b)(a + b)}})\\\frac{{\rm{5}}}{{\rm{3}}}{\rm{ = }}\frac{{{\rm{(2)(26 - 2l)}}}}{{{\rm{(2)(22 - 2l)}}}}\\\frac{{\rm{5}}}{{\rm{3}}}{\rm{ = }}\frac{{{\rm{13 - l}}}}{{{\rm{11 - l}}}}\end{aligned}\)

Cross Multiplying –

\({\rm{55 - 5l = 39 - 3l}}\)

Add \({\rm{5l}}\) on both the sides –

\({\rm{55 = 39 + 2l}}\)

Subtract \({\rm{39}}\) from both the sides –

\({\rm{16 = 2l}}\)

Divide both the sides by \({\rm{2}}\)–

\({\rm{8 = l}}\)

Therefore, the length is obtained as \({\rm{8cm}}\).