When a rubber band is strecthed by a distance x, it exerts a restoring force of magnitude F = ax +bx², where a and b are constants. The work done in stretching are unstretched rubber-band by L is

aL² +bL²

$1 half left parenthesis aL squared plus bL cubed right parenthesis$
$aL squared over 2 space plus bL cubed over 3$
$1 half space open parentheses aL squared over 2 plus bL cubed over 3 close parentheses$

Solution

aL squared over 2 space plus bL cubed over 3

straight U subscript straight f minus straight U subscript straight i space equals space minus straight W space equals space minus space integral subscript straight i superscript straight f straight F. dr

Given, F = ax +bx²
We know that work done in stretching the rubber band by L is
|dW|= |Fdx|

vertical line straight W vertical line space equals space integral subscript 0 superscript straight L left parenthesis ax space plus bx squared right parenthesis dx space equals space open square brackets ax squared over 2 close square brackets subscript straight O superscript straight L space plus space open square brackets bx cubed over 3 close square brackets subscript straight O superscript straight L equals open square brackets aL squared over 2 minus fraction numerator ax space left parenthesis 0 right parenthesis squared over denominator 2 end fraction close square brackets space plus space open square brackets fraction numerator straight b space straight x space straight L cubed over denominator 3 end fraction minus fraction numerator straight b space straight x space left parenthesis 0 right parenthesis cubed over denominator 3 end fraction close square brackets vertical line straight W vertical line space equals space aL squared over 2 plus bL cubed over 3

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