Question

Show that the elastic force of a spring is a conservative force.Hence write an expression for the potential energy of an elastic stretch of spray.

Answer 1

$$\begin{array}{c}\left(a\right)lettherebeaspringwithaforceconstantk\\ \\ sotheforceonalockattachedtothespringwhichhasbeenelongatedbyadistancexis=-kx\\ \\ sotheworkdonebythespringforceinelongatingittoadistancexis,\\ W={\int }_0^x-kxdx=-\frac{{kx}^2}{2}\\ \\ nowbydefinitionthepotentialenergystoredinthespringatanyelongationisthenegativeoftheworkdonebythespringforceinproducingtheelongation\\ sothestoredpotentialenergyinthespringis=-(-\frac{{kx}^2}{2})=\frac{{kx}^2}{2}\\ \\ \left(b\right)springforceisconservativeforce:\\ \\ Astheworkdonebythespringforcedependsonlyonthefinalpositionandtheinitialpositionsowecansaythatthespringforceisaconservativeforce.\\ theworkdonebythespringispathindependent\\ \\ foraparticularspringmasssystem\\ lettheelongationinthespringbex\\ \\ sothestoredspringpotentialenergyinit=\frac{{kx}^2}{2}\\ \left(c\right)nowspringisreleased\\ \\ whenthespringreturnstoitsmeanposition,theattachedmassgainssomevelocity,letthevelocitybevm\\ \\ sothekineticenergyinthebody=\frac{{mv}^2}{2}\\ thus,byconservationofenergy\\ \\ decreaseinthespringpotentialenergy=increaseinthekineticenergyoftheblock\\ so\\ \\ so\frac{{mv}^2}{2}=\frac{{kx}^2}{2}\\ \\ orv=x\sqrt{\frac{k}{m}}atthemeanposition\end{array}$$