Q) [a] Derive an expression to find the potential energy possessed by a spring when it is extended though a - Physics - Work Energy And Power

Question

Q) [a] Derive an expression to find the potential energy possessed
by a spring when it is extended though a maximum extension of
‘Xm’ [b]Also show that the spring force is
conservative force [c]Derive the expression to find the maximum
velocity of the spring when it crosses the mean position

Sushant · Accepted Answer

'Dear
student,

Please
find below the solution to the asked query:

(a) let there be a spring with a force constant  kso the force on a lock attached to the spring which has been elongated by a distance x is  = -kxso the work done by the spring force in elongating it to a distance x is,W  =  ∫0x-kxdx =- kx22now by definition the potential energy stored in the spring at any elongation is the negative of the work done by the spring force in producing the elongationso the stored potential energy in the spring is  = -- kx22 = kx22(b) spring force is conservative force :As the work done by the spring force depends only on the final position and the initial position so we can say that the spring force is a conservative force
the  work done by the spring is path independentfor a particular spring mass systemlet the elongation in the spring be xso the stored spring potential energy in it  =kx22(c) now spring is releasedwhen the spring returns to its mean position , the attached mass gains some velocity, let the velocity be v mso the kinetic energy in the body = mv22thus, by conservation of energydecrease in the spring potential energy = increase in the kinetic energy of the blocksoso mv22 = kx22or v = xkm  at the mean position

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Q) [a] Derive an expression to find the potential energy possessed by a spring when it is extended though a maximum extension of ‘Xm’.[b]Also show that the spring force is conservative force.[c]Derive the expression to find the maximum velocity of the spring when it crosses the mean position.

Q) [a] Derive an expression to find the potential energy possessed by a spring when it is extended though a maximum extension of ‘X_m’.[b]Also show that the spring force is conservative force.[c]Derive the expression to find the maximum velocity of the spring when it crosses the mean position.