Question

A string of length l and force constant k is stretched to obtain extension l. it is further stretched to obtain extension l1. the work done in second stretching is:

Answer 1

Work done = - change in potential energy .
Initial potential energy , Ui = -1/2 kl² [ here k is spring constant and l is obtained first extension .]
Further spring is stretched to obtain extension l₁
∴ total extension in 2nd time = (l + l₁)
final potential energy ,Uf = -1/2 k(l + l₁)² [ here l₁ is obtained second extension of spring ]

Now, work done = -change in potential energy
= -[Uf - UI]
= -[-1/2k(l + l₁)² + 1/2 kl² }
= - 1/2k{l² - l² - l₁² - 2ll₁}
= - 1/2 k{-(l₁² + 2ll₁)}
= 1/2 kl₁(l + 2l)

Hence, Workdone in the second stretching is 1/2 kl₁(l₁ + 2l)