Three spring mass systems are shown in figure. Assuming gravity free spece, find the time period of oscillations in each case. What should be the answer if space is not gravity free ?

Answer

(a) `2pisqrt((m)/(k_(1) + k_(2))), k_(eq.) = k_(1) + k_(2) ;`
`2pisqrt((m)/(k_(1) + k_(2)))`
`k_(eq.) = k_(1) + k_(2) ;` (c) `2pisqrt((m(k_(1) + k_(2)))/(k_(1)k_(2)))`
`k_(eq.) = (k_(1)k_(2))/(k_(1) + k_(2))`

Explanation

(a) When spring is stretched by `x` then restoring force.
`F = K_(1) x + K_(2) x`
`F = K_(eq) x`
`K_(eq) x = K_(1) + K_(2) x`
`K_(eq) = K_(1) + K_(2)`
`T = 2pisqrt((m)/(K_(eq))) = 2pisqrt((m)/(K_(1) + K_(2)))`
(b) When bock is displaced by `x` from mean position then restoring force.
`F = K_(1) x + K_(2) x`
`K_(eq) x = K_(1) x + K_(2) x`
`K_(eq) = K_(1) + K_(2)`
`T = 2pisqrt((m)/(K_(eq))) = 2pisqrt((m)/(K_(1) + K_(2)))`
(c) When block is displaced `y x` and extension is upper spring is `x_(1)`, extension in lower spring is `x_(2)` then `F = K_(1)x_(1) rArr x_(1) = (F)/(K_(4))`
`F = K_(2)x_(2) rArr x_(2) = (F)/(K_(2))`
`F = K_(eq)x rArr x = (F)/(K_(eq))`
`x = x_(1) + x_(2) rArr (F)/(K_(eq)) = (F)/(K_(1)) + (F)/(K_(2))`
`K_(eq) = (K_(1)K_(2))/(K_(1) + K_(2))`
`T = 2pisqrt((m)/(K_(eq))) = 2pisqrt((m(K_(1) + K_(2)))/(K_(1)K_(2)))`
When space in not gravity free than answer do not changes as time period of spring mass system is independent of gravity.