Three spring mass systems in vertical plane are shown in the figure. The mass of all the pulleys and connecting strings and springs are negligible and friction at all contacts is absent. `[g = 10 m//s^(2)]`

Calculate `T_(1), T_(2) and T_(3)` which are the time periods of small vertical oscillation of mass m in system, l system - II and system -III respectively.

Explanation

`T_(1) = 2pi sqrt((3m)/(2k)), T_(2) = 2pi sqrt((9m)/(8k)), T_(3) = 2pi sqrt((3m)/(4k))`

From constraint, `x =2 (x_(1) + x_(2))`
`(T)/(k_(eq)) = 2 ((2T)/(8k) + (2T)/(4k))`
`(1)/(k_(eq)) = (1)/(2k) + (1)/(k) = (3)/(2k)`
`:. k_(eq) = (2k)/(3)`
Time period `T = 2pi sqrt((3m)/(2k))`
From constraint,
`x = 2x_(1) + x_(2)`
`(T)/(k_(eq)) = 2 ((2T)/(4k)) + (T)/(8k)`
`(1)/(k_(eq)) = (1)/(k) + (1)/(8k) = (9)/(8k)`
`:. k_(eq) = 8k//g`
`rArr T = 2pi sqrt((9m)/(8k))`
From constraint,
`x= 2x_(1) + x_(2)`
`(T)/(k_(eq)) = 2 ((2T)/(8k))+ (T)/(4k)`
`(1)/(k_(eq)) = (1)/(2k) + (1)/(4k) = (3)/(4k)`
`k_(eq) = (4k)/(3)`
`:. T = 2pi sqrt((3m)/(4k))`