Five identical springs are used in the following three configurations. The time periods of vertical oscillations in configurations (i), (ii) and (iii) are in the ratio

Options

A. `(1)/(sqrt(2)) : 2 : 1`
B. `2 : sqrt(2) : (1)/(sqrt(2))`
C. `2 : (1)/(sqrt(2)) : 1`
D. `1 : sqrt(2) : (1)/(sqrt(2))`

Answer

D

Explanation

Each spring has a spring constant K, as they are identical. In fig. (a) with only one spring, the spring constant `K_(1)=K`.
In fig. (b) the springs are in series.
`therefore` The spring constant `(K_(2))=(KxxK)/(K+K)=(K^(2))/(2K)=(K)/(2)`
In fig. (c ), the springs are in parallel. `therefore K_(3)=2K`
For a loaded spring, `T=2pi sqrt((m)/(K))`
`therefore T_(1) : T_(2) : T_(3)=(1)/(sqrt(K_(1))) : (1)/(sqrt(K_(2))) : (1)/(sqrt(K_(3)))`
`= (1)/(sqrt(K)) : (1)/(sqrt(K//2)) : (1)/(sqrt(2K))`
`=1 : sqrt(2) : (1)/(sqrt(2))`