The figure shows the arrangement of block of mass m with one or more identical springs in three different ways . If Force cosntant of each springs is K and `T_(1)`, `T_(2)` and `T_(3)` are the time periods of oscillations of the arrangement 1, 2, and 3, respectibely, then

Options

A. `T_(2)=(1)/(sqrt2T_(1)`
B. `T_(3)=sqrt2Tt_(1)`
C. `T_(3)=2T_(2)`
D. `T_(3)=sqrt2T_(2)`

Answer

A, B, C

Explanation

For parallel combination in (2).
Equivalent force constant = 2K
`therefore T_(2) = 2pisqrt((m)/(2K))`
For combination (1)
`T_(1) = 2pisqrt((m)/(K))`
For series combination in (3)
Equivalent force constant = K/2
`therefore T_(3) = 2pisqrt((m)/(K//2)) = 2pisqrt((2m)/(K))`
`therefore T_(2) = (1)/(sqrt(2))T_(I)`
`T_(3) = sqrt(2)T_(1)`
`T_(3) = 2T_(2)`