A mass is suspended separately by two springs and the time periods in the two cases are `T_(1)` and `T_(2)`. Now the same mass is connected in parallel `(K = K_(1) + K_(2))` with the springs and the time is suppose `T_(P)`. Similarly time period in series is `T_(S)`, then find the relation between `T_(1),T_(2)` and `T_(P)` in the first case and `T_(1),T_(2)` and `T_(S)` in the second case.

Answer

A, B

Explanation

`T = 2pisqrt((m)/(K))`
`:. T = (alpha)/sqrt(K)`
(where, `2pi sqrt(m) =alpha` = constant)
`rArr K = alpha^(2)/T^(2)`...(i)
or `(1)/(K) = T^(2)/alpha^(2)`
First case In parallel,
` K = K_(1) + K_(2)`
Using Eq. (i), `alpha^(2)/T_(p)^(2) = alpha^(2)/T_(1)^(2) + alpha^(2)/T^(2)`
`:. T_(p)^(-2) = T_(1)^(-2) + T_(2)^(-2)`
This is the desired relation.
Second case In series, `(1)/(K) = (1)/(K_(1)) + (1)/(K_(2))`
Now, using Eq. (ii) we have, `(T_(s)^(2))/(alpha^(2)) = (T_(1)^(2))/(alpha^(2)) + (T_(2)^(2))/(alpha^(2))`
`:. T_(s)^(2) = T_(1)^(2) + T_(2)^(2)`
This is the desired relation in this case.