Let `T_(1)" and "T_(2)` be the time periods of spring A and B when mass M is suspended from the end of the spring. If both springs are taken in series and the same mass M is suspended from the series combination, the time period is T, then

Options

A. `T= T_(1) +T_(2)`
B. `(1)/(T) = (1)/(T_1) +(1)/(T_2)`
C. `T^(2)= T_(1)^(2) +T_(2)^(2)`
D. `(1)/(T^2) = (1)/(T_(1)^(2)) +(1)/(T_(2)^(2))`

Answer

C

Explanation

From `T= 2pi sqrt((M)/(k)), k= (4pi^(2) M)/(T)`
`therefore k_(1)= (4pi^(2)M)/(T_(1)^(2))" and "k_(2)= (4pi^(2)M)/(T_(2)^(2))`
For series combination,
`(1)/(k)=(1)/(k_1)+(1)/(k_2)= (T_(1)^(2))/(4pi^(2)M)+(T_(2)^(2))/(4pi^(2)M)= (T_(1)^(2)+T_(2)^(2))/(4pi^(2)M)`
`therefore k= (4pi^(2)M)/(T_(1)^(2)+T_(2)^(2))`
`therefore T_(1)^(2)+ T_(2)^(2)= 4pi^(2) (M)/(k)`
`therefore sqrt(T_(1)^(2)+T_(2)^(2))= 2pi sqrt((M)/(k))` is the resultant periodic time.