question_answer

A particle at the end of a spring executes simple harmonic motion with a period\[{{t}_{1}},\]while the corresponding period for another spring is\[{{t}_{2}}\]. If the period of oscillation with the two springs in series is T, then [AIEEE 2004]

A)  \[T={{t}_{1}}+{{t}_{2}}\]                 

B)  \[{{T}^{2}}=t_{1}^{2}+t_{2}^{2}\]

C)  \[{{T}^{-1}}=t_{1}^{-1}+t_{2}^{-1}\]

D)       \[{{T}^{-2}}=t_{2}^{-2}+t_{2}^{-2}\]

Correct Answer: B

Solution :

[b] Time period of spring

k, being the force constant of spring.

For first spring                     ...(i)

For second spring

...(ii)

The effective force constant in their series combination of springs is

Time period of combination

..(iii)

From Eqs. (i) and (ii), we obtain

[from Eq. (iii)]