Question

A particle at the end of a spring executes simple harmonic motion with a period t₁, while the corresponding period for another spring is t₂. If the period of oscillation with the two springs in series is T, then ______.

Options

• T  = `t_1 + t_2`

• `T^2 = t_1^2 + t_2^2`

• `T^-1 = t_1^-1 + t_2^-1`

• `T^2 = t_1^-2 + t_2^-2`

Answer 1

A particle at the end of a spring executes simple harmonic motion with a period t₁, while the corresponding period for another spring is t₂. If the period of oscillation with the two springs in series is T, then `underlinebb(T^2 = t_1^2 + t_2^2)`.

Explanation:

`t_1 = 2pisqrt(m/k_1)` ...............(i),

`t_2 = 2pisqrt(m/k_2)` ...............(ii),

when springs are in series then,

T = `2pisqrt((m/(k_1k_1))/(k_1 + k_2))`

= `2pisqrt((m(k_1 + k_2))/(k_1k_2))`

Squaring and adding (1) and (2) we get,

`t_1^2 + t_2^2 = 4pi^2 m/k_1 + 4pi^2 m/k_2`

= `4pi^2m((k_1 + k_2)/(k_1k_2))`

or `t_1^2 + t_2^2 = T^2`