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 ______.

विकल्प

• T  = ${\displaystyle t_1+t_2}$

• ${\displaystyle T^2=t_1^2+t_2^2}$

• ${\displaystyle T^{-1}=t_1^{-1}+t_2^{-1}}$

• ${\displaystyle T^2=t_1^{-2}+t_2^{-2}}$

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 ${\displaystyle \underline{T^2=t_1^2+t_2^2}}$.

Explanation:

${\displaystyle t_1=2\pi \sqrt{\frac{m}{k_1}}}$ ...............(i),

${\displaystyle t_2=2\pi \sqrt{\frac{m}{k_2}}}$ ...............(ii),

when springs are in series then,

T = ${\displaystyle 2\pi \sqrt{\frac{\frac{m}{k_1{k}_1}}{k_1+k_2}}}$

= ${\displaystyle 2\pi \sqrt{\frac{m(k_1+k_2)}{k_1{k}_2}}}$

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

${\displaystyle t_1^2+t_2^2=4{\pi }^2\frac{m}{k_1}+4{\pi }^2\frac{m}{k_2}}$

= ${\displaystyle 4{\pi }^{2}m\left(\frac{k_1+k_2}{k_1{k}_2}\right)}$

or ${\displaystyle t_1^2+t_2^2=T^2}$