Question

$T_1,T_2$ are time periods of oscillation of a block individually suspended to spring of force constants $K_1,K_2$ respectively same block is suspended to parallel combination of same two springs. Its time period is

• A. $T_1+T_2$
• B. $\frac{T_1+T_2}{2}$
• C. $\frac{T_1T_2}{T_1+T_2}$
• D. $\frac{T_1T_2}{\sqrt{T^2_1+T_2^2}}$

Answer 1

Answer: (D)

for a parallel combination $k = k _{1}+ k _{2}$
$T =2 \pi \sqrt{\frac{ m }{ k }}$
$=2 \pi \sqrt{\frac{ m }{\left( k _{1}+ k _{2}\right)}}$
$T _{1}=2 \pi \sqrt{\frac{ m }{ k _{1}}}$
$\Rightarrow k _{1}=(2 \pi)^{2} \frac{ m }{ T _{1}^{2}}$
$k _{2}=(2 \pi)^{2} \frac{ m }{ T _{2}^{2}}$
$k _{1}+ k _{2}=(2 \pi)^{2}\left(\frac{ m }{ T _{1}^{2}}+\frac{ m }{ T _{2}^{2}}\right)$
$\therefore \frac{ T _{1} T _{2}}{\sqrt{ T _{1}+ T _{2}}}= T$