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. $\frac{T_1{T}_2}{T_1+T_2}$
• D. $\frac{T_1{T}_2}{\sqrt{T_1^2+T_2^2}}$

Answer 1

Answer: (D)

for a parallel combination $k=k_1+k_2$
$T=2\pi \sqrt{\frac{m}{k}}$
$=2\pi \sqrt{\frac{m}{\left(k_1+k_2\right)}}$
$T_1=2\pi \sqrt{\frac{m}{k_1}}$
$\Rightarrow k_1=(2\pi {)}^2\frac{m}{T_1^2}$
$k_2=(2\pi {)}^2\frac{m}{T_2^2}$
$k_1+k_2=(2\pi {)}^2\left(\frac{m}{T_1^2}+\frac{m}{T_2^2}\right)$
$\therefore \frac{T_1{T}_2}{\sqrt{T_1+T_2}}=T$