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. If 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 $k=k_1+k_2$
$T=2\pi \sqrt{\frac{m}{k}}$

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

$T_1=2\pi \sqrt{\frac{m}{k_1}}$

$\Rightarrow k_1=(2\pi {)}^2\frac{m}{T_1^2}$

$k_2=(2\pi {)}^2\frac{m}{T_2^2}$

$k_1+k_2=\left(2\pi {)}^2\right(\frac{m}{T_1^2}+\frac{m}{T_2^2})$

$\therefore \frac{T_1{T}_2}{\sqrt{T_1+T_2}}=T$