Question

The ratio of time period of three identical springs if they are first joined in parallel and then in series is (assume same mass is suspended from them)

• A. $1:3$
• B. $2:1$
• C. $2:3$
• D. $9:1$

Answer 1

The correct option is A $1:3$

Effective spring constant when connected in series,
$\frac{1}{K_S}=\frac{1}{K}+\frac{1}{K}+\frac{1}{K}\Rightarrow K_s=\frac{K}{3}$

In parallel,
$K_P=K+K+K=3K$

Time period in series $T_s=2\pi \sqrt{\frac{m}{\left(\frac{K}{3}\right)}}=2\pi \sqrt{\frac{3m}{K}}$
Time period in parallel
$T_P=2\pi \sqrt{\frac{m}{3K}}$

$\Rightarrow \frac{T_P}{T_s}=2\pi \sqrt{\frac{m}{3K}}\times \frac{1}{2\pi }\sqrt{\frac{K}{3m}}$
$\Rightarrow \frac{T_P}{T_s}=\frac{1}{3}$