Question

If two similar springs, each of spring constant $K_1$ are joined in series, the new spring constant and the time period would be changed by a factor of:

• A. $\frac{1}{2},\sqrt{2}$
• B. $\frac{1}{4},2\sqrt{2}$
• C. $\frac{1}{2},2\sqrt{2}$
• D. $\frac{1}{4},\sqrt{2}$

Answer 1

The correct option is A $\frac{1}{2},\sqrt{2}$

Before connecting in series,
Spring constant,
$K=K_1$

Time period,
$T=2\pi \sqrt{\frac{M}{K_1}}$

After connecting in series,

Spring constant, $K^{{}^{\prime }}=K_{\text{eq}}=\frac{K_1\times K_1}{K_1+K_1}=\frac{1}{2}{K}_1=\frac{1}{2}K$

Time period,
$T^{{}^{\prime }}=2\pi \sqrt{\frac{M}{K_{\text{eq}}}}=2\pi \sqrt{\frac{M}{K_1/2}}$

$\Rightarrow T^{{}^{\prime }}=\sqrt{2}\times 2\pi \sqrt{\frac{M}{K_1}}=\sqrt{2}T$

Hence, option $\left(A\right)$ is the correct answer.