Question

Two springs are connected to a block of mass $M$ placed on a frictionless surface as shown below. If both the springs have a spring constant $k$, then the frequency of oscillation of the block is

• A. $\frac{1}{2 \pi} \sqrt{\frac{k}{M}}$
• B. $\frac{1}{2 \pi} \sqrt{\frac{k}{2 M}}$
• C. $\frac{1}{2 \pi} \sqrt{\frac{2 k}{M}}$
• D. $\frac{1}{2 \pi} \sqrt{\frac{M}{k}}$

Answer 1

Answer: (B)

The frequency of oscillation of a vibrating spring-block, system is given by
$v=\frac{1}{2 \pi} \sqrt{\frac{k_{\text {eff }}}{M} \ldots \text { (i) }}$
Since, the springs are attached in series, each having spring constant $k$, so we have
$\frac{1}{k_{ eff }}=\frac{1}{k}+\frac{1}{k}=\frac{2}{k}$
$\Rightarrow \frac{1}{k_{ eff }}=\frac{k}{2}$
From Eq. (i), we get
$v=\frac{1}{2 \pi} \sqrt{\frac{k_{\text {eff }}}{M}}=\frac{1}{2 \pi} \sqrt{\frac{k}{2 M}}$