Question

Two springs are connected to a block of mass $M$ placed on a frictonless surface as show 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. $\frac{1}{2\pi }\sqrt{\frac{k}{2M}}$
• C. $\frac{1}{2\pi }\sqrt{\frac{2k}{M}}$
• D. $\frac{1}{2\pi }\sqrt{\frac{M}{k}}$

Answer 1

The correct option is B $\frac{1}{2\pi }\sqrt{\frac{k}{2M}}$

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