Question

A block of mass $m$ is connected to two indentical springs of spring constant $k$ which are in turn connected to fixed supports as shown in the figure. Find the time period for small oscillations of the block.

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

Answer 1

Answer: (B)

(a)

(b)
Let the mass $m$ be displaced by a small distance $x$ to the right from its mean position as shown in figure $(b)$ . Due to it the spring on the left side gets starched by a length $x$ while that on the right side gets compressed by the same length. The forces acting on the mass are
$F_1=kx$ towards left hand are
$F_2=kx$ towards left hand are
The net forces acting on the mass is, $F=F_1+F_2=-2kx$
Here, $F\propto x$ and $-ve$ sign shows that force is towards the mean position, therefore the motion executed by the particle is simple harmonic.
Its acceleration is
$a=\frac{F}{m}=-\frac{2kx}{m}$
The standard equation of $SHM$ is
$a=-{\omega }^{2}x$
Comparing $(i)$ and $\left(ii\right),$ we get
${\omega }^2=\frac{2k}{m}$ or $\omega =\sqrt{\frac{2k}{m}}$
Time period, $T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{m}{2k}}$