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

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 $\left(\right.b\left.\right)$ . 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
$a=\frac{F}{m}=-\frac{2 k x}{m}$
The standard equation of $SHM$ is
$a=-\omega ^{2}x$
Comparing $\left(\right.i\left.\right)$ and $\left(i i\right),$ we get
$\omega ^{2}=\frac{2 k}{m}$ or $\omega =\sqrt{\frac{2 k}{m}}$
Time period, $T=\frac{2 \pi }{\omega }=2\pi \sqrt{\frac{m}{2 k}}$