Question

An object suspended from a spring exhibits oscillations of period $T$. Now, the spring is cut in two halves and the same object is suspended with two halves as shown in figure. The new time period of oscillation will become

• A. $\frac{T}{2 \sqrt{2}}$
• B. $\frac{T}{2}$
• C. $\frac{T}{\sqrt{2}}$
• D. $2 T$

Answer 1

Answer: (B)

Let, $k$ be the spring constant of each half part of the spring. For a complete spring, the spring constant $k'=k / 2$ (spring in series). When two splitted parts of a spring are connected to the body, then spring are in parallel. Their effective spring constant, $k'=k+k=2\,k$.
As $T=2 \pi \sqrt{\frac{m}{k}}$
or $T \propto \frac{1}{\sqrt{k}}$ (for a fixed value of $m$ )
$\therefore \frac{T^{\prime}}{T}=\sqrt{\frac{k / 2}{2 k}}=\frac{1}{2}$
Or $T' =\frac{T}{2}$