Question

Let ${{T}_{1}}$ and ${{T}_{2}}$ be the time periods of springs $A$ and $B$ when mass $M$ is suspended from one end of each spring. If both springs are taken in series and the same mass $M$ is suspended from the series combination, the time period is $T$, then:
A. ${{T}_{1}}+{{T}_{2}}+{{T}_{3}}$
B. $\dfrac{1}{T}=\dfrac{1}{{{T}_{1}}}+\dfrac{1}{{{T}_{2}}}$
C. ${{T}^{2}}={{T}_{1}}^{2}+{{T}_{2}}^{2}$
D. $\dfrac{1}{{{T}^{2}}}=\dfrac{1}{{{T}_{1}}^{2}}+\dfrac{1}{{{T}_{2}}^{2}}$

Answer 1

Hint: Recall the formula for time period of oscillation $\left( T \right)$ of a spring i.e., $T=2\pi \sqrt{\dfrac{M}{k}}$. Where,

$T$ = time period
$M$ = mass of the block
$k$ = spring constant

and the formula for series combination of springs,

$\dfrac{1}{{{k}_{net}}}=\dfrac{1}{{{k}_{1}}}+\dfrac{1}{{{k}_{2}}}$.

Where symbols have their usual meaning.

Complete step by step answer:
Mathematically, time period of oscillation for the spring is given by,

$T=2\pi \sqrt{\dfrac{M}{k}}$

Where symbols have their usual meaning.

Now, on squaring both sides, spring constant can be given by
$\Rightarrow k=\dfrac{4{{\pi }^{2}}M}{{{T}^{2}}}$
In the same manner the spring constant for first spring will be,
${{k}_{1}}=\dfrac{4{{\pi }^{2}}M}{{{T}_{1}}^{2}}$ --------(1)
and the spring constant for second spring will be,
${{k}_{2}}=\dfrac{4{{\pi }^{2}}M}{{{T}_{2}}^{2}}$ --------(2)
As we know that for series combination of springs,
$\dfrac{1}{{{k}_{net}}}=\dfrac{1}{{{k}_{1}}}+\dfrac{1}{{{k}_{2}}}$ --------(3)

Let the time period of oscillation for the series combination of springs is $T$.

On putting values of ${{k}_{1}}$ and ${{k}_{2}}$ from equation (1) and (2), in equation (3) we get

$\Rightarrow \dfrac{{{T}^{2}}}{4{{\pi }^{2}}M}=\dfrac{{{T}_{1}}^{2}}{4{{\pi }^{2}}M}=\dfrac{{{T}_{2}}^{2}}{4{{\pi }^{2}}M}$

On solving, we have

$\Rightarrow {{T}^{2}}={{T}_{1}}^{2}+{{T}_{2}}^{2}$

So, the relation between time periods, of series combination of spring and for separate springs will be

${{T}^{2}}={{T}_{1}}^{2}+{{T}_{2}}^{2}$
Hence, the correct option is C, i.e., ${{T}^{2}}={{T}_{1}}^{2}+{{T}_{2}}^{2}$.

Note: Students need to memorize the formula for time period $\left( T \right)$ of oscillation and they also need to know that how to find resultant spring constant $\left( {{k}_{net}} \right)$ when springs are connected in series. Students should be very careful while doing calculations.