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. $\frac{1}{T}=\frac{1}{T_1}+\frac{1}{T_2}$
• C. $T^2=T_1^2+T_2^2$
• D. $\frac{1}{T^2}=\frac{1}{T_1^2}+\frac{1}{T_2^2}$

Answer 1

Answer: (C)

$T^2=T_1^2+T_2^2$

Time period $T=2\pi \sqrt{\frac{M}{K}}$

$⟹$ $K=\frac{4{\pi }^{2}M}{T^2}$

Thus spring constant for first spring $K_1=\frac{4{\pi }^{2}M}{T_1^2}$ ............(1)

Spring constant for second spring $K_2=\frac{4{\pi }^{2}M}{T_2^2}$ .........(2)

For the series combination $\frac{1}{K_s}=\frac{1}{K_1}+\frac{1}{K_2}$

Let the time period for the series combination be $T$

$\therefore$ $\frac{T^2}{4{\pi }^{2}M}=$ $\frac{T_1^2}{4{\pi }^{2}M}+$ $\frac{T_2^2}{4{\pi }^{2}M}$ (using (1) and (2))

$⟹$ $T^2=T_1^2+T_2^2$