Question

The time period of oscillations of a block attached to a spring is $t_1$ . When the spring is replaced by another spring, the time period of the block is $t_2$ . If both the springs are connected in series and the block is made to oscillate using the combination, then the time period of the block is

• A. $\text{T}=t_1+t_2$
• B. ${\text{T}}^2=t_1^2+t_2^2$
• C. $T^{-1}=t_1^{-1}+t_2^{-1}$
• D. $T^{-2}=t_1^{-2}+t_2^{-2}$

Answer 1

Answer: (B)

When springs are in series, $k=\frac{k_1{k}_2}{k_1+k_2}$
For first spring, $t_1=2\pi \sqrt{\frac{\text{m}}{k_1}}$
For Second spring, $t_2=2\pi \sqrt{\frac{\text{m}}{k_2}}$
$\therefore$ $t_1^2+t_2^2=\frac{4{\left(\pi \right)}^2\text{m}}{k_1}+\frac{4{\left(\pi \right)}^2\text{m}}{k_2}=4{\left(\pi \right)}^2\text{m}\left(\frac{k_1+k_2}{k_1{k}_2}\right)$
or $t_1^2+t_2^2={\left(\left[2\pi \sqrt{\frac{\text{m}\left(k_1+k_2\right)}{k_1{k}_2}}\right]\right)}^2$
or $t_1^2+t_2^2={\text{T}}^2$ .