Physics - SHM Question with Solution | TestHub

In case of cutting of a spring,$l\times k=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$

$k_1{l}_1=k_2{l}_2$

$k_1\left(\eta l_2\right)=k_2{l}_2 \Rightarrow \eta k_1=k_2$...............(1)

Now both the springs when connected in series should give the same effect as that of the original spring. Hence, by applying the concept of springs connected in series, we obtain,

$\frac{1}{k}=\frac{1}{k_1}+\frac{1}{k_2}$...............(2)

$\frac{1}{k}=\frac{1}{k_1}+\frac{1}{\eta k_1} \Rightarrow \frac{k_1}{k}=\frac{\eta +1}{\eta }$(using (1))

$k_1=\left(\frac{\eta +1}{\eta }\right)k$............(3)

Using (3) again in (2), we obtain,

$k_2=\left(\eta +1\right)k$