Question

A spring is cut into two equal parts. What will be the difference in a time period of the spring pendulum thus formed any one of the cut part and from the original spring pendulum?

Answer 1

original string of length $L$ cross sectional

area $A$ $K=\frac{YA}{l}....\left(1\right)$

after cutting $K^1=\frac{YA}{l/2}=\frac{2YA}{L}=2K$

Finst case $T=2\pi \sqrt{\frac{m}{k}}$

second case $T=2\pi \sqrt{\frac{m}{k^1}}=2\pi \sqrt{\frac{m}{2k}}$

So the difference is $T-T=2\pi \sqrt{\frac{m}{k}}[1-\frac{1}{\sqrt{2}}]$

$\Rightarrow \Delta T-T\frac{(\sqrt{2}-1)}{\sqrt{2}}$