A helical coil spring of stiffness k is cut to two equal halves and then these are connected in parallel to support a vibrating mass m. The angular frequency of vibration, ω_n is

Concept:

When a spring of constant ko and length Lo is cut into pieces of length, say l1, l2, l3, ……. Ln, their respective spring constants can be calculated as,

k1l1 = k2l2 = k3l3 = ……. = knln = kolo

Here the spring is cut into two equal parts: l1 = l2 = L/2

k1l1 = k2l2 = KL

\({k_1}\frac{L}{2} = {k_2}\frac{L}{2} = KL\)

k1 = k2 = 2K

When arranged in parallel, the equivalent stiffness will be 4K

Therefore, the angular frequency will be 

\(\omega_{n}=\sqrt {\frac{k_{eq}}{m}}= \sqrt {\frac{4k}{m}} \)