The frequency of oscillations of a mass m suspended by a spring is f1

1.	The frequency of oscillations of a mass m suspended by a spring is f1. If the length of the spring is cut to one-half, the same mass oscillates with frequency f2. Determine the value \(\frac{f_2}{f_1}\).
Answer» Let the full spring be the combination of two springs in series, each of force constant k. then in case (I), the effective spring constant (k₁) is given by k₁ = \(\frac{k\times k}{k\times k}=\frac{k}{2}\) Frequency of oscillation, f₁ = \(\frac{1}{2\pi}\sqrt{\frac{k_1}{m}}\) = \(\frac{1}{2\pi}\sqrt{\frac {k}{2m}}\) In case (ii); when the spring is cut to one-half, the effective spring constant k₂ = k. Frequency of oscillation, f₂ = \(\frac{1}{2\pi}\sqrt{\frac{k_2}{m}}\) = \(\frac{1}{2\pi}\sqrt{\frac{k}{m}}\) ∴ \(\frac{f_2}{f_1}=\sqrt 2\) The frequency of oscillations of mass m and force constant k is, f1 = 1/2π under root k/m when the spring is cut to one-half of its length, its force constant is doubles (2k). Then frequency of oscillation of mass m will be f2=1/2π under root 2k/m = f2/f1 = under root2`

The frequency of oscillations of a mass m suspended by a spring is f1. If the length of the spring is cut to one-half, the same mass oscillates with frequency f2. Determine the value \(\frac{f_2}{f_1}\).

Answer»

Let the full spring be the combination of two springs in series, each of force constant k. then in case (I), the effective spring constant (k₁) is given by

k₁ = \(\frac{k\times k}{k\times k}=\frac{k}{2}\)

Frequency of oscillation,

f₁ = \(\frac{1}{2\pi}\sqrt{\frac{k_1}{m}}\)

= \(\frac{1}{2\pi}\sqrt{\frac {k}{2m}}\)

In case (ii); when the spring is cut to one-half, the effective spring constant k₂ = k. Frequency of oscillation,

f₂ = \(\frac{1}{2\pi}\sqrt{\frac{k_2}{m}}\)

= \(\frac{1}{2\pi}\sqrt{\frac{k}{m}}\)

∴ \(\frac{f_2}{f_1}=\sqrt 2\)

The frequency of oscillations of mass m and force constant k is, f1 = 1/2π under root k/m
when the spring is cut to one-half of its length, its force constant is doubles (2k).
Then frequency of oscillation of mass m will be