91Ó°ÊÓ

A spring's spring constant is written as,

$k=\frac{F}{x}$

F denotes the applied force, and x denotes the spring's displacement.

Assume that there are no-mass springs, and that the spring constants of the two springs are and. The frequencies of the two springs and the frequency of the block are b f

In the figure, $F_{1w}$and $F_{w1}$are the forces exerted by spring 1 on wall and wall on spring 1

Spring number two is in contact with the mass. As a result, the net force on the mass is $F_m=F_{2m}$

Newton's third law states that the force exerted by spring 2 on mass is equal and opposite to the force exerted on mass by spring 2, and the force exerted on spring 1 by spring 2 is equal to the force exerted on spring 2 by spring 1.

The massless spring has no net force acting on it..

There fore,

$F_{w1}=k_1\mathrm{Δ}{x}_1\text{and}{F}_{m2}=k_2\mathrm{Δ}{x}_2\text{.}$

Combine the two realities mentioned above.

$F_m=k_1\mathrm{Δ}{x}_1=k_2\mathrm{Δ}{x}_2$

The mass's net displacement is

$\mathrm{Δ}x=\mathrm{Δ}{x}_1+\mathrm{Δ}{x}_2$

$=\frac{F_m}{k_1}+\frac{F_m}{k_2}$

$=F_m\left[\frac{1}{k_1}+\frac{1}{k_2}\right]$

$=\left(\frac{k_1+k_2}{k_1{k}_2}\right)F_m$

The mass's net displacement is

$F_m=\left(\frac{k_1+k_2}{k_1+k_2}\right)\mathrm{Δ}x$

As a result, the effective spring constant is

$k=\frac{k_1{k}_2}{k_1+k_2}$

The expression for the mass's angular frequency is

$\mathrm{Ó\neg }=\sqrt{\frac{k}{m}}$

$=\sqrt{\frac{1}{m}\left(\frac{k_1{k}_2}{k_1+k_2}\right)}$

$=\sqrt{\frac{\left(\frac{k_1}{m}\right)\left(\frac{k_2}{m}\right)}{\frac{k_1}{m}+\frac{k_2}{m}}}$

$\mathrm{Ó\neg }=\sqrt{\frac{{\mathrm{Ó\neg }}_1^2{\mathrm{Ó\neg }}_2^2}{{\mathrm{Ó\neg }}_1^2+{\mathrm{Ó\neg }}_2^2}}$

In the above expression, ${\mathrm{Ó\neg }}_1=\frac{k_1}{m}$and ${\mathrm{Ó\neg }}_2=\frac{k_2}{m}$.

As a result, the actual frequency is simply a multiple of the angular frequency, which equals the value of $f=\sqrt{\frac{f_1^2{f}_2^2}{f_1^2+f_2^2}}$, where $f_1$and $f_2$are the frequencies of oscillations of the springs $1$and $2$.