k1 Con si der the sy stem shown in the diagram, which consi st s of two springs with constants k and k2 and a mass m. oscillation fre quency v of the mass? If all surfaces are frictionless, what is the

If a mass m is attached to a spring with spring constant k and is displaced from the equilibrium position, a restoring elastic force opposite to the displacement is exerted by the spring.

The equation of motion for such a mass on a spring, is given by using Hooke's law-
$\mathbf{F}\mathbf{=}\mathbf{m}\stackrel{\mathbf{.}\mathbf{.}}{\mathbf{x}}\mathbf{=}\mathbf{-}\mathit{k}\mathbf{x}$

During the oscillation, one spring is stretched and other is compressed and vice-versa.

Let x be the displacement of mass m from its equilibrium position, and this is also the compression and expansion in the springs, $k_1 and k_2$.

The restoring force developed in the two springs is
$$\begin{gathered} {\mathbf{F}}_{\mathbf{1}}\mathbf{=}\mathbf{-}{\mathit{k}}_{\mathbf{1}}\mathbf{x} \\ \mathbf{and} \\ {\mathbf{F}}_{\mathbf{2}}\mathbf{=}\mathbf{-}{\mathit{k}}_{\mathbf{2}}\mathbf{x} \end{gathered}$$

Since both these forces acts on the mass m in the same direction (towards left in our case of motion of mass). Therefore-
$\begin{array}{rcl}\mathrm{F}& =& {\mathrm{F}}_1+{\mathrm{F}}_2\\ \mathrm{F}& =& -k_1\mathrm{x}-k_2\mathrm{x}\\ \mathbf{F}& \mathbf{=}& \mathbf{-}\mathbf{(}{\mathit{k}}_{\mathbf{1}}\mathbf{+}{\mathit{k}}_{\mathbf{2}}\mathbf{)}\mathbf{x}\end{array}$

or $\mathbf{F}\mathbf{=}\mathbf{m}\stackrel{\mathbf{.}\mathbf{.}}{\mathbf{x}}\mathbf{=}\mathbf{-}\mathbf{(}{\mathit{k}}_{\mathbf{1}}\mathbf{+}{\mathit{k}}_{\mathbf{2}}\mathbf{)}\mathbf{x}$

Thus the force constant of the system is (${\mathit{k}}_{\mathbf{1}}\mathbf{+}{\mathit{k}}_{\mathbf{2}}$)

Since angular frequency is given by: $\mathbf{\omega }\mathbf{=}\sqrt{\frac{{\mathbf{k}}_{\mathbf{system}}}{\mathbf{m}}}$
$$\begin{gathered} \Rightarrow \omega =\sqrt{\frac{k_1+k_2}{m}} \\ \Rightarrow 2\pi \nu \mathit{=}\sqrt{\frac{k_{\mathit{1}}\mathit{+}{k}_{\mathit{2}}}{m}} \\ \mathit{\Rightarrow }\mathit{\nu }\mathit{=}\frac{\mathit{1}}{\mathit{2}\mathit{\pi }}\sqrt{\frac{{\mathit{k}}_{\mathit{1}}\mathit{+}{\mathit{k}}_{\mathit{2}}}{\mathit{m}}} \end{gathered}$$