In the given figure, a body of mass $$M$$ is held between two massless springs, on a smooth inclined plane. The free ends of the springs are attached to firm supports. If each spring has spring constant $$k$$, the frequency of oscillation of given body is:

If the mass is displaced by a small distance x up the incline:

• Left spring stretches by x, giving restoring force

kx

down the incline.

• Right spring compresses by x, also gives restoring force

kx

down the incline.

So net restoring force is

F=−2kx

Thus effective spring constant is

$$k_{\text{eff}}=2k$$

(Weight component Mgsin⁡α only shifts equilibrium position; it does not affect frequency.)

Equation of SHM:

$$Mx¨+2kx=0$$

So angular frequency

$$\omega=\sqrt{\frac{2k}{M}}$$

Frequency is

$$f=\frac{\omega}{2\pi}$$