A mass $$m$$ is attached to two springs as shown in figure. The spring constants of two springs are $$K_1$$ and $$K_2$$. For the frictionless surface, the time period of oscillation of mass $$m$$ is

If the mass is displaced by x toward right:

• right spring gets compressed by x
• left spring gets stretched by x

Both exert restoring forces toward equilibrium.

Force by right spring:

$$F_1=-K_1x$$

Force by left spring:

$$F_2=-K_2x$$

Net restoring force:

$$F=-(K_1+K_2)x$$

Comparing with SHM form

$$F=-K_{\text{eff}}x$$

effective spring constant is

$$K_{\text{eff}}=K_1+K_2$$

Angular frequency:

$$\omega=\sqrt{\frac{K_1+K_2}{m}}$$

Therefore time period

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

so

$$2\pi\sqrt{\dfrac{m}{K_1+K_2}}$$