Question 25

A block of mass 2 kg is attached with two identical springs of spring constant 20 $$\text{N m}^{-1}$$ each. The block is placed on a frictionless surface and the ends of the springs are attached to rigid supports (see figure). When the mass is displaced from its equilibrium position, it executes a simple harmonic motion. The time period of oscillation is $$\frac{\pi}{\sqrt{X}}$$ in SI unit. The value of $$X$$ is _____.

Correct Answer: 5

When block is displaced by x,

each spring contributes restoring force kx.

So net restoring force

$$F=-(k+k)x=-2kx$$

Thus effective spring constant

$$k_{\text{eff}}=2k=40\text{ N/m}$$

Given mass

m=2 kg

Time period of SHM:

$$T=2\pi\sqrt{\frac{m}{k_{\text{eff}}}}$$

$$=2\pi\sqrt{\frac{2}{40}}$$

$$=2\pi\sqrt{\frac{1}{20}}$$

$$=\frac{2\pi}{2\sqrt{5}}$$

$$=\frac{\pi}{\sqrt{5}}$$

Given

$$T=\frac{\pi}{\sqrt{\ X}}$$

so X=5

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