When a mass is hung from the lower of a spring of negligible mass, an extension x is produced in spring. The mass is set into vertical oscillations. The time period of oscillation is :

Question

Accepted Answer

To determine the time period of oscillation for a mass-spring system, we need to consider the fundamental principles of simple harmonic motion (SHM). When a mass $ m $ is hung from a spring, it causes an extension $ x $ in the spring. This extension is due to the force of gravity acting on the mass, which is balanced by the spring force at equilibrium.

1. **Explanation for Option 1: $ T = 2\pi \sqrt{\frac{x}{g}} $**

The time period $ T $ of a mass-spring system in vertical oscillation is given by the formula for simple harmonic motion:
   $$
   T = 2\pi \sqrt{\frac{m}{k}}
   $$
   where $ k $ is the spring constant. From Hooke's Law, the spring force $ F $ is given by:
   $$
   F = kx
   $$
   At equilibrium, this force is equal to the gravitational force $ mg $, so:
   $$
   kx = mg \quad \Rightarrow \quad k = \frac{mg}{x}
   $$
   Substituting $ k $ into the formula for $ T $, we get:
   $$
   T = 2\pi \sqrt{\frac{m}{\frac{mg}{x}}} = 2\pi \sqrt{\frac{x}{g}}
   $$
   Thus, Option 1 is correct as it accurately represents the time period of oscillation for the given system.

2. **Explanation for why the other options are incorrect:**

- **Option 2: $ T = 2\pi \sqrt{\frac{2x}{g}} $**
     This option suggests that the time period is based on an extension of $ 2x $, which is not consistent with the equilibrium condition derived from Hooke's Law and the balance of forces. The factor of 2 is not justified by the physics of the system.

- **Option 3: $ T = 2\pi \sqrt{\frac{x}{2g}} $**
     This option implies that the gravitational acceleration $ g $ is effectively halved in the calculation of the time period. There is no physical basis for this reduction in $ g $ in the context of vertical oscillations of a mass-spring system.

- **Option 4: $ T = \sqrt{2\pi} \sqrt{\frac{x}{g}} $**
     This option introduces an incorrect modification of the constant $ 2\pi $ to $ \sqrt{2\pi} $. The standard derivation of the time period for a mass-spring system in SHM does not support this alteration.

In conclusion, Option 1 is the correct answer as it is derived directly from the principles of simple harmonic motion and the equilibrium condition of the mass-spring system. The other options do not align with the established physical laws governing such systems.