In figure (A), mass $$2m$$ is fixed on mass $$m$$ which is attached to two springs of spring constant $$k$$. In figure (B), mass $$m$$ is attached to two springs of spring constant $$k$$ and $$2k$$. If mass $$m$$ in (A) and (B) are displaced by distance $$x$$ horizontally and then released, then time period $$T_1$$ and $$T_2$$ corresponding to (A) and (B) respectively follow the relation.

We need to find the relationship between the time periods $$T_1$$ and $$T_2$$ for the two spring-mass systems shown in the figures.

1. Analyze Figure (A)

• Total Mass ($$m_1$$): A mass $$2m$$ is fixed on top of mass $$m$$. Therefore, they move together as a single unit:

  $$m_1 = m + 2m = 3m$$

• Equivalent Spring Constant ($$k_{eq1}$$): The mass is connected between two springs of spring constant $$k$$. When the mass moves horizontally, one spring compresses while the other stretches by the exact same amount. This means the springs are in parallel configuration:

  $$k_{eq1} = k + k = 2k$$

The time period ($$T_1$$) for system A is given by:

$$T_1 = 2\pi \sqrt{\frac{m_1}{k_{eq1}}} = 2\pi \sqrt{\frac{3m}{2k}}$$

2. Analyze Figure (B)

• Total Mass ($$m_2$$): Only a single mass $$m$$ is attached:

  $$m_2 = m$$

• Equivalent Spring Constant ($$k_{eq2}$$): Similar to system A, the mass is connected between two springs ($$k$$ and $$2k$$) that work in opposition/parallel during horizontal displacement:

  $$k_{eq2} = k + 2k = 3k$$

The time period ($$T_2$$) for system B is given by:

$$T_2 = 2\pi \sqrt{\frac{m_2}{k_{eq2}}} = 2\pi \sqrt{\frac{m}{3k}}$$

3. Calculate the Ratio ($$\frac{T_1}{T_2}$$)

Divide the expression for $$T_1$$ by $$T_2$$:

$$\frac{T_1}{T_2} = \frac{2\pi \sqrt{\frac{3m}{2k}}}{2\pi \sqrt{\frac{m}{3k}}}$$

Cancel out common terms ($$2\pi$$, $$m$$, and $$k$$):

$$\frac{T_1}{T_2} = \frac{\sqrt{\frac{3}{2}}}{\sqrt{\frac{1}{3}}} = \sqrt{\frac{3}{2} \times \frac{3}{1}} = \sqrt{\frac{9}{2}} = \frac{3}{\sqrt{2}}$$

Conclusion

The correct relationship is $$\frac{T_1}{T_2} = \frac{3}{\sqrt{2}}$$, which corresponds to Option A .