A light string passing over a smooth light pulley connects two blocks of masses \(m_1\) and \(m_2\) (where \(m_2>m_1\)). If the acceleration of the system is $\frac{g}{\sqrt{2}}$, then the ratio of the masses $\frac{m_1}{m_2}$ is:
- $\frac{\sqrt{2} - 1}{\sqrt{2} + 1}$
- $\frac{1 + \sqrt{5}}{\sqrt{5} - 1}$
- $\frac{1 + \sqrt{5}}{\sqrt{2} - 1}$
- $\frac{\sqrt{3} + 1}{\sqrt{2} - 1}$
The Correct Option is A
Solution and Explanation
The acceleration of the system is given by:
\[ a = \frac{m_2 - m_1}{m_1 + m_2} g \]
Given \(a = \frac{g}{\sqrt{2}}\), substitute in the equation:
\[ \frac{g}{\sqrt{2}} = \frac{m_2 - m_1}{m_1 + m_2} g \]
Simplifying:
\[ \frac{1}{\sqrt{2}} = \frac{m_2 - m_1}{m_1 + m_2} \]
Cross-multiplying:
\[ \sqrt{2}(m_2 - m_1) = m_1 + m_2 \]
Rearranging:
\[ m_1(\sqrt{2} + 1) = m_2(\sqrt{2} - 1) \]
The ratio of masses is:
\[ \frac{m_1}{m_2} = \frac{\sqrt{2} - 1}{\sqrt{2} + 1} \]
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