Question

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:

• A. $\frac{\sqrt{2} - 1}{\sqrt{2} + 1}$
• B. $\frac{1 + \sqrt{5}}{\sqrt{5} - 1}$
• C. $\frac{1 + \sqrt{5}}{\sqrt{2} - 1}$
• D. $\frac{\sqrt{3} + 1}{\sqrt{2} - 1}$

Answer 1

The Correct Option is A

The system's acceleration is expressed as:

\[ a = \frac{m_2 - m_1}{m_1 + m_2} g \]

Given that \(a = \frac{g}{\sqrt{2}}\), this value is substituted into the acceleration equation:

\[ \frac{g}{\sqrt{2}} = \frac{m_2 - m_1}{m_1 + m_2} g \]

After simplification:

\[ \frac{1}{\sqrt{2}} = \frac{m_2 - m_1}{m_1 + m_2} \]

Performing cross-multiplication yields:

\[ \sqrt{2}(m_2 - m_1) = m_1 + m_2 \]

Rearranging the terms results in:

\[ m_1(\sqrt{2} + 1) = m_2(\sqrt{2} - 1) \]

The resulting ratio of the masses is:

\[ \frac{m_1}{m_2} = \frac{\sqrt{2} - 1}{\sqrt{2} + 1} \]