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
Approach Solution - 1
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:
\[ \frac{m_1}{m_2} = \frac{\sqrt{2} - 1}{\sqrt{2} + 1} \]
Approach Solution -2
To solve this problem, we will analyze the motion of the two blocks connected by a string passing over a pulley. The masses of the blocks are \( m_1 \) and \( m_2 \) (where \( m_2 > m_1 \)), and the acceleration of the system is given as \( \frac{g}{\sqrt{2}} \).
Let's begin by applying Newton's second law to both masses:
- For mass \( m_1 \):
- The force acting on \( m_1 \) is \( T - m_1g = m_1a \). (Equation 1)
- For mass \( m_2 \):
- The force acting on \( m_2 \) is \( m_2g - T = m_2a \). (Equation 2)
Here, \( T \) is the tension in the string, and \( a = \frac{g}{\sqrt{2}} \) is the acceleration of the system.
From the two equations, we have:
- From Equation 1: \( T = m_1g + m_1a \)
- From Equation 2: \( T = m_2g - m_2a \)
Equating the two expressions for tension \( T \), we get:
\(m_1g + m_1a = m_2g - m_2a\)
Plug in the value of \( a = \frac{g}{\sqrt{2}} \) into the equation:
\(m_1g + m_1\left(\frac{g}{\sqrt{2}}\right) = m_2g - m_2\left(\frac{g}{\sqrt{2}}\right)\)
Simplify the equation:
\(g(m_1 + \frac{m_1}{\sqrt{2}}) = g(m_2 - \frac{m_2}{\sqrt{2}})\)
Cancel out the common factor \( g \):
\(m_1(1 + \frac{1}{\sqrt{2}}) = m_2(1 - \frac{1}{\sqrt{2}})\)
Rearrange to express the ratio of the masses:
\(\frac{m_1}{m_2} = \frac{1 - \frac{1}{\sqrt{2}}}{1 + \frac{1}{\sqrt{2}}}\)
Simplify the expression using the conjugate:
\(\frac{1 - \frac{1}{\sqrt{2}}}{1 + \frac{1}{\sqrt{2}}} \times \frac{\sqrt{2} - 1}{\sqrt{2} - 1} = \frac{(\sqrt{2} - 1)(1 - \frac{1}{\sqrt{2}})}{(\sqrt{2}^2 - 1^2)}\)
This simplifies to:
\(\frac{\sqrt{2} - 1}{\sqrt{2} + 1}\)
Thus, the correct option for the ratio \(\frac{m_1}{m_2}\) is:
\(\frac{\sqrt{2} - 1}{\sqrt{2} + 1}\)
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