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. \( \frac{\sqrt{3} + 1}{\sqrt{2} - 1} \)

Hint:

For Atwood’s machine, the acceleration is given by: \[ a = \frac{(m_2 - m_1)}{m_1 + m_2} g \] This equation helps determine the ratio of masses when acceleration is known.

Answer 1

The Correct Option is A

Step 1: {Equation of motion for the system}
The system's acceleration is derived as: \[ a = \frac{(m_2 - m_1)}{m_1 + m_2} g \] Step 2: {Equating given acceleration}
Setting the derived acceleration equal to the given acceleration: \[ \frac{g}{\sqrt{2}} = \frac{(m_2 - m_1)}{m_1 + m_2} g \] Step 3: {Solve for \( \frac{m_1}{m_2} \)}
Simplifying the equation: \[ \sqrt{2} (m_2 - m_1) = m_1 + m_2 \] Rearranging to find the ratio of masses: \[ \frac{m_1}{m_2} = \frac{\sqrt{2} - 1}{\sqrt{2} + 1} \] Therefore, the correct answer is (A) \( \frac{\sqrt{2} - 1}{\sqrt{2} + 1} \).