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Physics Question on Newton's Laws of Motion

A light string passing over a smooth light pulley connects two blocks of masses m1m_1​ and m2m_2​ (where m2​>m1m_2​>m_1​). If the acceleration of the system is g2\frac{g}{\sqrt{2}}, then the ratio of the masses m1m2\frac{m_1}{m_2} is:

A

2−12+1\frac{\sqrt{2} - 1}{\sqrt{2} + 1}

B

1+55−1\frac{1 + \sqrt{5}}{\sqrt{5} - 1}

C

1+52−1\frac{1 + \sqrt{5}}{\sqrt{2} - 1}

D

3+12−1\frac{\sqrt{3} + 1}{\sqrt{2} - 1}

Answer

2−12+1\frac{\sqrt{2} - 1}{\sqrt{2} + 1}

Explanation

Solution

The acceleration of the system is given by:

a=m2−m1m1+m2ga = \frac{m_2 - m_1}{m_1 + m_2} g

Given a=g2a = \frac{g}{\sqrt{2}}, substitute in the equation:

g2=m2−m1m1+m2g\frac{g}{\sqrt{2}} = \frac{m_2 - m_1}{m_1 + m_2} g

Simplifying:

12=m2−m1m1+m2\frac{1}{\sqrt{2}} = \frac{m_2 - m_1}{m_1 + m_2}

Cross-multiplying:

2(m2−m1)=m1+m2\sqrt{2}(m_2 - m_1) = m_1 + m_2

Rearranging:

m1(2+1)=m2(2−1)m_1(\sqrt{2} + 1) = m_2(\sqrt{2} - 1)

The ratio of masses is:

m1m2=2−12+1\frac{m_1}{m_2} = \frac{\sqrt{2} - 1}{\sqrt{2} + 1}