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 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:

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

Answer 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:

1. For mass \( m_1 \):
   • The force acting on \( m_1 \) is \( T - m_1g = m_1a \). (Equation 1)
2. 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:

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