Question

A light string passing over a smooth light fixed pulley connects two blocks of masses m1 and m2. If the acceleration of the system is g/8, then the ratio of masses is

• A. \(\frac{9}{7}\)
• B. \(\frac{8}{1}\)
• C. \(\frac{4}{3}\)
• D. \(\frac{5}{3}\)

Answer 1

The Correct Option is A

To find the ratio of masses \( m_1 \) and \( m_2 \), we can use the concepts of mechanics involving pulleys and blocks.

Given:

• Masses: \( m_1 \) and \( m_2 \)
• Acceleration of the system: \( a = \frac{g}{8} \)
• Acceleration due to gravity: \( g \)

Consider the forces acting on each mass:

• For mass \( m_1 \):
  • The tension in the string \( T \) and gravitational force \( m_1g \) act on it.
  • Equating forces: \( T = m_1g - m_1a \)
• For mass \( m_2 \):
  • The tension in the string \( T \) and gravitational force \( m_2g \) act on it.
  • Equating forces: \( T = m_2g + m_2a \)

Setting these two expressions for tension \( T \) equal gives:

\(m_1g - m_1a = m_2g + m_2a\)

Substituting \( a = \frac{g}{8} \):

\(m_1g - m_1\left(\frac{g}{8}\right) = m_2g + m_2\left(\frac{g}{8}\right)\)

Simplifying further:

\(m_1g\left(1 - \frac{1}{8}\right) = m_2g\left(1 + \frac{1}{8}\right)\)

\(\frac{7m_1g}{8} = \frac{9m_2g}{8}\)

Cancelling \( g \) and simplifying gives:

\(7m_1 = 9m_2\)

Thus, the ratio of the masses is:

\(\frac{m_1}{m_2} = \frac{9}{7}\)

Therefore, the correct answer is \(\frac{9}{7}\).

Answer 2

The acceleration \( a \) of the system is given by:

\[ a = \frac{(m_1 - m_2)g}{m_1 + m_2} = \frac{g}{8}. \]

This implies:

\[ 8m_1 - 8m_2 = m_1 + m_2. \]

Rearrange terms:

\[ 7m_1 = 9m_2. \]

Thus, the ratio of \( m_1 \) to \( m_2 \) is:

\[ \frac{m_1}{m_2} = \frac{9}{7}. \]

Therefore, the answer is:

\[ \frac{9}{7}. \]