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
- \(\frac{9}{7}\)
- \(\frac{8}{1}\)
- \(\frac{4}{3}\)
- \(\frac{5}{3}\)
The Correct Option is A
Approach Solution - 1
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 \(\frac{9}{7}\).
Approach Solution -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:
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