Question

A light unstretchable string passing over a smooth light pulley connects two blocks of masses \(m_1\) and \(m_2\). If the acceleration of the system is \(\frac{g}{8}\), then the ratio of the masses \(\frac{m_2}{m_1}\) is:

• A. 9 : 7
• B. 4 : 3
• C. 5 : 3
• D. 8 : 1

Answer 1

The Correct Option is A

To determine the mass ratio \(\frac{m_2}{m_1}\), the forces on two blocks of masses \(m_1\) and \(m_2\), connected by a string over a pulley, are analyzed.

1. Block \(m_1\) is acted upon by gravity \(m_1g\) downwards and tension \(T\) upwards.
2. Block \(m_2\) is acted upon by gravity \(m_2g\) downwards and tension \(T\) upwards.
3. Newton's second law yields the equations of motion: \(m_1g - T = m_1a\) (Equation 1) and \(T - m_2g = m_2a\) (Equation 2).
4. With acceleration \(a = \frac{g}{8}\) given:
5. From Equation 1: \(T = m_1g - m_1\frac{g}{8}\).
6. From Equation 2: \(T = m_2g + m_2\frac{g}{8}\).
7. Equating the expressions for \(T\):
   • \(m_1g - m_1\frac{g}{8} = m_2g + m_2\frac{g}{8}\)
   • This simplifies to: \(m_1g(1 - \frac{1}{8}) = m_2g(1 + \frac{1}{8})\)
   • Further simplification gives: \(m_1 \cdot \frac{7}{8} = m_2 \cdot \frac{9}{8}\)
   • This leads to: \(m_1 \cdot 7 = m_2 \cdot 9\)
   • The mass ratio is: \(\frac{m_2}{m_1} = \frac{9}{7}\).

The calculated ratio \(\frac{m_2}{m_1}\) is 9:7.