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 find the ratio of the masses \(\frac{m_2}{m_1}\), we start by analyzing the forces acting on each block. Consider two blocks of masses \(m_1\) and \(m_2\) connected by a light unstretchable string passing over a smooth pulley.

1. The block with mass \(m_1\) experiences a gravitational force \(m_1g\) downward and a tension \(T\) upward.
2. The block with mass \(m_2\) experiences a gravitational force \(m_2g\) downward and a tension \(T\) upward.
3. According to Newton's second law, for each block, the equations of motion are given as follows: \(m_1g - T = m_1a\) (Equation 1) and \(T - m_2g = m_2a\) (Equation 2).
4. Given that the acceleration \(a = \frac{g}{8}\), we can substitute it into the equations:
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 \(T\) from both equations, we have:
   • \(m_1g - m_1\frac{g}{8} = m_2g + m_2\frac{g}{8}\)
   • Simplifying, we get: \(m_1g(1 - \frac{1}{8}) = m_2g(1 + \frac{1}{8})\)
   • This simplifies to: \(m_1 \cdot \frac{7}{8} = m_2 \cdot \frac{9}{8}\)
   • Therefore, \(m_1 \cdot 7 = m_2 \cdot 9\).
   • Thus, \(\frac{m_2}{m_1} = \frac{9}{7}\).

Hence, the correct ratio of the masses \(\frac{m_2}{m_1}\) is 9:7, which matches the given correct option.

Answer 2

Step 1: Equation for acceleration The acceleration of the system is given by:

\[ a_{\text{sys}} = \frac{(m_2 - m_1)}{m_1 + m_2} \cdot g. \]

Substitute \(a_{\text{sys}} = \frac{g}{8}\):

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

Cancel \(g\) from both sides:

\[ \frac{m_2 - m_1}{m_1 + m_2} = \frac{1}{8}. \]

Step 2: Solve for \(\frac{m_2}{m_1}\) Rearrange the equation:

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

Simplify:

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

Combine like terms:

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

\[ 7m_2 = 9m_1. \]

Take the ratio:

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

Final Answer: \(\frac{m_2}{m_1} = 9 : 7\).