Problem 147 - A Complete Resource Book in Physics for JEE Main

Question

A light string passing over a smooth light pulley connects two blocks of masses \(m_{1}\) and \(m_{2}\) (vertically). If the acceleration of the system is \(g / 8\), then the ratio of the masses is (A) \(8: 1\) (B) \(9: 7\) (C) \(4: 3\) (D) \(5: 3\)

Step-by-Step Solution

Answer

The ratio of the masses is (B) \(9: 7\).

Step 1: Analyze the forces

To start solving this problem, we first need to analyze the forces acting on the two blocks. The forces acting on block \(m_1\) are the tension in the string \(T\) and the force due to gravity \(m_1g\). The forces acting on block \(m_2\) are the tension in the string \(T\) and the force due to gravity \(m_2g\).

Step 2: Write the equations of motion

Now, we will write the equations of motion for the blocks using Newton's second law. Since the acceleration is \(g/8\), where \(g\) is the acceleration due to gravity, we will use that value in our equations. For block \(m_1\): \(m_1\frac{g}{8} = T - m_1g\) For block \(m_2\): \(m_2\frac{g}{8} = m_2g - T\)

Step 3: Solve the system of equations

To find the ratio of the masses, we will now solve the system of equations. First, let's add the two equations of motion to eliminate the tension \(T\). \(m_1\frac{g}{8} + m_2\frac{g}{8} = -m_1g + m_2g\) Now, let's simplify the equation and solve for the ratio. \(\frac{m_1 + m_2}{8} = m_2 - m_1\) \(8(m_2 - m_1) = m_1 + m_2\) \(8m_2 - 8m_1 = m_1 + m_2\) \(9m_1 = 7m_2\) Now we can see that the ratio of the masses is: \(\frac{m_1}{m_2} = \frac{7}{9}\) So the correct answer is: (B) \(9: 7\)