Question

Two masses \(M_1\) and \(M_2\) are tied together at the two ends of a light inextensible string that passes over a frictionless pulley. When the mass \(M_2\) is twice that of \(M_1\), the acceleration of the system is \(a_1\). When the mass \(M_2\) is thrice that of \(M_1\), the acceleration of the system is \(a_2\). The ratio \(\frac{a_1}{a_2}\) will be

• A. \(\frac{1}{3}\)
• B. \(\frac{2}{3}\)
• C. \(\frac{3}{2}\)
• D. \(\frac{1}{2}\)

Answer 1

The Correct Option is B

To solve this problem, we need to understand the mechanics of a pulley system with two masses connected by a light inextensible string passing over a frictionless pulley. The given problem involves calculating the acceleration ratios when the mass \(M_2\) is twice and thrice the mass of \(M_1\).

1. When \(M_2 = 2M_1\), let's denote the acceleration as \(a_1\).
   • Using Newton's Second Law for each mass, we have:
   • For \(M_1\): \(T - M_1g = M_1a_1\)
   • For \(M_2\): \(M_2g - T = M_2a_1\)
2. Substitute \(M_2 = 2M_1\) into the equation:
   • \(2M_1g - T = 2M_1a_1\)
   • Equate the tension \(T\) from both masses' equations and solve for \(a_1\):
   • \(T - M_1g = M_1a_1 \Rightarrow T = M_1g + M_1a_1\)
   • Substitute this \(T\) in the second equation:
   • \((2M_1g - (M_1g + M_1a_1)) = 2M_1a_1\)
   • Simplify to get: \(M_1g = 3M_1a_1\).
   • \(a_1 = \frac{g}{3}\)
3. When \(M_2 = 3M_1\), let's denote the acceleration as \(a_2\).
   • For \(M_1\): \(T - M_1g = M_1a_2\)
   • For \(M_2\): \(M_2g - T = M_2a_2\)
4. Substitute \(M_2 = 3M_1\) into the equation:
   • \(3M_1g - T = 3M_1a_2\)
   • Equate the tension \(T\) from both masses' equations and solve for \(a_2\):
   • \(T - M_1g = M_1a_2 \Rightarrow T = M_1g + M_1a_2\)
   • Substitute this \(T\) in the second equation:
   • \((3M_1g - (M_1g + M_1a_2)) = 3M_1a_2\)
   • Simplify to get: \(2M_1g = 4M_1a_2\).
   • \(a_2 = \frac{g}{2}\)
5. Calculate the ratio \(\frac{a_1}{a_2}\):
   • \(\frac{a_1}{a_2} = \frac{\frac{g}{3}}{\frac{g}{2}} = \frac{2}{3}\)

The ratio \(\frac{a_1}{a_2}\) is \(\frac{2}{3}\). Therefore, the correct answer is \(\frac{2}{3}\).