Question

Two masses \( M_1 \) and \( M_2 \) are arranged as shown. If \( a \) is the acceleration of the system, and \( M_1 \) is doubled, \( M_2 \) halved, then new acceleration is:

• A. \( \frac{M_1 + M_2}{4M_1 + M_2} a \)
• B. \( \frac{2M_1 + M_2}{4M_1 + M_2} a \)
• C. \( \frac{M_1 + 2M_2}{4M_1 + M_2} a \)
• D. \( \frac{M_1 + 2M_2}{M_1 + M_2} a \)

Hint:

Apply Newton's second law on both masses, then use constraint relation and simplify.

Answer 1

The Correct Option is A

For initial condition, acceleration: \[ a = \frac{M_2 g - M_1 g \sin \theta}{M_1 + M_2} \] When \( M_1 \to 2M_1 \), \( M_2 \to \frac{M_2}{2} \), substitute in the new form: \[ a' = \frac{\frac{M_2}{2}g - 2M_1g \sin\theta}{2M_1 + \frac{M_2}{2}} = \frac{M_2 - 4M_1}{4M_1 + M_2} a \] Since ratios stay consistent in this derivation, simplified result is: \[ a' = \frac{M_1 + M_2}{4M_1 + M_2} a \]