# Acceleration of masses hanging from a system of two pulleys

"Masses $M_1$ and $M_2$ are connected to a system of strings and pulleys as shown. The strings are massless and inextensible, and the pulleys are massless and frictionless. Find the acceleration of M1. (Clue if $M_2 = M_1$, A(acceleration)=g/5.)” (The problem is from “An Introduction to Mechanics” – Kleppner & Kolenkow) My try: Let T be the tension of the rope connected to $M_2$. So the tension of the rope connected to $M_1$ will be 2T. The acceleration of both the masses is A. Now, $M_2g – T = M_2A$ … (i)

$2T – M_1g = M_1A$ … (ii)

From (i) and (ii) ,

$$A=\frac{g(2M_2-M_1)}{(2M_2+M_1)}$$ Now if $M_2 = M_1$, I get, A = g/3.

But the answer is A=g/5.

Where am I wrong? Will the accelerations of $M_1$ and $M_2$ not be the same? or, are there anything about the tensions ?

• The tensions are correct. The problem is that you assume $a_1 = a_2$. If $M_2$ descends 1cm, how far does pulley 2 descend? – mmesser314 May 18 '14 at 13:47
• I think, if $M_2$ descends 1cm, then pulley 2(i.e the movable one) also descends 1cm and $M_1$ ascends 1cm as well. Isn't it? But how are the accelerations different? – Siddhartha May 18 '14 at 13:58