How Do You Calculate Acceleration in a Double Atwood Machine?

[Calpalned]

Homework Statement

The double Atwood machine shown in the figure has frictionless, mass-less pulleys and cords.
Determine the acceleration of m_A, m_B, m_C.

Homework Equations

F = ma

The Attempt at a Solution

http://photo1.ask.fm/779/725/638/-259996976-1shhdoh-qms4ifsem3hthf/preview/IMG_4082.jpg
http://photo1.ask.fm/905/398/673/-59997000-1shhdp1-2726aartaqghh2f/preview/IMG_4083.jpg
http://photo1.ask.fm/916/959/786/-239996977-1shhdpk-78i75a0215n926k/preview/IMG_4084.jpg
http://photo1.ask.fm/333/391/062/-249996981-1shhdrb-eeemt2tkfheqfp0/preview/IMG_4085.jpg

What did I do wrong?

 

[BvU]

First I see a_a = a_b , later on it becomes a_a = -a_b ?

 

[ehild]

The accelerations of A and B are equal in magnitude with respect to the hanging pulley, which accelerates, too.

 

[BvU]

Oops

 

[Calpalned]

Here is the answer given in my solutions manual.

Because the pulleys are massless, the net force on them must be 0. Because the cords are massless, the tension will be the same at both ends of the cords. Use the free-body diagrams to write Newton’s second law for each mass. We take the direction of acceleration to be positive in the direction of motion of the object. We assume that m_C is falling, m_B is falling relative to its pulley, and m_A is rising relative to its pulley. Also note that if the acceleration of m_A relative to the pulley above it is a_R , then a_A = a_R + a_C. Then, the acceleration of m_B is a_B = a_R - a_C since a_C is in the opposite direction of a_B .

Even though m_A and m_B are moving in opposite directions, how can they both be considered moving in the positive direction?

The acceleration of m_A relative to the free pulley should be the acceleration of the free pulley plus the acceleration of m_A. The free pulley is rising and m_A is rising too. Therefore, a_R = a_C + a_A. Why is it necessary to consider the free pulley in the first place? I don't see any problem with ignoring it (my original attempt doesn't seem to have an errors) but I somehow got the wrong answers.

Finally, what's the error in my original attempt in the starting post in this thread?

 

[Nathanael]

Even though m_A and m_B are moving in opposite directions, how can they both be considered moving in the positive direction?

m_a and m_b are not necessarily moving in opposite directions. It's only with respect to the lower pulley that they move in opposite directions. So in reality, they can both be moving in the same direction.

Finally, what's the error in my original attempt in the starting post in this thread?

Your error was the equation a_a = -a_b... This is not true! Imagine a_a = a_b ... They should both be at rest with respect to the lower pulley, right? According to your equation, a_a+a_b=0... But this is clearly not true because both a and b will moving with some nonzero acceleration in the same direction (they will move with the same acceleration that the lower pulley is moving with.)