The double Atwood machine has frictionless, massless pulleys

[berkdude022]

Homework Statement

The double Atwood machine has frictionless, massless pulleys and cords. Determine (a) the acceleration of masses m_a, m_b, and m_c, and (b) the tensions F_ta and F_tc in the cords.

Homework Equations

F=ma

The Attempt at a Solution

So I drew free body diagrams for the mass A, mass B, mass C, and the lower hanging pulley.

I then computed these equations from F=ma:F_ta - m_a = m_aa_a
F_ta - m_b = m_ba_b
2F_ta = F_tc
F_tc - m_c = m_ca_c

This is all assuming that there is currently no relationship between the accelerations. And this is where I am stuck. I do not know how to represent a_c as that of a_a and a_b. Or maybe that isn't the correct way to solve the system either. The only thing that I assume is that the tensions for ma and mb are the same. Any guidance would be appreciated. Thanks!

 

[Nathanael]

Can you relate the acceleration of the lower pulley to the acceleration of mass C?

There is a simple way to relate the acceleration of mass A and mass B to the acceleration of the lower pulley. I don't want to spoil it, yet I'm not sure how to hint at it

Have another go at it, just remember that the length of each string is constant.

 

[berkdude022]

Can you relate the acceleration of the lower pulley to the acceleration of mass C?

There is a simple way to relate the acceleration of mass A and mass B to the acceleration of the lower pulley. I don't want to spoil it, yet I'm not sure how to hint at it

Have another go at it, just remember that the length of each string is constant.

So I believe that the acceleration of mass B is the negative of mass A; or a_a = -a_b.
But I have looked into how the system would react to m₃ accelerating downwards with a constant string length, and I just cannot understand it. Like, I want to believe that as m₃ moves down a distance L, that the length of the lower hanging pulley also moves at that same L as do the masses m_a and m_b.

 

[Nathanael]

So I believe that the acceleration of mass B is the negative of mass A; or a_a = -a_b.

This would be true for a single atwood-machine with two blocks A and B, but it is not quite right in our situation.
(For example, if m_a=m_b, then they will have the same acceleration in the same direction and it will not necessarily be zero.)

But I have looked into how the system would react to m₃ accelerating downwards with a constant string length, and I just cannot understand it. Like, I want to believe that as m₃ moves down a distance L, that the length of the lower hanging pulley also moves at that same L as do the masses m_a and m_b.

m₃ is mass C?

Anyway, consider the acceleration of the center of the lower pulley. What is it's relationship to a_c?

When you think about the acceleration of A and B, try to imagine it as being from two parts: partly from the acceleration of the lower pulley and partly from the movement of the string.

 

[berkdude022]

This would be true for a single atwood-machine with two blocks A and B, but it is not quite right in our situation.
(For example, if ma=mb, then they will have the same acceleration in the same direction and it will not necessarily be zero.)

Wait, why exactly would the accelerations not be opposite for these masses? If the pulley system is moved up, wouldn't they be affected in opposite directions?

Anyway, consider the acceleration of the center of the lower pulley. What is it's relation ship to ac?

I would think that the lower pulley's acceleration would be the opposite of the acceleration of mass c.

 

[Nathanael]

Wait, why exactly would the accelerations not be opposite for these masses? If the pulley system is moved up, wouldn't they be affected in opposite directions?

A special case to imagine, pretend A and B have the same mass: if the lower-pulley moves up then they will both move up with it, so the accelerations are not opposite.

I would think that the lower pulley's acceleration would be the opposite of the acceleration of mass c.

Right, because the length of string can't change (if C moves up an amount, then the center of the pulley must move down by the same amount).

Consider the reference frame of the lower-pulley (just ignore the upper pulley and mass C for now). In this reference frame it is a normal atwood-machine, and so A and B have equal and opposite accelerations. Let's call the accelerations of A and B in this reference frame +a_D and -a_D. Now when we switch back to the original frame, we must add the acceleration of the lower-pulley, which I'll call a_p. This means the accelerations of A and B are a_p+a_D and a_p-a_D.