Euler-Lagrange equation: pulley system

[bookworm031]

Homework Statement

Determine, using Euler-Lagrange's equation, the acceleration for B when the weights are moving vertically.

Relevant Equations

$\frac{d}{dt}\bigg(\frac{\partial L}{\partial \dot{y}}\bigg) = \frac{\partial L}{\partial y}$, $L = T - V$

$m_{A} = 3 kg$
$m_{B} = 2 kg$

$y_{A} + y_{B} = c \Leftrightarrow y_{A} = c - y_{B}$, where c is a constant.
$\Rightarrow \dot{y_{A}} = -\dot{y_{B}}$

The Lagrangian:
$$L = T - V$$
$T =\frac{1}{2}m_{A}\dot{y_{B}}^{2} + \frac{1}{2}m_{B}\dot{y_{B}}^{2}$
$V = m_{A}g(c - y_{B}) + m_{B}gy_{B}$
$\Leftrightarrow L = \frac{1}{2}m_{A}\dot{y_{B}}^{2} + \frac{1}{2}m_{B}\dot{y_{B}}^{2} - (m_{A}g(c - y_{B}) + m_{B}gy_{B})$

Applying Euler-Lagrange's equation:

$\frac{d}{dt}\bigg(\frac{\partial L}{\partial \dot{y}}\bigg) = \ddot{y_{B}}(m_{B} + m_{A})$
$\frac{\partial L}{\partial y} = g(m_{A} - m_{B})$

Solving for $\ddot{y_{B}}$:
$\ddot{y_{B}} = \frac{g(m_{A} - m_{B})}{(m_{B} + m_{A})} = 1.9604 \frac{m}{s^{2}}$

The answer is supposed to be $1.78 \frac{m}{s^{2}}$. What am I doing wrong? I'm completely lost.

Thanks!

 

[haruspex]

$y_{A} + y_{B} = c$, where c is a constant.

Sure?

 

[bookworm031]

Sure?

No. However, the rationale was that, since one weight moves down as the other moves up, and vice versa, the difference should always be a constant. Do you think this is wrong? If so, what's the relationship between $y_{A}$ and $y_{B}$?

Edit: Now that I think about it, $y_{A} + y_{B} = c$ doesn't make much sense, though I'm still not sure how to set up a relationship.

 

[haruspex]

No. However, the rationale was that, since one weight moves down as the other moves up, and vice versa, the difference should always be a constant. Do you think this is wrong? If so, what's the relationship between $y_{A}$ and $y_{B}$?

Edit: Now that I think about it, $y_{A} + y_{B} = c$ doesn't make much sense, though I'm still not sure how to set up a relationship.

Express the total length of the string in terms of the three straight parts. Don't worry about the semicircular arcs since those are constant.

 

[bookworm031]

Express the total length of the string in terms of the three straight parts. Don't worry about the semicircular arcs since those are constant.

I don't know why I find this so difficult, I feel very stupid right now. I should express the total length only in terms of $y_{A}$ and $y_{B}$, right? I've been staring myself blind at this figure.

 

[haruspex]

I don't know why I find this so difficult, I feel very stupid right now. I should express the total length only in terms of $y_{A}$ and $y_{B}$, right? I've been staring myself blind at this figure.

Let the lengths of the string sections, numbered from the left, be L1, L2, L3.
Allow also a constant Lf for the short fixed string supporting the upper pulley.
What equations can you write relating these to $y_{A}$ and $y_{B}$?
Can you then find L1+L2+L3 in terms of $y_{A}$, $y_{B}, L_f$?