Kleppner/Kolenkow Pulley Example

[Cosmophile]

Homework Statement

[/B]
Hey, all. I am working through Kleppner/Kolenkow and encountered a problem when trying to follow the example pictured below. My issues comes when they say "Equations (1)-(3) are easily solved..."

As it turns out, they are not so easily solved for me! So, it appears I've found a hole in my mathematical training that needs to be filled (K&K have proved to be good at exposing these weak spots).

Homework Equations

All pictured above.

The Attempt at a Solution

Frankly, I'm not sure where to begin. I've tried expanding the $\ddot{y_1}$ and $\ddot{y_2}$ terms using $(3)$, but the equations quickly become quite long. I haven't had the time to really sit down and see if they eventually clean up nicely and give the desired result, but truth be told, I don't feel too confident going into this and would love some help.

 

[BvU]

Build confidence by starting off with A = 0 !
Take your time to really sit down and try another tack if things still become too complicated.

 

[SammyS]

Homework Statement

[/B]
Hey, all. I am working through Kleppner/Kolenkow and encountered a problem when trying to follow the example pictured below. My issues comes when they say "Equations (1)-(3) are easily solved..."

As it turns out, they are not so easily solved for me! So, it appears I've found a hole in my mathematical training that needs to be filled (K&K have proved to be good at exposing these weak spots).
[ IMG]http://puu.sh/mWaau/e8e583d28b.png[/PLAIN]

Homework Equations

All pictured above.

The Attempt at a Solution

Frankly, I'm not sure where to begin. I've tried expanding the $\ddot{y_1}$ and $\ddot{y_2}$ terms using $(3)$, but the equations quickly become quite long. I haven't had the time to really sit down and see if they eventually clean up nicely and give the desired result, but truth be told, I don't feel too confident going into this and would love some help.

First of all, you may have noticed that the x_p in the figure should have been y_p .

Solve Eq.(3) for either $\ \ddot y_1\$ or $\ \ddot y_2\ .\$ Plug the result into Eq (1) or (2).

etc.

 

[Cosmophile]

Update: Got it! I can type up my solution if you guys would like to see it.

 

[SammyS]

Update: Got it! I can type up my solution if you guys would like to see it.

Sure. Why not?

 

[Cosmophile]

Sure. Why not?

I actually solved it right before you posted your advice (I solved it in the same way):

I solved Eq. (3) for $\ddot {y_2}$ and plugged that solution into Eq. (2). This gave T = W_2 + M_2 (2A - \ddot{y_1})
I then set the right-hand sides of Eq. (1) and Eq. (2) together: W_1 + M_1 \ddot {y_1} = W_2 + M_2 \ddot {y_2}
From here, it was just a matter for breaking down the $W$ terms into their corresponding $M_ig$ terms and rearranging:

M_1g+M_1 \ddot{y_1} = M_2g + 2A M_2 - M_2 \ddot {y_1}
\ddot{y_1} =(2A+g) \frac {M_2 - M_1g}{M_1 + M_2}

To solve for $T$, I simply plugged this result into Eq. (1), and the desired result followed immediately.