Kleppner/Kolenkow Pulley Example

  • #1
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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).
e8e583d28b.png


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.
 

Answers and Replies

  • #2
BvU
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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.
 
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  • #3
SammyS
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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 xp in the figure should have been yp .

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

etc.
 
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  • #4
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Update: Got it! I can type up my solution if you guys would like to see it.
 
  • #5
SammyS
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Update: Got it! I can type up my solution if you guys would like to see it.
Sure. Why not?
 
  • #6
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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 [tex] T = W_2 + M_2 (2A - \ddot{y_1}) [/tex]
I then set the right-hand sides of Eq. (1) and Eq. (2) together: [tex] W_1 + M_1 \ddot {y_1} = W_2 + M_2 \ddot {y_2} [/tex]
From here, it was just a matter for breaking down the ##W## terms into their corresponding ##M_ig## terms and rearranging:

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

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

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