Unequal Accelerations in Double Atwood Machine

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SUMMARY

The discussion centers on the Double Atwood machine, specifically addressing the unequal accelerations of blocks A and B. The accelerations, denoted as a_a and a_b, differ due to the configuration of the pulleys, where the movement of one pulley affects the other. This results in additional terms in the equations of motion, confirming that the assumption of a massless rope does not account for the complexities introduced by the pulley system.

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  • Understanding of Newton's laws of motion
  • Familiarity with pulley systems and their mechanics
  • Basic knowledge of kinematics and acceleration
  • Concept of massless ropes in physics
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  • Study the dynamics of pulley systems in classical mechanics
  • Learn about the equations of motion for multiple connected bodies
  • Explore the implications of massless vs. massive ropes in mechanical systems
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id the sloth
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I have a simple question about the Double Atwood machine.


220px-DoubleAtwood.jpg


I am able to follow the solution but one thing makes no sense to me. The accelerations for block A and block B are:

a6bf5b8c63c161a91d457195a827344a.png


431f4b7dd585fb56b793ab9272923c01.png


Why aren't these equal? If they aren't, doesn't that mean the rope stretches? Or is this just an effect of assuming a massless rope?
 
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In the few minutes since posting this, I have realized my mistake.

a_b and a_a are not equal because that pulley is attached to a second pulley. When that pulley A and B are attached to moves up and down, it will add a term to both a_b and a_a, making them unequal.
 

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