Dynamics -- Velocity of a Block in a system of Pulleys

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SUMMARY

The discussion focuses on calculating the velocity of block C in a pulley system after 6 seconds, given specific relative accelerations. The relative acceleration of block C with respect to collar B is 60 mm/s² upward, while block D relative to block A is 110 mm/s² downward. The user attempts to derive equations based on the lengths of cables connecting the blocks but struggles to relate these to the accelerations needed for the solution. The key equations referenced include the relative acceleration formula and the kinematic equation for velocity.

PREREQUISITES
  • Understanding of kinematics, specifically the equations of motion.
  • Familiarity with relative acceleration concepts in mechanics.
  • Knowledge of pulley systems and their dynamics.
  • Ability to differentiate equations to find acceleration and velocity.
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  • Study the principles of relative motion in pulley systems.
  • Learn how to derive equations for cable lengths in mechanical systems.
  • Explore kinematic equations in detail, focusing on their applications in dynamics.
  • Investigate methods for solving complex acceleration problems in multi-block systems.
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Students studying mechanics, particularly those focusing on dynamics and pulley systems, as well as educators looking for practical examples of relative motion and acceleration calculations.

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Hey guys, I have been a long time lurker here but this problem has eluded me for a few hours now and my frustration at not being able to figure out what it is I am missing prompted me to finally pull the trigger on making an account here. I hope the way I wrote everything out is clear to understand but if it is not I would be happy to do my best to elaborate further. Thank you!

1. Homework Statement

The system shown starts from rest and each component moves with a constant acceleration. The relative acceleration of block C with respect to collar B is 60mm/s^2 upward and the relative acceleration of block D with respect to block A is 110mm/s^2 downward. Determine the velocity of block C after 6s.
13252705286065231206.tp4?REQUEST=SHOWmedia&conId=13252704214052800&media=59-a.jpg


2. Homework Equations
Relative Acceleration: ac = ac/b + ab
v = v0 + a*t


The Attempt at a Solution


My first thought to solve this problem involved deriving an equation for the length of the cable that connects blocks A B and C that I could differentiate twice to get an acceleration equation and an equation for the length of the cable that connects blocks A B and D to do the same with. For the length of cable 1 I got 2SA + 2SB + SC = Constant taking my reference point to be the line that goes through the center of the 3 top pulleys. For the other cable I wasn't 100% certain what it should be though for what the question is asking I don't believe it is necessary to know. I can't seem to find any way to utilize this equation for the length of the cable to give me an acceleration for block B that I can use combined with the relative velocity equation to give me the acceleration for block C that I can use with the kinematic equation to find the velocity. My next thoughts for this problem was to find some way to use the given relative acceleration to simply obtain accelerations for the blocks directly however if this is possible then despite my best efforts I haven't found a way to do so. Any help at solving this would be GREATLY appreciated. I feel like it is easier than I am making it out to be which makes it all the more frustrating that I cannot solve it.[/B]
 
Physics news on Phys.org
Let ##\delta_A##, ##\delta_B##, and ##\delta_D## be the downward displacements of A, B, and D respectively. In order for the bottom string to remain constant in length, $$(\delta_D-\delta_A)+(\delta_D-\delta_B)=0$$
 

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