B Displacement of Hanging Mass - Simple Pulley System

AI Thread Summary
The discussion centers on understanding the displacement of a hanging mass in a simple pulley system when the left cord is pulled. The user initially struggles to connect the change in length of the cord, denoted as ##l_x##, to the expected movement of the mass, which should be ##\frac{\delta}{2}##. A clarification is provided that differentiates between the distance pulled up, ##\delta##, and the resultant change in the mass's height. By introducing a fixed coordinate system with a reference point from the ceiling, the relationship between the lengths becomes clearer, allowing for the correct interpretation of how the mass moves. Ultimately, defining a fixed coordinate system resolves the confusion regarding the displacement calculations.
erobz
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I'm having some kind of mental block.

2 to 1 - Pulley.jpg

If I extend ##l_x## by ##\delta## ,I expect the hanging mass to move ##\frac{ \delta}{2}##.

I can't figure out how this is the case from the constraint:

$$ l_x+l_1=C $$

##C## is an arbitrary length

I keep getting that ##l_1## changes by ##\delta##, but that must mean the height of the mass changes by ##\delta##...

:oldgrumpy:
 
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I suspect you need to differentiate between the distance you pull the left hand cord up, ##\delta##, and the increase in distance from that point to the pulley, ##\frac \delta 2##.
 
Call the distance of the left end of the cord from the ceiling ##y##. Then ##l_x=l_1-y## and ##l_x=C-l_1## from the constraint given by the fixed length of the entire cord. Then you can eliminate the somewhat inconvenient quantity ##l_1## in favor of ##y## which has a well defined meaning of a coordinate, i.e., it's a length measured from a fixed reference point (the ceiling):
$$C-l_1=l_1-y \; \Rightarrow \; L_1=\frac{1}{2}(C+y).$$
So if you change ##y## by ##\delta## (which is the same as changing ##l_x## by ##-\delta##) ##L_1## (the distance of the pulley from the ceiling) changes by ##\delta/2##.
 
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vanhees71 said:
Then you can eliminate the somewhat inconvenient quantity ##l_1## in favor of ##y## which has a well defined meaning of a coordinate, i.e., it's a length measured from a fixed reference point (the ceiling):
The addition of the coordinate ##y## and the weirdness seems to vanish right in front of my eyes. Amazing!
 
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My key failure was that I had not defined a fixed coordinate system. I basically was trying to measure distances relative the datum at the center of the pulley...a transformation not so obvious to me.
 
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