Displacement of Hanging Mass - Simple Pulley System

In summary, the conversation revolved around a physical problem where the length of a cord and the position of a hanging mass were related by a constraint. The speaker struggled to understand why a change in the length of one part of the cord resulted in a change in the position of the mass. Another person explained that this was due to the lack of a fixed coordinate system, and once a coordinate system was established, the problem became clearer.
  • #1
erobz
Gold Member
3,858
1,656
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:
 
Last edited:
Physics news on Phys.org
  • #2
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##.
 
  • Like
Likes topsquark
  • #3
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##.
 
  • Like
Likes topsquark and erobz
  • #4
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!
 
Last edited:
  • Like
Likes vanhees71
  • #5
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.
 
  • Like
Likes vanhees71

FAQ: Displacement of Hanging Mass - Simple Pulley System

1. What is the displacement of a hanging mass in a simple pulley system?

The displacement of a hanging mass in a simple pulley system refers to the distance it moves from its initial position to its final position. This can be calculated by measuring the length of the rope or string connecting the mass to the pulley.

2. How does the displacement of the hanging mass affect the tension in the rope?

The displacement of the hanging mass affects the tension in the rope by increasing it as the mass moves further away from the pulley. This is due to the force of gravity acting on the mass, causing it to pull down on the rope and create tension.

3. What factors can affect the displacement of a hanging mass in a simple pulley system?

The displacement of a hanging mass can be affected by several factors, including the mass of the object, the length of the rope, the angle of the pulley, and any external forces acting on the system. These factors can influence the amount of tension in the rope and the overall displacement of the mass.

4. How is the displacement of a hanging mass related to the mechanical advantage of a simple pulley system?

The displacement of a hanging mass is directly related to the mechanical advantage of a simple pulley system. The mechanical advantage is the ratio of the output force (tension in the rope) to the input force (force applied to the rope). As the displacement of the hanging mass increases, the mechanical advantage also increases.

5. Can the displacement of a hanging mass in a simple pulley system be calculated?

Yes, the displacement of a hanging mass in a simple pulley system can be calculated using basic trigonometry and the length of the rope. By measuring the angle of the pulley and the length of the rope, the displacement can be determined using the formula: displacement = length of rope x sin(angle of pulley).

Similar threads

Replies
30
Views
1K
Replies
3
Views
1K
Replies
8
Views
4K
Replies
35
Views
2K
Replies
7
Views
3K
Replies
22
Views
4K
Back
Top