Leverage and Forces on a Pivot System

In summary: angle will be aprrox. 45 degrees, and once its been pulled down it will end up at approx. 90 degrees.
  • #1
Saints-94
63
1
The diagram drawn below shows a lever system fixed to a point on the left with a pivot point (circle) on the line which attaches to vertical line. I have dimensions for Y and Z and the forces pulling the lever down on the right, and the force acting upwards from below. I am trying to work out the dimension for X to determine how long the lever needs to be to compress the upwards force, but I am unsure due to the varying angle as the lever is pulled down. Any help is greatly appreciated.
Forces Sketch.png

Before lever is pulled down.
upload_2017-2-21_16-13-3.png

After lever is pulled down.

Thanks.
 
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  • #2
Hi Saints,

I sense some contradiction between the Y (FIxed Dim) and the Z piece seemingly staying vertical. Y is clearly shorter in the lower picture. Is the point where XY touches Z fixed on XY (and X also a 'Fixed Dim') or not ?
 
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  • #3
Y is a fixed dim (may not be to scale on the image). The point on the left of line Y is fixed, and the point where XY touches Z is fixed. However, X is a variable dim as I am trying to work out the length of leverage required to compress the force acting up.

I hope this clears it up a bit better.
Thanks.
 
  • #4
Point on the left of Y is fixed. Y is a fixed dim ##\Rightarrow## point where XY touches describes a circle when the 'varying angle' varies. Correct ?
 
  • #5
Yes. The angle varies when the lever is pulled down, as Z always stays vertical. The circle represents a pivot point.
 
  • #6
You miss my point. The x coordinate of the pivot point moves in a circle, so Z can not stay vertical

upload_2017-2-22_15-24-44.png
 
  • #7
I understand the point moves in a circle, however, where Z touches XY it is a pivot point. The pivot allows Z to stay vertical as it moves around the red circle marked on.
 
  • #8
Let me try again. In the left picture the arm is at 45 degrees, in the right it is horizontal. Do you see the distance between the left fixed point and the vertical Z has changed ? So either the leeft fixed point has moved to the left or Z has moved to the right and down.

upload_2017-2-22_15-40-10.png
 
  • #9
I can see that the length of the arrows have changed as the lever moves down. However, the Y dimension has not changed- the length of the yellow line has stayed the same. From my understanding the pivot allows Z to stay at the same length and vertical, and also allows Y to stay at a fixed length?

upload_2017-2-22_14-49-0.png
 
  • #10
The point on the left is always fixed.
 
  • #11
Has anyone got any other suggestions as to how this would be solved? Thanks.
 
  • #12
The problem as stated can only be solved for the case where the lever almost doesn't move . Applying down pressure on the lever end can then apply down force on the column without the point of contact moving significantly .
 
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  • #13
I'm struggling to work out what the equation would be. I want to find out the minimum length X would have to be to move the lever down, with a specified force pulling the lever down.
 
  • #14
For what you describe - and assuming the small movement idea is ok - and if the 'variable angle' shown in your diagram is set to be somewhere near 90 degrees - the essential mechanism is just that of a simple lever .

Solution can be obtained by taking moments about the fixed pivot point at left end of lever .

Force down on column multiplied by distance Y = Force on end of lever multiplied by distance ( X + Y )

Example :

X = 2 metre , Y = 1 metre and force on end of lever = 100 N

Force on column * 1 = 100 * ( 2 +1 ) . Therefore force on column = 300 N
 
  • #15
Thanks for your help.

However, the varying angle can be between 45 and 90 degrees. Does that have an affect on the force on the Z column? And also does the length of Z change the force created?
 
  • #16
I am struggling now . Perhaps we have to define the mechanism more clearly .

When you say variable angle do you mean :

(a) that the lever has to swing through that angle when the force is applied ?
or
(b) that the lever could be preset to different angles but doesn't then swing significantly when force is applied ?
 
  • #17
The column Z has to stay horizontal when the lever is pulled down. Therefore when the lever is up the angle will be aprrox. 45 degrees, and once its been pulled down it will end up at approx. 90 degrees.

The mechanism is waste compactor, similar to the one below.
41wF6gK1mQL._SX300_.jpg
 
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  • #18
So the vertical link is not actually fixed in position after all . Mystery solved .

For the lever being horizontal the calculation in #14 is still ok .

You could do a calculation for the lever at other angles but since the reaction force from squashing the waste is least when the lever is 45 deg up and greatest when it is somewhere near horizontal I think you could just use that one answer for horizontal for design purposes .

Any calculations are going to be approximate anyway due to the varying properties of waste material .
 
  • #19
That's great. So as the force will be greatest at 90 degrees I can assume that it would be worst case?

Yes, they will be approximate. I have done some testing to get an approximate force required to compact household waste.
 
  • #20
Yes - I think you'd be quite safe to use lever being horizontal as worst case .
 
  • #21
Okay, thanks. If I were to do a calculation for the lever when it is at 45 degrees, how would I incorporate the angle into the equation?
 
  • #22
Saints-94 said:
That's great. So as the force will be greatest at 90 degrees I can assume that it would be worst case?
You should be careful here. The ratio of the two (parallel) vertical forces is the same for all angles of the arm - if the system really can be treated as implied in the diagrams at the top of the thread. So the Mechanical Advantage (or at least the Velocity Ratio) will be independent of angle.
 
  • #23
sophiecentaur said:
You should be careful here. The ratio of the two (parallel) vertical forces is the same for all angles of the arm - if the system really can be treated as implied in the diagrams at the top of the thread. So the Mechanical Advantage (or at least the Velocity Ratio) will be independent of angle.

The two vertical forces will always be the same.

So does the angle need to be taken into consideration in the equation or not?
Thanks.
 
  • #24
In practice there are two problems :

(a) Human operators tend to push or pull on long levers in a direction at right angles to the lever rather than just always straight downwards ,

(b) The drop link may swing from being vertical to an inclined position .

(a) and/or (b) can alter the calculated mechanical advantage figure considerably .

If working with all known quantities an exact calculation might be useful but in this case probably not worth the trouble .
 
  • #25
(a)The lever will be pulled down from overhead, I can't see how that will affect the calculation? (b)Also I am making an assumption that the 'drop link' will stay vertical for the purpose of the equation.

Going back to the formula discussed previously, is there a way to incorporate the change in the 'varying angle', or is it not necessary?
 
  • #26
Saints-94 said:
(a)The lever will be pulled down from overhead, I can't see how that will affect the calculation? (b)Also I am making an assumption that the 'drop link' will stay vertical for the purpose of the equation.
Going back to the formula discussed previously, is there a way to incorporate the change in the 'varying angle', or is it not necessary?
Have you done any reading around levers and moments? The principle of moments uses the phrase 'perpendicular distance' in the definition of the moment of a force around a fulcrum. This link gives you a way into the topic if you haven't already read about it. The simple, lower school, treatment of moments is only restricted to horizontal levers / see-saws etc. but the angle is important if you want to understand it more fully.
 
  • #27
I have done some work on moments previously.

I understand that when the line is at 90 degrees: Force on the vertical line multiplied by Y = Force on end of lever multiplied by ( X + Y )

However, when the lever is at 45 degrees I am struggling to create a formula that incorporates the angle.
 
  • #28
Why do you need to "create" one? It's in any A level textbook and many other places. Try the link I gave you
You don't have to invent Physics all your own.
 
  • #29
Sorry, maybe 'create' was the wrong terminology.

I am trying to write out the formula that incorporates the angle.

For the two instances - line at 45 and 90 degrees- I have written the formulas below.

When the line is at 90 degrees: Force on the vertical line multiplied by Y = Force on end of lever multiplied by ( X + Y )

When the line is at 45 degrees: Force on the vertical line (sin45) multiplied by Y = Force on end of lever multiplied by ( X + Y )

Are these correct?
 
  • #30
Nidum said:
Human operators tend to push or pull on long levers in a direction at right angles to the lever rather than just always straight downwards
For good reasons: the handlebar is also constrained to move along an arc with radius X+Y. I don't think a component along the X Y arm is doing anything useful -- anyone ?

So I drew ##F_1## as Nidum suggests

Moment balance: $$F_1 \times (X+Y) = F_2 \times X$$

Decomposition of ##F_2##: $$F_{2,\rm v} = F_2 \sin 45^\circ$$

So your formula should be $$F_{2,\rm v} = F_1 {X+Y\over X} \sin 45^\circ$$

Again, if Z starts moving, the top is constrained to move along the dashed arc, but that shouldn't be a big issue: let go, press again is the recipe.

I do see trouble for big strong guys: they can easily compress the stuff so tightly that it doesn't come out any more when the bin is collected...

upload_2017-2-24_22-51-5.png
 
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  • #31
BvU said:
I do see trouble for big strong guys: they can easily compress the stuff so tightly that it doesn't come out any more when the bin is collected...
This is a common problem and there's no sure fire solution. A slight taper on the container can help. The same solution that works to let moulds (molds) release their contents.
The arrangement in the diagram immediately above can produce a lateral force that can bend or bind the mechanism. Getting this just right in a design makes the difference between working and breaking. A cam mechanism can sometimes ensure that the vertical force stays vertical and better controlled. in these situations. But that puts the manufacturing cost up.
 
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  • #32
Firstly, thanks for your help.
upload_2017-2-27_10-16-6.png


The only unknown I have is the distance you have labelled Y. I have added 'F3' to the diagram to show the force acting against mechanism (representing the resistance of the waste).

How do I find the value of F2? Do I just substitute F2v for F3?
 
  • #33
Saints-94 said:
Firstly, thanks for your help.
View attachment 113820

The only unknown I have is the distance you have labelled Y. I have added 'F3' to the diagram to show the force acting against mechanism (representing the resistance of the waste).

How do I find the value of F2? Do I just substitute F2v for F3?
I notice that you now have the applied force acting at right angles to the lever. Would this be likely - as it would tend to tip the bin over. I would think that the natural direction to push would be vertically downward. The bottom line is that the force on the rubbish would be twice the applied force (more accurately [X+Y]/X plus or minus any variation in applied angel and friction). The angle of the bar to the horizontal will have no effect because the ratio of the perpendicular distances is the same.
Apart from any weight force from the press arms, the force on the rubbish is the same as the magnified force from the operator. F2,v = F3)
 
  • #34
BvU said:
Decomposition of F2F2F_2:
F2,v=F2sin45∘F2,v=F2sin⁡45∘​
F_{2,\rm v} = F_2 \sin 45^\circ

So your formula should be
F2,v=F1X+YXsin45∘F2,v=F1X+YXsin⁡45∘​
F_{2,\rm v} = F_1 {X+Y\over X} \sin 45^\circ
So the equation above is not relevant as the angle does not need to be considered as the force is worked out using a ratio of X and Y?
 
  • #35
The compactor is going to be prone to tipping over anyway . Excessive lever length / lever ratio will make this problem worse .

Useful pictures here .

The picture with the lady operator and the handle at highest level is particularly interesting .

Lever ratio appears to be about 2.5 : 1
 
Last edited:
<h2>What is a pivot system?</h2><p>A pivot system is a type of mechanical device that allows for rotation around a fixed point, known as the pivot point. It consists of a pivot point, a lever arm, and a load, and is used to apply force and create movement in a controlled manner.</p><h2>What is the principle of leverage?</h2><p>The principle of leverage states that a small force applied at a greater distance from the pivot point can produce a larger force at a shorter distance from the pivot point. This is because the longer lever arm creates a greater torque, or rotational force, at the pivot point.</p><h2>How does the force on a pivot system change with distance?</h2><p>The force on a pivot system is directly proportional to the distance from the pivot point. This means that the farther away the force is applied from the pivot point, the greater the force will be.</p><h2>What is the relationship between the length of the lever arm and the force applied?</h2><p>The length of the lever arm and the force applied are inversely proportional. This means that as the length of the lever arm increases, the force applied decreases, and vice versa.</p><h2>What factors affect the stability of a pivot system?</h2><p>The stability of a pivot system is affected by the position of the pivot point, the length of the lever arm, and the distribution of weight on the lever arm. A longer lever arm and a lower pivot point will result in a more stable system, while a shorter lever arm and a higher pivot point will make the system less stable.</p>

What is a pivot system?

A pivot system is a type of mechanical device that allows for rotation around a fixed point, known as the pivot point. It consists of a pivot point, a lever arm, and a load, and is used to apply force and create movement in a controlled manner.

What is the principle of leverage?

The principle of leverage states that a small force applied at a greater distance from the pivot point can produce a larger force at a shorter distance from the pivot point. This is because the longer lever arm creates a greater torque, or rotational force, at the pivot point.

How does the force on a pivot system change with distance?

The force on a pivot system is directly proportional to the distance from the pivot point. This means that the farther away the force is applied from the pivot point, the greater the force will be.

What is the relationship between the length of the lever arm and the force applied?

The length of the lever arm and the force applied are inversely proportional. This means that as the length of the lever arm increases, the force applied decreases, and vice versa.

What factors affect the stability of a pivot system?

The stability of a pivot system is affected by the position of the pivot point, the length of the lever arm, and the distribution of weight on the lever arm. A longer lever arm and a lower pivot point will result in a more stable system, while a shorter lever arm and a higher pivot point will make the system less stable.

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