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The Lever Paradox
4

The Lever Paradox

The Lever Paradox

(OP)
I ran across this thing called the "lever paradox" on page 19 of the book "1800 Mechanical Movements" by Gardner D. Hiscox. I think the same lever mechanism is shown in Gardner's other books on mechanical movements, some of which can be downloaded free via google's book search function.   

I have included three links below showing pictures of the lever mechanism in three different positions, i.e., unloaded, loaded on one side only, and balanced with loads at each end.

The paradox here is that the lever will always balance as long as equal weights are placed on each side, regardless of the distance of the weights from the center lever pivot point.

I had a couple of notions as to what may be going on here, but I'm not 100% sure. What is actually going on & why does this lever always balance even when equal weights are placed at different distances from the lever pivot point ?

Thanks
John

http://files.engineering.com/getfile.aspx?folder=2754b790-ab5b-46dd-a651-50dc5485b135&file=lever_1.jpg

http://files.engineering.com/getfile.aspx?folder=31d48790-7039-42de-958e-b865fd591dbc&file=lever_2.jpg

http://files.engineering.com/getfile.aspx?folder=e616c9cc-3be2-4636-a02d-500737540d93&file=lever_3.jpg

RE: The Lever Paradox

Since the "lever" is a rectangle, the line of action of the shear force is through the edge of the rectangle (the support holding the weight is a cantilever, so the shear reaction at the fixed side of the cantilever doesn't depend on where the load is placed).  Since the line of action is always at the same place on each side of the rectangle, it will always balance, as long as the weight is the same on either side of the rectangle.

RE: The Lever Paradox

...and the sum of the moments about the "pivot" need not equal zero, since the "pivot" can (and does) react a moment.
 

RE: The Lever Paradox

There are top and bottom levers that support platforms at their ends. The position of the weight does not affect the vertical force applied to the ends of the levers. It only changes the horizontal force on the levers, which are equal and opposite so the net torque around the pivot points is zero. Platform balances use similar linkages so that it doesn't matter where the weight is on the platform.

Ron's explanation is another way to look at it

RE: The Lever Paradox

(OP)
Hi Ron, thanks for your reply.

Your explanation seems similar to one of the notions I had, it seemed to be a situation where the weight forces were acting more linearly, as if the  moment or torque forces of the weights were not coming into play as far as balance is concerned, as they would on a simple teeter-totter.

After reading your explanation, it seems the center pivot support holds (supports) the torque or moment forces produced by the weights on each side, so you only have linear or shear forces that effect balance. The torque or moment loads produced by the weights have no effect as far as balance is concerned, so for the purpose of balance,  there is no real effective "moment arm" that the weights are acting through, and thus the location of the weights relative to the pivot has no effect on balance as long as the weights are equal.  

Does it sound like I understood your explanation correctly ?

Thanks again,
John  

RE: The Lever Paradox

(OP)
Thanks for your replies as well Ivymike and Compositepro, I did not see them before posting my reply to Ron. The opera browser has this annoying glitch where you have to reload the page before you can see the updates to the page. If you come back and forget to reload, you don't see the updates.  

RE: The Lever Paradox

I was just at the fish house were they were using a platform balance with a mechanism almost exactly as shown to weigh out and box five pounds of shrimp. The upper arm has a pointer attached that swings in arc to show on a scale the approach or overrun to the tare weight. It wasn't quite setup right as they had a 5 pound tare weight on the right pan and box and enough shrimp to equal 5 pounds on the left. At around $8.00 a pound every little ounce helps

RE: The Lever Paradox

Get out the old statics textbook and do some math.  It makes much more sense once you start balancing the forces.

RE: The Lever Paradox


EITHER:  the "pivot" on the middle bottom of the rectangular array is locked to the frame and the base so the external forces are the 2 weights and the bottom platform which to me will produce the the vertical force at the CM of the 2 weights.This explanation would allow any 2 weights so long as the CM falls inside the base.

OR: the "pivot" at the bottom cannot be a simple pivot, but one that not only supplies the force vector but also a moment such as a torsion spring or something more clever.

Have you seen it?
If so, then examine the "pivot" point to look for the lock or the moment mechanism.
Has to be one or the other. No other explanation is possible.
 

RE: The Lever Paradox

Correction:
Strike my post since I didn' see the post sticking up; I will submit an update after I take another look.

RE: The Lever Paradox

2
Use energy method.  By inspection (due to parallelogram) the platforms stay flat during motion.  Therefore, vertical displacement at any point on either platform is equal.  Same vertical displacement, same potential energy.

-handleman, CSWP (The new, easy test)

RE: The Lever Paradox

John2004,

   Draw a free body diagram.   

               JHG

RE: The Lever Paradox

Hi John2004

Very interesting device.
The out of balance moment from the two equal weights is distributed as axial forces down the horizontal members ie putting the top lever in tension and the bottom one in compression. Because the forces are horizontal and pass through the lever fulcrums they create no moment.
So that leaves the two vertical forces of the respective weights which are equal, acting through the vertical bars connected to the horizontal levers at each side of the scale and therefore they cancel out resulting in the balance we are seeing.
Also worth noting that if the top beam is moved from the horizontal equilibruim is still maintained.

desertfox

RE: The Lever Paradox

No need to go any further. Handleman's got it.
Very elegant.

RE: The Lever Paradox

The horizontal forces at the ends of the levers do not pass through the pivot points. I, too, was going to say that at first but they do actually create a moments around the pivots. Its just that the moments on opposite sides cancel each other.

RE: The Lever Paradox

Hi Compositepro

I think you will find the forces act in line with the pivots if you do the maths.

desertfox

RE: The Lever Paradox

The horizontal forces are just that. Only when the levers are horizontal do the forces pass through the pivot point. In all other positions the the horizontal forces try to return the top lever to horizontal and the bottom lever is being pushed away from horizontal by an equal amount so all positions are in balance.

What your question made me realize is that if the top pivot arm is slightly longer than the bottom one there will be a slight force to make the lever arms horizontal in the balanced position. This would tend to happen naturally if there is any slop at the end pivots.

This reminds me of a discovery that I made years ago. If you have two cylindrical rods of different diameters it is possible to balance the smaller one on the larger one (crossing at 90 degrees) but it is not possible to balance the larger on the smaller. It has to do with how the center of mass of the pivoting rod moves relative to the movement of the pivot point. It is a useful concept for making a stable balance with a low friction (rolling) pivot, where the pivot point is easily adjustable (to achieve balance) by rolling the lower rod. Try it with cylindrical objects on your desk. It's fun.

RE: The Lever Paradox

Thanks, desertfox, I love old books like that. But that simply presents a different way of looking at the same problem, like the the energy method. In my statements I was referring to the horizontal components of of the forces on the pivots at the ends of the levers. The vertical components are your balance weights. The vector sum of these two components must pass through the lever pivots or else the lever will rotate.

What I was responding to was your statement "Because the forces are horizontal and pass through the lever fulcrums they create no moment." Because you said horizontal I thought you were talking about the horizontal components of force. If you take "horizontal" out of your statement you are correct. The "horizontal" is not really correct. There is no moment on the levers only because the forces pass through the fulcrum, whether the levers are horizontal or not. This geek has to log-off now. I'll be back tomorrow.

RE: The Lever Paradox

(OP)
Thanks for all the feedback guys, all clear now.

I think handleman cut right to the chase. The weight platforms stay horizontal and are moving linearly not arcing, so torque is not a factor & there is no effective moment arm for balance purposes. If the platforms had an arcing motion it would be a different situation.

At first glance, the two horizontal beams look so much like teeter-totters it kind of tricks you into thinking the weights should behave as they would on a teeter-totter.

John

RE: The Lever Paradox

My take is ( forgive me if it was already said) that the forces that each weight exerts on the frame consist of a shear force on either side that are equal to the weights.
But each of these weights induces a moment at the point where the support bars meet  the vertical members. These moments are removed by the upper and lower members in the form of tension and compression and cannot contribute to the balancing.
So, the system now behaves like a lever where the weights, no matter where they are placed, act as if they were placed at  points at the juncture of the support bars and the vertical members.
 

RE: The Lever Paradox

Hi Compositepro

Amazing what you can find on google!

When I said horizontal I meant the forces were acting along the levers of the device which in that case happen to be in the horizontal.

desertfox

RE: The Lever Paradox

The cans produce equal, downward forces on the side verticals, these forces are balanced with respect to pivot point (sum M = 0), thus no effect on the system.

However, the cans introduce imbalanced moments on the verticals, in turn the moments resolve into horizontal forces and result in equal, but opposite forces on the top and lower horizontal members.

The net horizontal forces are acting on the top center and bottom center pivoit points and produce a couple on the middle support rod, the couple is then resisted by internal moment at the interface of the center rod and its base. If a pin is introduced at this interface, the system will be deformed in the manner similar to pic #2.  

 

RE: The Lever Paradox

(OP)
I think another way to state what is going on is that the ends of the horizontal loading platforms are rigidly fixed to the  vertical beams that are located at each end of the two pivotal horizontal beams.

In order for the transmission of torque to take place from one loading platform to the other, the vertical beams connected to the loading platforms would have to be able to rotate in response to the torque forces produced by the load weights. Since the vertical beams engaged with the loading platforms are prevented from rotating due to their pinned connection to the ends of the upper an lower horizontal beams, the two pivotal horizontal beams *block* or prevent the transmission of torque from one loading platform to the other.

The ends of the upper and lower horizontal members are arcing, which is what can fool you at first glance.

The connection points between the loading platforms and the vertical members, are moving vertically and horizontally at the same time (like a ramp) but in a straight line, not an arc. If the connection point between the loading platforms and the vertical beams were simply located to the ends of either the upper or lower horizontal members, (not connected to the vertical members) the device would no longer work and you would simply have a teeter-totter.

Are there any other likely applications for a system like this besides a balance scale ? Most likely not many at all, but you never know & it seemed like a natural question to ponder.

RE: The Lever Paradox

It makes sense to me, partly because I've been immersed in analyzing 4-bar parallel lift mecanisms for the past few weeks, using all the tools, includncil and paper (but  o slide rule).

Intuition is the innate sum of one's experience and intellect.  When a phenomenon defies one's intuition, it reveals a hole in one's knowledge and presents an opportunity to refine one's knowledge and sharpen that intuition.

RE: The Lever Paradox

in shade tree engineering speak, the opposing forces are located to pivot points that are equal distance from the center pivot point.  

RE: The Lever Paradox

The two situations are slightly different, as the original case has unbalanced moment (locally & globally), the latter does not.  

RE: The Lever Paradox

Drawoh,

Your assumption of horizontal position of the upper and lower members is unnecessarily restrictive.
The free body diagram would be essentially the same except that at any angle you would remove the moment with a couple created by the tension compression members.

RE: The Lever Paradox

Missing the upward reaction, VR = 2W, at joint C (on rod).

RE: The Lever Paradox

Interestingly, there is still a moment in the end of the center post from unequal horizontal reactions, therefore such scale must be sturdy enough to minimize tip deflection for the scale to work.

RE: The Lever Paradox

zekeman,

   I was analyzing the model in the original post which shows horizontal levers.  I only analyzed the mount on the one side to show that the force on the levers equals the weight of the load.  The rest should be obvious.  Given that the levers can take a bending load, they can be at any angle.  Also, you can analyze unclesyd's Wikipedia model by entering a negative value for LH.

cntw1953,

   Point_C on my diagram is where the load contacts the structure.  There is no reaction force there.  Are you refering to the centre rod?  As Chicopee notes, the centre rod sees a side load, and the load on the base is eccentric.   

               JHG

RE: The Lever Paradox

chicopee:

Yes, the elonged base services to stabilize the whole system. If stands on the center rod alone (similar to inser a pin between the rod and its triangle base pedestal), unless perfectly balanced, the whole thing wouldn't work.
Structural stability and material strength are keys to this genius trick.

RE: The Lever Paradox

drawoh:

I was pointing to my diagram. Point c is at the junction of the middle rod and the support (trianglar shapped) pedestal.

Oh, I forgot to "submit" my response, sorry, I will post tomorrow.

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