The Lever Paradox 4

[John2004]

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/getfil...-ab5b-46dd-a651-50dc5485b135&file=lever_1.jpg

http://files.engineering.com/getfil...-7039-42de-958e-b865fd591dbc&file=lever_2.jpg

http://files.engineering.com/getfil...-3be2-4636-a02d-500737540d93&file=lever_3.jpg

 

[zekeman]

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.

 

[desertfox]

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

 

[cntw1953]

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.

 

[John2004]

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.

 

[TheTick]

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.

 

[unclesyd]

Gentleman, what we have here is a Roberval Balance mechanism from 1669.

http://en.wikipedia.org/wiki/Roberval_Balance

 

[John2004]

Thanks for the link unclesyd.

At the link below, which I got from the wiki page you provided, it shows a gear-balance that behaves like the Roberval balance.

http://www.lhup.edu/~dsimanek/museum/roberval.htm

 

[ChrisDuncan]

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

 

[cntw1953]

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

 

[drawoh]

John2004,

Here is the free body diagram of the platform the mass sits on. Note how the vertical force equals the weight of the mass. The distance to the mass affects the tension force on the two levers.

JHG

 

[zekeman]

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.

 

[cntw1953]

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

 

[chicopee]

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.

 

[drawoh]

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 L_H.

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

 

[cntw1953]

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.

 

[cntw1953]

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.

 

[cntw1953]

Here is my missing response with diagram (responsed before posting stated "Missing the upward reaction, VR = 2W, at joint C (on rod)."

Please comments on the attached diagram. Thanks.