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

 

[Ron]

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.

 

[ivymike]

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

 

[Compositepro]

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

 

[John2004]

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

 

[John2004]

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.

 

[unclesyd]

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

 

[TheTick]

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

 

[zekeman]

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.

 

[zekeman]

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

 

[handleman]

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)

 

[drawoh]

John2004,

Draw a free body diagram.

JHG

 

[desertfox]

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

 

[zekeman]

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

 

[Compositepro]

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.

 

[desertfox]

Hi Compositepro

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

desertfox

 

[Compositepro]

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.

 

[desertfox]

Hi Compositepro

I found this book which you can view in pdf

http://www.lulu.com/items/volume_43/657000/657559/5/print/657559.pdf

go to the page in the book 125 however it says page 133 in the toolbar of the pdf reader.

desertfox

 

[Compositepro]

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.

 

[John2004]

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