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The handle has a pivot because you have to apply force on the handle at a specific point for the wrench to be accurate. . . So, when turning a nut or bolt the handle should be balanced on it's pivot point at all times. This style of wrench works off the idea that the metal in the thick part of the wrench will bend when force is applied at the handle end. If the handle were pivoted so its edge touched the main, thick section of the wrench you would be changing the effective length of that section and it could bend slightly less.
I had thought this to be true for some time -- I've been using beam type torque wrenches with pivoting handles (as well as clickers) for years. But this thread got me thinking more about the idea, so I set up an experiment.

In addition to the torque wrenches, I also have a digital torque adapter similar to this HF version.

I mounted a 3/4-inch bolt horizontally in a vise, fitted the digital adapter to it, and then fitted my 1/2-inch Craftsman beam-type torque wrench to the adapter. I then hung a 20 pound weight on the wrench handle right over the pivot. The beam, of course, flexed down, and both the beam indicator/pointer, and the digital adapter read slightly less than 30 ft-lbs. [The distance from the pivot to the center of the square socket fitting is 17-inches. 20 lbs X 1.42 ft = 28.3 lb-ft.]

I then moved the weight so that it was hanging on the beam right where the handle would be pressing if it were pivoted fully. The readings on both the wrench and adapter were now closer to 25 lbs-ft. [The distance was reduced by about 2-inches, so 20 lbs X 1.25 ft = 25.]

[I then moved the weight in another 2-inches, and again the reading on both dropped proportionally.]

This was interesting. The digital adapter reads to fractions of a lb-ft, whereas the beam scale has to be interpolated to read between markings, but both were quite acceptably close to each other, and consistent with the mathematical determination of what the torque reading should be.

The significance of the change in the readings between the two weight positions, however, is that if the weight, or hand pressure, is applied other than at the handle pivot, then the amount of weight or hand pressure will have to be different to achieve the same torque at the working end. (Or put differently, if the exact same weight or pressure is applied at different points along the beam, the torque at the far end will change proportionally.)

Nevertheless, the accuracy of the tool does not appear to be compromised by the degree to which the handle is pivoted, even if all the way, at least not within the few inches between the handle pivot and its end.
 
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