Sorry, badly phrased. Was just using examples. Fully aware that what we are talking about are proportional changes, units only matter in that you need to be consistent when you try to solve for a particular set of values.
I also get your “extreme” examples. However if I apply force right where the torque wrench meets the extension, I will never arrive at the “click” and not get the measurement of the force I am applying.
But at the point where the Torque wrench clicks, the force at that point (where torque wrench meets the extension) must be the same regardless of where I apply the force, yes? (if not, this is one place where my logic needs improving).
And if it is, the force at the bolt-end of the extension must be proportional to the measured force.
For some of the other replies:
I am fully aware that where I apply a given force influences how much torque I get for that force value.
I know that if I am trying to calculate the torque at the nut for a given applied force (without a torque wrench) then the distance from the force to the nut matters.
With the extension, I know that that torque at the nut will be greater than the indicated setting on the torque wrench. But I can’t see why the force at the nut at the time/point where the torque wrench clicks should depend on the lever length before the torque wrench head.
It might be easier to think in terms of an expensive digital torque wrench, that always shows you the torque value, instead of a cheap one, that just either clicks or doesn’t. Apply force right at the torque head, and the digital wrench will display a value just fine, and that value is 0.
Let’s try 2 options, torque wrenches with lengths 1 and 2 units, which will click at a torque of 200 units. Let’s also hope that I don’t muck it all up.
Without crow’s foot:
Wrench 1, you apply 200 units of force before it clicks (200 force x 1 length = 200 torque)
Wrench 2, you apply 100 units of force before it clicks (100 force x 2 length = 200 torque)
Now we add a crow’s foot with a length of 1 unit, this increases the length of the wrench by one unit, but doesn’t change the torque at the head when it clicks:
Wrench 1, as before, you apply 200 units of force before it clicks (200 force x 1 length = 200 torque), the torque at the end of the crow’s foot is 400 (200 force x 2 length = 400 torque)
Wrench 2, as before, you apply 100 units of force before it clicks (100 force x 2 length = 200 torque), the torque at the end of the crow’s foot is 300 (100 force x 3 length = 300 torque)
You have a new lever arm “E” between the torque wrench measuring axis and the bolt. Torque is Force X Distance, right? You have distance(E). You need a force. Knowing the torque at the measuring axis is useless, by itself. That extra “E” is just hanging out there, needing something to multiply with that has the right units and acting perpendicular to “E”.
What’s the force acting at the wrench measuring axis? It’s the same as the force you applied. In fact, if you chopped off the entire handle of the torque wrench, and applied a torque(twisting motion) right at the measuring axis of the same amount as the wrench was reading before, and applied the same force as you did previously, only this time pushing right at the location of the measuring axis (the wrench isn’t going to read this part at all), the bolt/joint will see this as completely equivalent, and react the exact same way.
You can ignore the lever arm “L”, but to do so you have to know the force. But you don’t know the force, so you have to make an assumption about “L”, and calculate it using the torque reading.
None of this matters if E=0. (i.e, a normal torque wrench, even one with a cheater). applied force can be whatever (assuming the torque reading at the wrench is still what you want) it’s still getting multiplied by zero length. Sure, the reaction force at the joint may be different, but the bolt doesn’t care about that, only the total torque.
OK, light bulb is flickering on.
What I was missing is that the whole length (L+E) is acting a rigid system, while the original Torque wrench measured it’s own result. Or something like that.