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I've been seeing on the web how the total work (energy) inputted into a system, or a simple machine, is the same as the output work (energy), assuming no friction or any other stuff like that.

If I'm correct, torque is defined as force applied to the circle multiplied by the radius, multiplied by sin of the angle between force and effort arm. This means that it's measured in newton meters, or joules, and therefore torque is a measurement of work done. In simpler terms, it should be the tendency of the force applied to cause a load to rotate.

So my first question here is: in a wheel and axle system, the torque should be the same, right? Since the work cannot change in a frictionless system, when you apply work to a larger wheel connected to a smaller axle, the torque should be the same on both the axle and the wheel. To clarify, this means that the force you exert on the wheel is less than the force exerted by the axle, and the axle exerts a greater force but its linear velocity is less. So the axle would apply a greater force to something attached to it, but cover less distance than the wheel. And vice versa, so if you apply force to an axle, the torque stays the same, but the force decreases as the distance/velocity increases. On wikipedia, it says

"...the larger the ratio [of the effort radius to the load radius] the greater the force (torque) multiplication achieved.

So my clarification here is - is the torque changed, or the force changed? I understand that the smaller axle would apply a greater turning force over less distance, but that doesn't mean that the torque is changed, right?

My second question - in a gear system, is the torque changing? Say you have a big gear and a small gear, connected without friction or other stuff. For sake of calculation purposes, assume that the small gear is half the radius of the big gear. You apply x amount of force to the small gear, to turn it. Therefore it turns y degrees, covering z circumference distance. The big gear turns y/2 degrees, covering z circumference distance as well. I understand this.

The force x you apply to the small gear should all be transmitted to the big gear, right? Therefore even though the degrees turned by the gears are different, the force and the circumference covered remain the same.

Now, you apply the same x force to y/2 degrees of turning on the big gear, and so, therefore, the turning force exerted by the big gear should increase, as you exert x force to turn it half the degrees - so if an axle would be connected to the big gear, the axle would exert twice the force than an axle connected to the small gear (x force on y degrees for the small gear, x force on y/2 degrees for the big gear, force is concentrated into half the turning).

So my question - is the torque changing in a gear system? The circumference covered is the same for both gears, so the distance moved is the same. However, it seems that the force is magnified by 2 for the big gear compared to the small gear! I know that the big gear moves half the speed in degrees/second, but the linear velocity is still the same. So where is that tradeoff happening? The work applied to the system should remain the same, and torque seems to be a measure of work. Wikipedia says that there is a torque ratio, instead of a force ratio. Torque is radius times force, and in the big gear both seem to be amplified - force is concentrated into half the degrees, and radius is doubled. Where is the tradeoff? Why is torque changing?

An axle connected to the big wheel will rotate at half the speed than an axle connected to the small wheel, therefore covering less distance. Is that the tradeoff? Force concentrated into a small angular change by the gear system? For the small gear, you exert x force for y angular change with some radius, and the tendency of it to rotate an object, in this case an imaginary axle attached, would be x force times the radius. In the large gear, you have twice the force for the same angular change, and twice the radius. The torque should therefore be 4 times as much, and angular change only half as much? Is it the angular change that is distance in the work = force x distance equation, or is it the circumference covered that is the distance? The torque seems to be magnified exponentially with changing radius of the driven gear - force is concentrated into less degrees and the radius of the force point to center increases. The force is obviously magnified with the big gear, but is the torque changing? Since the torque = work? Please help, any answer is much appreciated!

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This means that it's measured in newton meters, or joules, and therefore torque is a measurement of work done.

Just because two things have the same units does not make them the same thing.

Torque is not a measure of work done.

The force is obviously magnified with the big gear, but is the torque changing?

Yes, In a car, when you change from second gear to first gear, you exert greater torque on the driving wheels. It is equivalent to changing the length of a lever.

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