For screwdrivers, a significant limit is the amount of angle which can be tolerated before the screwdriver rocks out of the slot. When exerting a large force, humans cannot keep the centre of rotation completely steady. A longer driver means the angle for a given wobble is less, so the driver stays in its slot at higher torque.

Tappingstainless steel

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I personally have found that I can sometimes get better torque with a high quality miniature screwdriver ( with a narrow handle that has a rotating end ) held with one hand turning and the other hand pressing rather than a larger driver in one hand - preventing the head 'camming out' is more important than the leverage. I also haven't found the same effect with hex or socket drives, only flat and cross head screws which can cam out.

Threading stainless steelpipe

If you have a firm grip on the screwdriver, then your forearm and the screwdriver act as a unit, and the torque then depends on how far your elbow is from the axis of rotation. If the screw in question is, say, inside a computer case, a longer screwdriver may let you get your forearm at a right angle to the shaft and thus as far away as possible; a short screwdriver, conversely, may restrict you so that not only is your arm at a shallower angle, but you might need to apply force with your wrist instead of your whole arm, combining the reduced force of a weaker set of muscles with the reduced lever arm, and producing much less torque.

Also this might be due to the ability of the longer shaft to twist slightly. This elasticity seems to soak up the irregularities in the force that you can apply with hand, and enables you to apply a steady high torque without so accidentally twisting the screwdriver head out of the slot.

Everyone who deals with screws and screwdrivers knows that long screwdrivers are stronger than short ones. However, I can't find any relationship between length of a screwdriver and mechanical advantage. For a wrench, it's obvious: Long arms produce more torque. But in case of a screwdriver, it is not so simple.

In terms of a long shaft driven by a motor, then it won't have much effect if the shaft is driving something with it is firmly attached to. Adding a flexible coupling between the the motor and the shaft though will have a somewhat similar effect in that it will allow the motor to rotate in its bearings' axis and the shaft in its axis even if there is a small misalignment.

Torque is in units of force times distance, but the distance of course is not distance along the axis of rotation (the screwdriver shaft) but instead distance away from the axis. This is one reason screwdrivers have handles; the bigger the handle, the more torque to the shaft is provided by a given force. (though actually the more important issue might be the friction force that your hand can produce, which is proportional to the surface contact area, which is itself of course determined by the length and radius of the handle.)

For a long screwdriver the inclination angle may be smaller, so you do not loose your force due to applying a part of it in a wrong (perpendicular to the rotation axis) direction.

Threading stainless steel rodby hand

Often you would find that longer screwdrivers are used commonly to loosen/tighten larger screws and they come with thicker cross-section. Which is one of the reason why they are stronger. They also provide higher bending resistance due to thicker cross section so you will find it hard to snap them. But that being said longer screwdrivers with thinner cross sections do exist to tighten/loosen up screws at a larger depth. Also a larger screwdriver comes with a thicker handle, this allows more torque to be applied. This is because you apply the force on the periphery of the handle and that slight distance from periphery to centre of shaft does make a difference.

I'm reasonably sure the torsional strength of a screwdriver shaft does not depend on length. At least not appreciably so within the range of screwdriver shaft lengths I have encountered (say 1cm to 20 cm).

Maximum torsional stress $ T_{max} = ( \frac \pi {16}) \tau_{max} D^3 $ where $\tau_{max}$ is maximum shear stress of material and D is diameter of a solid rod with circular cross-section. Ref

For this question, my point is not originally about screwdrivers. I'm actually going to design a simple machine that has a relatively long arm connected to an electric motor. It's intuitive that the long arm weakens the motor (Or maybe I'm wrong?). The answer of my question about screwdrivers will help me undrestand the mechanism behind this case, and solve the problem.

It is not necessarily true that long screwdrivers have more strength in all cases. It all depends on the cross section of the shaft, length of the shaft and material property. In fact shorter screwdrivers are stronger for a similar cross section and flexural stiffness. You can refer torsional stiffness, polar moment of inertia, bending stresses and shear strength of a material to get a better understanding how it influences the strength of an object.

Your hand may be able to exert more torque on a larger handle but ungripped shaft length has no significant effect on the torque you can deliver to the bit end. In general I'd expect shorter drive shafts to be stronger, not weaker.