Say I have a motor with a certain amount of torque, $T.$ It is turning a ball screw mechanism like this .
Say, I mount something on the nut, I want to calculate the mass of an object I can lift upwards, i.e. the force that the actuator can generate.
I think this equation is what I want: $$T=\frac{Fl}{2\pi\nu}$$ source
But I don't understand:
$\bullet$ What is the "lead" $l\,?$ Is it the millimeter pitch distance between grooves?
If so, using that tiny motor above (I have a similar one) it seems that $F = 2\cdot \pi \cdot 0.26~\mathrm{Nm / 8mm}$ which is $204$ Newtons, so equating to $F = mg$ it seems I could lift $204~\mathrm N/9.8~\mathrm{m/s^2} = 20.8~\mathrm{kg}\,?$ This seems really heavy for that small motor, so I figure something is wrong with my interpretations and/or calculations.
$\bullet$ How is the wikipedia formula derived?
Best Answer
lead is the pitch of the screw,it probably needs to be in meters if the rest of the equation is SI units.
Yes you can lift a very big load with a screw and a small force - that's why your car has a screw thread jack to change wheels.
Really a screw is just a slope (inclined plane in physics speak) so the equation should be pretty easy to derive.
edit: An easy way to check is to consider the energy. In fact whenever you aren't sure in a physics calculation ALWAYS consider the energy, it's often the simplest way.
One turn of the thread moves the object 8mm vertically.
If you have a Torque of 0.26Nm then one turn of the motor is like providing a force of 0.26N at a radius of 1m, and energy is force * distance.
So a force of 0.26N around a circle of 1m radius is 2*pi*1.0m*0.26N = 1.6J
The energy to lift 20kg vertically 8mm = 20kg * g * 0.008m = 1.6J