Understanding required torque for a motor lifting a weight

Question

This is a continuation of me trying to understand torque and stepper motors in my other question.  I'm trying to understand the torque a motor would be required to generate to lift a small weight, and the formulas involved.

The first part of my question is to verify if I am calculating this correctly:
Let's say I have a 450 g mass (roughly one pound) then the force of gravity pulling it down is:
$begin{align}
F &= ma 
&= 0.450 :mathrm{kg} * 9.8 :mathrm{m}/mathrm{s}^2 
&= 4.41 :mathrm{N} 
end{align}$
If I have a stepper motor with a spindle for my string that pulls up my motor with a radius of 5 cm.  I think my torque needed would be:
$begin{align}
T &= Fr 
&= F * 0.05 
&= 0.22 :mathrm{Nm} 
end{align}$
So now if I want to move that mass I need to find a stepper motor that can output more than 0.22 Nm of torque, right?

The follow-on to my question is that if I want to see how fast I can move it then I need to look at a Torque speed curve, right?
My confusion is this: do I have to ensure that I'm moving slow enough to get the torque I need, or does that curve say if you need this torque you won't be able to go above this speed because the motor won't let you?

Olin Lathrop · Accepted Answer

You have the right concept, but slipped a decimal point.  5 cm = 0.05 m.  The gravitational force on your 450 g mass is 4.4 N as you say, so the torque just to keep up with gravity is (4.4 N)(0.05 m) = 0.22 Nm.

However, that is the absolute minimum torque just to keep the system in steady state.  It leaves nothing for actually accellerating the mass and for overcoming the inevitable friction.

To get the real torque required, you have to specify how fast you want to be able to accellerate this mass upwards.  For example, let's say you need at least 3 m/s&sup2;.  Solving Newton's law of F = ma:

(0.450 kg)(3 m/s&sup2;) = 1.35 N

That, in addition to the 4.4 N just to balance gravity means you need 5.8 N upwards force.  At 0.05 m radius, that comes out to a torque of 0.28 Nm.  There will be some friction and you want some margin, so in this example a 0.5 Nm motor would do it.

Note also that torque isn't the only criterion for a motor.  Power is another important one.  For that you have to decide what's the fastest speed you want to be able to pull the mass upwards at.  Let's say 2 m/s for sake of example.  From above, we know the highest upwards force is 5.8 N.

(5.8 N)(2 m/s) = 11.6 Nm/s = 11.6 W

After accounting for some losses due to friction and leaving a little margin, the motor should be rated for about 15 W minimum.

Understanding required torque for a motor lifting a weight

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