Troque calculations for a direct coupledmotor and ball screw

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SUMMARY

The discussion centers on calculating torque for a direct coupling between a stepper motor and a ball screw. The formula T=F*D is utilized, where T represents torque, F is force, and D is diameter. The motor in question has a torque rating of 400 oz-in with a 0.5" diameter shaft coupled to a 0.75" diameter ball screw. It is established that the torque remains constant regardless of the shaft radius, thus the torque at the 0.25" radius does not change, and the force calculation should consider the same torque value.

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I know the formula T=F*D but having problems wrapping my head around this problem.
I want to direct couple a stepper motor with a .5"D shaft to a .75"D ballscrew.
The motor has 400 oz inch torque but since the motor shaft is .5"D or .25" radius How do I calculate the Torque at .25" radius? I would think the force at .25" would be greater than the torque at 2"D
In this case is the torque at .25 also the force F? Please provide the math in your answer.
 
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Same torque. Difference in shaft radius does not matter.
 

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