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I have a 337 watt motor, and want to lift ten pounds. What kind of gear ratio will I need for a gearbox I attach to this in order to lift the ten pounds at a constant speed?

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    $\begingroup$ We need to know quite a bit more than this to answer the question? $\endgroup$ – joojaa Jan 15 at 5:17
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Without RPM answering this with a simple number is impossible. 337 watt is 337 Joules/second. Potential energy of 1kg lifted 1 meter grows by 9.8 Joules; for 10 pounds (4.5 kg) it's 44.1J/m, so the payload without resistance needs to move upwards at 7.64m/s to maintain constant speed.

Now what gear ratio and rotary to linear conversion (a pulley?) will convert the rotary motion of your motor to linear motion of the payload at that speed will depend eon RPM of the motor which is completely independent from its power.

Given motor RPM $\omega$ (in RPM) and assuming radius of gear coupled to its shaft ('input') $r_1 = 1$, you have $2 \pi \omega_1$ of linear speed of the gear's teeth. For the desired output's linear speed to be 7.64 m/s, or 458.4 m/minute, $v = {458.4 \over {2 \pi \omega}}$ or about 73/$\omega$ - where $\omega$ is your motor's RPM.

This is obviously assuming no friction, which in case of such mechanisms will in practice be quite significant.

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