# Motor and Pulley torque contradiction

Consider this scenario, with a motor (grey) connected to a pulley (yellow):

The motor is joined to the pulley at the pulley's center. I know the motor outputs some torque but I do not understand how to then determine the torque the pulley can apply at a radius $$r$$.

Will the torque that the pulley can apply, vary with the radius of the pulley? I feel like it must, because I don't understand how torque could be constant no matter how wide the pulley is.

I would appreciate an explanation as to why and how the output torque varies with the radius of pulley.

• @Bernhard I'm not too sure what you mean Dec 5, 2020 at 11:28

The torque delivered by an ideal motor is constant (real motors deliver a variable torque, depending on their speed, but if you only have one torque value you can assume you have an ideal motor to a first approximation anyway). What varies is the force delivered by the motor. This is inversely proportional to the radius of the pulley, because

$$\text{force delivered}\times \text{radius}=\text{torque delivered}=\text{constant}$$

You can appreciate why the equation is correct by considering the sum of the moments of the tangential force. If you increase the radius the force decreases proportionately for the same torque. For the same reason a screwdriver with a big handle needs less force than a small one, but delivers the same torque to the screw. The energy supplied by the screwdriver operator - or by the motor - which is the force x the distance it moves ( = 2x pi x R per revolution) is the same.

If you can control the power supplied to the motor, then you can adjust the torque from the motor to match the torque required to do what you want to do with the hanging mass. The required torque will depend on the radius of the pulley.

You probably messed it up. Torque depends upon the distance between the point of application of force and the pivoted point. In your diagram the motor is joined to the centre of the pulley. Therefore no matter what be the size, (I mean radius) of the pulley, it won't affect the torque. Yeah, maybe the weight of the pulley might decrease it. Here, the torque is being applied by the motor so it has got nothing to do with the pulley.