# 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.

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