no smaller diameter does not mean spinning faster.

I'm an engineering major, and here's rotational dynamics 101.

the rotational speed, in terms or complete revolutions per unit of time (for this instance rpm, or revolutions per minute) is still the exact same, however the linear velocity is what changes, this is because the linear speed on a rotational body is the cross product of angular speed and the distance from the instantaneous center of zero velocity.

to put it shortly, the speed at which the belt is going is the linear velocity of the edge of the pulley, which is equal to the angular speed multiplied by radius of the pulley.

so v=w*r (w=the angular speed), since w is constant, and r decrease's by whatever ratio the smaller pulley is to the stock one, the velocity at which it drives the belt must also be decreased by that much.

all the smaller ratio means is it has to spin faster (in terms of angular speed or rpm) to have the same output as before with the stock pulley.

/nerd rant

 

^is right! But I think I follow you amist the engineering technical jargon-and believe me, there's a couple of jargons in the above statement.

I would've said:

Man A with short arms throws a disc spinning a given RPM throws the disc a given distance.

Man B with longer arms than Man A, throws the same disc, spinning at the same RPM, but throws the disc farther because his arms are longer and because JUPITER WAS ALIGNED WITH MARS!