Calculating Lever Arm Length for Electric Motor Pulley System with String and Rock

if i attach 1 end of a string to an electric motor shaft and the other end to rock, creating a pulley, is the length of the string the length of the lever arm. i know torque equals lever arm times force and lever arm equals radius times sine angle.

No, the radius of the motor shaft is the lever arm, otherwise known as the moment arm.

Assuming you are lifting the rock (the string stays vertical) the point at which the string joins the shaft does not move (with respect to the ground) so the lever arm is the horizontal line from the center of the shaft to that point. It does not change its angle and the angle is 0 degrees thus the sine of the angle is 0. Thus torque equals the weight of the rock times the radius of the shaft. This also assumes the lifting rate is constant. During acceleration, the force will be higher than the weight.

Best, -JD

So if I make the string longer, the lever arm will be the same?

Yes. The length of the string has no bearing on the length of the lever arm. It is the radius of the shaft.

The lever arm is the perpendicular distance from the axis of rotation to the line of action of the force. and the magnitude of the torque is τ = N m. The direction of the torque associated with the illustration above is toward you. That is, it is along the rotation axis in the direction out of the screen.