Passenger side windshield wiper doesn't move when I turn them on

Needs new motor

If it were the motor neither side would move. Chances are its the linkage.
You'll probably need to remove the panel under the windshield to access the linkage and check it out.

Either the wiper arms are loose or stripped on the pivots or the wiper linkage is broken. Try tightening the arm to pivot bolt. If that doesn't work, remove the plastic cowling and examine the linkage. You can buy new linkage for it.

You will need to find where in the linkage the problem is. I would start taking it apart to see what is and what isn't working correctly. Autozone website has free repair manuals available online.

bent/broken wiper arms under the base of the windshield..remove the vent cover and service/replace broken parts.

The wiper motor must be directly linked to the passenger side wiper, and the driver side wiper gets its action from the passenger motion. probably the linkage from passenger wiper to the driver side wiper must be the problem. Under the dashboard you can check it, the linkage must be snapped off. (safety first)

First remove the complete passenger side wiper check and make sure splines on wiper are not stripped. If this is not the problem It could be a linkage bar from the wiper motor to the wiper off or disconected. Since both wipers share the same motor if one works the other should. Check this by removing both wipers and then "Cawl" (little black piece between the hood and windshield underneath the wipers) Underneath the cawl you should be able to see the windshield wiper motor, and linkage bar to both of the wipers. See if the bar is disconected from the passenger side. If it is than reconnect it. These are the only two possible scenarios.

replace the arm and the wiper transmission.

It will not be the motor, it will be the linkage or the wiper is slipping on the shaft, the lingkage is accsessible under the cowl, Hope this solves your problem and Merry Christmas

tighten the nut under the cap, of the wiper