How to move an object through 30 degree rotation?
If the plastic "floor" of the bird feeder is not too big and the fairly light weight you may very well be able to make this work as follows. 

First hinge one edge of the "floor" to the corresponding side of the bird feeder framework.

Next create a linkage rod that connects to the "floor" away from the hinge that will be suitable for pushing the floor between its two positions. With appropriate design considerations and placement of the linkage rod you can connect its opposite end to a hobby servo that is equipped with a lever arm. Here is a picture of the servo device. 



Control of the servo is done by feeding it a pulse train that changes in width according to the lever position you wish to attain. For a typical servo the pulse train would be a 30 to 50Hz waveform with the high time of the waveform ranging from 1 to 2msec pulse width.

Most servos operate with a voltage from 5 to 6 volts and the control signal swinging from 0 to 5V.