Catapults are powered by springs or the like, as compared to a trebuchet which uses a weight and gravity.
Some ancient catapults used several strands of heavy rope, attached to the frame and the arm, which were tensioned by turning a ratcheting mechanism on the frame. You can see the twisted rope surrounding the axle in this image.
Other catapults used the energy stored in the bending of the arm. See how this wooden arm is bent after being pulled down by the rope?
In ANY CASE, be sure to consider what will happen when the arm breaks. Release mechanisms must be operable from a distance, and much stronger than you think. Make sure you stay safe please.