Why in Kater's pendulum we have two cylinders of different material , like for example : wooden and metal?

Answer 1

Here's a reference that will explain the answer:

http://www.inference.phy.cam.ac.uk/teaching/dynamics/images/Kater.html

And I assume you know the configuration for the experiment, but anyway here it is

http://lecturedemo.ph.unimelb.edu.au/mechanics/pendula_shm/kater_s_reversible_pendulum