I do not get the idea that how does the torsion balance result in a damped oscillation. Two big and small masses separately attract each other. I naively think that after a while big and small masses stick together, which stops the motion. But it is wrong. Where is my mistake?
I this experiment you have a torsional pendulum in which the restoring couple is provided by the twisted suspension equivalent to the spring when you have an oscillating mass at the end of the spring.
There will be frictional forces acting and so the motion of the torsional pendulum will be damped ie as time progresses the amplitude of motion will decrease.
The motion will eventually stop but not instantly.
The damped oscillations are used to determine the torsion constant of the suspension. The measurements are made with the small masses initially at rest. The larger masses (on a rotating mount) are brought in slowly and their position adjusted so that the small masses are again at rest with the gravitational force between masses balanced by the torque from the suspension.