I don't understand how (h_i - h_f) turned into (1-cos x).

With the aid of Figure 1, prove Total Energy Lost =AU =mgl(1cos 6) where l is the distance from the pivot point to the center of mass of the coil wand and 49 is the angular position of the coil wand. d , diameter of the coil N, Number of turns -.......... V XXXXXXX XXXX- - -id Uniform X X X X X X.......XXX magnetic field X.X-x.X XXXXXXXX XXXXXXXX X X pointing into X X the paper XX X X . X XXXXXXXXX Figure 1. Position of center of massEnergy and Power If the center of mass of the pendulum starts from rest at an initial height h: , its potential energy is U i = m 9 hi (17) As the pendulum swings and passes through the magnet, some energy is lost to mechanical frictional heat and some energy is converted to electrical energy and then to thermal energy in the resistor. Thus the center of mass of the pendulum does not rise to the same height but rather to a lower nal height, hf. See Figure 11. The total energy lost by the pendulum is equal to its change in potential energy: Total Energy Lost = AU = mg(hi hf) (18) 01' Total Energy Lost : AU : mg] (1 cos :9) (19) where l is the distance from the pivot point to the center of mass of the coil wand. and I9 is the angular position of the coil wand. Centre of mass Figure 11: Coil Height Decreases Figure 12: Coil and Resistor Circuit