A motorized mechanism for operating a window shade (5, 7) for controlling the amount of light admitted through a window, comprising: a window shade (5, 7) adapted to be extended and compressed relative to the window in accordance with motion within a housing of a rail assembly (25, 45) attached to one end of said window shade (5, 7), wherein said rail assembly (25, 45) includes an axle (33, 53) extending between opposite ends of said rail assembly (25, 45); a motor (69, 69a) secured to the housing and coupled to a motor-driven first pulley (65, 65a); a cable (63, 63a) looped between said motor-driven first pulley (65, 65a) and a second pulley (85, 86), said second pulley (85, 86) being secured to the housing remotely from said first pulley (65, 65a); and characterized in that first and second components (35, 36; 37, 38; 55, 56; 57, 58) are respectively coupled to said axle at opposite ends of said rail assembly, with only one of said first and second components (35, 36; 37, 38; 55, 56; 57, 58) comprising a driven component that is coupled to said cable to be movable within the housing between the first and second pulleys (65, 85, 86) with motion of the cable in response to motor driven rotation of said first pulley to extend or compress the window shade (5, 7), with the other one of said first and second components (35, 36; 37, 38; 55, 56; 57, 58) comprising a passive component (35, 57) that is movable within the housing in response to the movement of the driven component (37, 55), wherein said first component comprises a carrier (36, 58) having a gear (35, 57) rotatably coupled thereto, and wherein said cable (63, 63a) is secured to said carrier (36, 58). a window shade (5, 7) adapted to be extended and compressed relative to the window in accordance with motion within a housing of a rail assembly (25, 45) attached to one end of said window shade (5, 7), wherein said rail assembly (25, 45) includes an axle (33, 53) extending between opposite ends of said rail assembly (25, 45); a motor (69, 69a) secured to the housing and coupled to a motor-driven first pulley (65, 65a); a cable (63, 63a) looped between said motor-driven first pulley (65, 65a) and a second pulley (85, 86), said second pulley (85, 86) being secured to the housing remotely from said first pulley (65, 65a); and characterized in that first and second components (35, 36; 37, 38; 55, 56; 57, 58) are respectively coupled to said axle at opposite ends of said rail assembly, with only one of said first and second components (35, 36; 37, 38; 55, 56; 57, 58) comprising a driven component that is coupled to said cable to be movable within the housing between the first and second pulleys (65, 85, 86) with motion of the cable in response to motor driven rotation of said first pulley to extend or compress the window shade (5, 7), with the other one of said first and second components (35, 36; 37, 38; 55, 56; 57, 58) comprising a passive component (35, 57) that is movable within the housing in response to the movement of the driven component (37, 55), wherein said first component comprises a carrier (36, 58) having a gear (35, 57) rotatably coupled thereto, and wherein said cable (63, 63a) is secured to said carrier (36, 58).