Exercise 1: sample answer

Atomic time (TAI) is a very steady count of seconds made by averaging many atomic clocks, and it ignores the Earth. UTC runs at the same rate as atomic time but is kept within a second of the Earth's rotation, so that noon stays near the Sun's highest point. Local time is UTC plus or minus a set offset for the zone, plus daylight saving where used. A leap second is needed because the Earth's rotation is slightly irregular and slowing, so without an occasional extra second UTC would slowly drift away from the Sun's time.