If the angular speed of the Earth as it spins about, its axis is 7.27 x 10^{-5}rad/s. How many rotations the Earth can make in 24 hours?

One rotation. If full rotation makes 2pi radians, or 6.28 radians, divide this number by number of seconds a day (24 hours = 86400 sec), and you wll come up with the angular speed of the Earth given in the text.