The earth has a radius of 6380 km and turns around once on its axis in 24 h. (a) What is the radial acceleration of an object at the earth’s equator? Give your answer in m/s2 and as a fraction of g. (b) If arad at the equator is greater than g, objects will fly off the earth’s surface and into space. (We will see the reason for this in Chapter 5.) What would the period of the earth’s rotation have to be for this to occur?
Solution 25E The radial acceleration in a circular motion is given by a = r ….(r) rad Step 1 Earth completes one rotation, that is 2 rad in 24 hour. Hence the rotational speed of earth is 2 rad 2 rad 5 = 24 h = 86400 s= 7.3 × 10 rad/s Now the radius circular motion, which will be equal to the radius of earth is r = 6380 km = 6380 × 10 m 3 Hence the radial acceleration is given by a = (7.3 × 10 5 rad/s)(6380 × 10 m) = 3.4 × 10 2 m/s 2 rad Now we know that the acceleration due of gravity g is 9.8 m/s . Hence, in terms of fraction of g , the acceleration is a = (3.4 × 10 2 m/s ) 1 g = 3.5 × 10 g rad 9.8 m/s Hence the acceleration is 3.5 × 103 times the acceleration due to gravity.