The GRACE Satellite Mission measures changes in water on the

The GRACE Satellite Mission measures changes in water on the Earth’s surface indirectly by measuring local changes in Earth’s gravitational field. The satellite altitude is 440 km above the Earth’s surface. The satellite weighs 300 kg. Its observations apply to a 10,000 km2 area. In January, GRACE measured the gravitational force F over the Sierra Nevadas as 617640 N. In May, the measurement was repeated and F had declined by 1 x 10 -9 N. Newton’s law of universal gravitation states that: € F = G m1m2 r 2 where G is the gravitational constant 6.674×1011 N m2 kg2, r is the distance between the objects’ centers of mass and m1, m2 are the masses of the two objects. (a) Assume that r and the satellite’s mass are constant. What has been the change in the mass in the Sierra Nevadas? (b) Assume the mass change is due to water. What volume of water has been removed? (c) What is the average change in the local depth of stored water?

Solution

mass of earth = 5.972 × 10²⁴ kg

r= 440 km = 440000 m change in force is due to change in mass of earth let change in mass is m

1*10^-9 = 6.674×10¹¹ *300 m / 440000 ²

m= 1*10^-9* 440000 ²/(6.674×10¹¹ *300) = 9.6694*10⁹ kg

b)

1 m³ of water = 1000kg

volume of water = 9.6694*10⁹/ 1000 = 9.6694*10⁶ m³

average change in local depth = volume/ area = 9.6694*10⁶ / (10,000 *10⁶)= 0.00097 m= 0.97 mm

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