We defined the weight of a body in Section 4-5 as the attractive gravitational force exerted on it by the earth. We can now broaden our definition. The weight of a body is the total gravitational force exerted on the body by all other bodies in the universe. When the body is near the surface of the earth, we can neglect all other gravitational forces and consider the weight as just the earth’s gravitational attraction. At the surface of the moon we consider a body’s weight to be the gravitational attraction of the moon,
and so on. If we again model the earth as a spherically symmetric body with radius RE and mass , the weight w of a small body of mass m at the earth’s surface (distance RE from its center) & (weight of a body of mass m at the earth’s surface). (12-3) But we also know from Section 4-5 that the weight w of a body is the force that causes the acceleration g of free fall, so by Newton’s second law, w = mg. Equating this with Eq. (12-3) and dividing by m, we find GmE g = RE2 (acceleration due to gravity at the earth’s surface). (12-4) The acceleration due to gravity g is independent of the mass m of the body because m doesn’t appear in this equation. We already knew that, but we can now see how it follows from the law of gravitation. We can measure all the quantities in Eq. (12-4) except for, so this relation allows us to compute the mass of the earth. Solving Eq. (12-4) for ~ and using R = 6380 kID = 6.38 X 106m and g = 9.80 mJs2 , we find g RE2 mE 5.98 X 1024 kg. G Once Cavendish had measured G, he computed the mass of the earth in just this way. He described his measurements with the grandiose phrase “weighing the earth.” In fact he really determined the mass, not the weight, of the earth. At a point above the earth’s surface a distance r from the center of the earth (a distance r – RE above the surface), the weight of a body is given by Eq. (12-3) with RE replaced by r. w = F. = GmEm I r2 (12-5) The weight of a body decreases inversely with the square of its distance from the earth’s center. Figure 12-6 shows how the weight varies with height above the earth for an astronaut who weighs 700 N at the earth’s surface. The apparent weight of a body on earth differs slightly from the earth’s gravitational force because the earth rotates and is therefore not precisely an inertial frame in figure.