12 ORBITAL MOTION
12.4 Gravitational potential energy
where ^z is a unit vector pointing straight upwards (i.e., away from the Earth’s centre). Canceling the factor m on either side of the above equation, we obtain ¨r = −g⊕ ^z,
(12.7)
where G M⊕ (6.673 × 10−11 ) × (5.97 × 1024 ) g⊕ = = 9.79 m/s2 . = 2 6 2 R⊕ (6.378 × 10 )
(12.8)
Thus, we conclude that all objects on the Earth’s surface, irrespective of their mass, accelerate straight down (i.e., towards the Earth’s centre) with a constant acceleration of 9.79 m/s2 . This estimate for the acceleration due to gravity is slightly off the conventional value of 9.81 m/s2 because the Earth is actually not quite spherical. Since Newton’s law of gravitation is universal, we immediately conclude that any spherical body of mass M and radius R possesses a surface gravity g given by the following formula: g M/M⊕ = . (12.9) g⊕ (R/R⊕ )2
Table 6 shows the surface gravity of various bodies in the Solar System, estimated using the above expression. It can be seen that the surface gravity of the Moon is only about one fifth of that of the Earth. No wonder Apollo astronauts were able to jump so far on the Moon’s surface! Prospective Mars colonists should note that they will only weigh about a third of their terrestrial weight on Mars.
12.4 Gravitational potential energy We saw earlier, in Sect. 5.5, that gravity is a conservative force, and, therefore, has an associated potential energy. Let us obtain a general formula for this energy. Consider a point object of mass m, which is a radial distance r from another point object of mass M. The gravitational force acting on the first mass is of magnitude f = G M/r2 , and is directed towards the second mass. Imagine that the first mass moves radially away from the second mass, until it reaches infinity. What 265