Question

The existence of the dwarf planet Pluto was proposed based on irregularities in Neptune's orbit. Pluto was subsequently discovered near its predicted position. But it now appears that the discovery was fortuitous, because Pluto is small and the irregularities in Neptune's orbit were not well known. To illustrate that Pluto has a minor effect on the orbit of Neptune compared with the closest planet to Neptune:

Required:
a. Calculate the acceleration due to gravity at Neptune due to Pluto when they are 4.50x10^12 m apart, as they are at present. The mass of Pluto is 1.4x10^22 kg.
b. Calculate the acceleration due to gravity at Neptune due to Uranus, presently about 2.50x10^12 m apart, and compare it with that due to Pluto. The mass of Uranus is 8.62x10^25 kg.

Answer 1

4.61 * 10^-14 m/s ; 9.20 * 10^-10 m/s

Step-by-step explanation:

To acceleration due to gravity, a

a = GM / r²

Mass of pluto = 1.4x10^22 kg

Distance apart, r = 4.50x10^12 m

G = Gravitational constant = 6.67 * 10^- 11

a = (6.67*10^-11 * 1.4x10^22) / (4.50x10^12)²

a = 9.338 * 10^(-11 + 22) / 20.25x10^24

a = 0.461 * 10^(11 - 24)

a = 0.461 * 10^-13

a = 4.61 * 10^-14 m/s

B.)

Mass of Uranus = 8.62x10^25 kg

Distance apart, r = 2.50x10^12 m

a = (6.67*10^-11 * 8.62x10^25 ) / (2.50x10^12 )²

a = 57.4954 * 10^(-11 + 25) / 6.25x10^24

a = 9.199264 * 10^(14 - 24)

a = 9.199264 * 10^-10

a = 9.20 * 10^-10 m/s