The existence of the dwarf planet Pluto was proposed based on irregularities in

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, answer the following questions.

(a) Calculate the acceleration (in m/s2) due to gravity at Neptune due to Pluto when they are 4.50 1012 m apart, as they are at present. The mass of Pluto is 1.4 1022 kg.

________ m/s2

(b) Calculate the acceleration (in m/s2) due to gravity at Neptune due to Uranus, presently about 2.30 1012 m apart. The mass of Uranus is 8.62 1025 kg.

__________ m/s2

Give the ratio of this acceleration to that due to Pluto.

Explanation / Answer

Given

a) mass of pluto M = 1.4*10^22 kg ,

separation of Neptune and Pluto r = 4.5*10^12 m

from gravitational force between two bodies of masses m1,m2 separated by a distance r is  

F = Gm1*m2/r^2

m1*a = Gm1*m2/r^2

a = G*m2/r^2

so acceleration due to gravity at Neptune due to Pluto is

a_p = G*M/r^2

a_p = (6.67408*10^-11*1.4*10^22)/(4.5*10^12)^2 m/s2

a_p = 4.6141788*10^-14 m/s2

b)

mass of Uranus M = 8.62*10^25 kg ,

separation of Neptune and Pluto r = 2.3*10^12 m

acceleration due to gravity at Neptune due to Uranus is

a_U = G*M/r^2

a_U = (6.67408*10^-11*8.62*10^25)/(2.3*10^12)^2 m/s2

a_U = 1.08753440*10^-9 m/s2

the ratio is  

a_U/a_p = (1.08753440*10^-9)/(4.6141788*10^-14) = 23569.4031  

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