Every hundred years or so, most of the planets line up on the same side of the S

Question

Every hundred years or so, most of the planets line up on the same side of the Sun. a) Calculate the total force on the Earth due to Venus, Jupiter and Saturn, assuming all four planets are in a line. An amateur astronomer measures the masses to be MV = 0.791 ME, MJ = 299 ME, MS = 99.9 ME, and their distances from the Sun are 1.080×108 km, 1.500×108 km, 7.780×108 km and 1.430×109 km respectively. The mass of the earth ME is 5.980×1024 kg and the distance from the earth to the sun is 1.500×108 km. The mass of the sun is 1.99×1030 kg. b) What fraction of the Sun's force on the Earth is this for this configuration?

Explanation / Answer

gravitational force between two masses m1 and m2 at a distance of d is given by:

magnitude=G*m1*m2/d^2

where G=universal gravitational constant=6.674*10^(-11)

the gravitational force is always attractive in nature.

as per the question, the position of all the celestial bodies are as given below:

Venus, Earth, Jupiter, Saturn

let force due to venus is F1, force due to jupiter is F2 and force due to saturn is F3.

F1 will be directed towards Venus and F2 and F3 will be directed in oppoiste direction

hence total force=F2+F3-F1

distance between venus and earth=distance of earth from sun-distance of venus from sun

d1=4.2*10^7 km=4.2*10^10 m

distance between jupiter and earth=distance of jupiter from sun-distance of earth from sun

d2=6.28*10^8 km=6.28*10^11 m

distance between saturn and earth=distance of saturn from sun-distance of earth from sun

d3=12.8*10^8 km=12.8*10^11 m

F1=G*ME*MV/d1^2=1.07*10^18 N

F2=G*ME*MJ/d2^2=1.8094*10^18 N

F3=G*ME*MS/d3^2=1.4552*10^17 N

then total force=F2+F3-F1=1.8094*10^18+1.4552*10^17 -1.07*10^18=8.8492*10^17 N

part b:

force due to sum=G*ME*mass of sun/distance of earth from sun^2

=G*5.98*10^24*1.99*10^30/(1.5*10^11)^2=3.5292*10^22 N

hence fraction of sun's force=(8.8492*10^17)/(3.5292*10^22)=2.5074*10^(-5)

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