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. Calculate the total force on the Earth due to Venus, Jupiter and Saturn, assuming all four planets are in a line (see figure). An amateur astronomer measures the masses to be M_v = 0.807 M_E, M_J = 321 M_E, M_S = 93.2 M_E, and their distances from the Sun are 1.080 Times 10^8 km, 1.500 Times 10^8km, 7.780 Times 10^8 km and 1.430 Times 10^9 km respectively. The mass of the earth M_E is 5.980 Times 10^24 kg and the distance from the earth to the sun is 1.500 Times 10^8 km. The mass of the sun is 1.99 Times 10^30 kg. What fraction of the Sun's force on the Earth is this for this configuration?

Explanation / Answer

Force Due to venus is F1 = G*mv*mE/rvE^2 = (6.67*10^-11*5.98*10^24*0.807*5.98*10^24)/((1.5-1.08)*10^11)^2

F1 = 1091.2*10^15 N towards venus

due to Sun is F2 = G*ms*mE/rsE^2 = (6.67*10^-11*1.99*10^30*5.98*10^24)/(1.5*10^11)^2 = 26.6*10^21 N towqards Sun

due to Jupiter is F3 = G*mE*mJ/rEj = (6.67*10^-11*5.98*10^24*321*5.98*10^24)/((7.78-1.5)*10^11)^2

F3 = 1941.4*10^15 N towards jupiter

due to saturn is

F4 = G*mE*Msn/rESn = (6.67*10^-11*5.98*10^24*93.2*5.98*10^24)/((14.3-1.5)*10^11)^2

F4 = 135.7*10^15 N towards Saturn

total force on earth is F3+F4-F1-F2 = (135.7+1941.4-1091.2-26600000)*10^15 = 2.65*10^22 N towards the sun

B) (2.65*10^22)/(2.66*10^21) = 9.96

then the required fraction is 9.96 times to the force due to sun on Earth

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