I\'m having trouble with this questions so any help is appreciated! At what poin

Question

I'm having trouble with this questions so any help is appreciated!

At what point in space should a mass m=100kg be placed so that the Sun's gravitational force on it is equal in magnitude but opposite in direction to the Earth's gravitational force on it? What is the gravitational potential energy of the mass m at this location? You will have to include contributions from both the Sun and the Earth. At what point should the mass m be placed so that the gravitational force of the Sun is equal to and in the same direction as the Earth's gravitational force? How much work would be required to move the mass m from the location in (a) to the location in (c)?

Explanation / Answer

1)

force due to gravity = G*M1*M2/ R^2.

ratio of mass of sun to earth = 333,000.

so ratio of distance between the (sun and the mass) and the distance between ( earth and the mass) = sqrt(333,000.)

R1/R2 = 577.06.

R1+R2 = 149,600,000 km------distance between earth and the sun.

578.06 R2 = 149,600,000.

258,796.66 = R2 = distance between earth and the mass.

R1 = 149,341,203.3 km. = distance between sun and the object.

the object should be on the line joining the centres of sun and the earth with corresponding distances of R1 and R2.

2)

Gravitational potential energy of the mass due to sun = -G*M1*m/R1.

Gravitational potential energy of the mass due to sun = -G*M2*m/R2

Total gravitational potential energy of the mass = -Gm* [ (M1/R1) + (M2/R2) ].

= -6.67*10^-11*100*[ (1,988,500/149,341,203,300) + ( 5.9726 / 258,7966600)]

= 8.896*10^10 J.

3)

for the force to be in the same direction.

R1 / R2 = 577.06

R1 - R2 = 149,600,000 Km.------------------distance between earth and sun.

566.06R2 = 149,600,000

R2 = 264,282.94 Km.

R1 = 152,507,112.3 Km.

the object should be in the line joining the centres of the earth and the sun away from both.the earth would lie between the object and the sun.

4)

the gravitational potential energy in the new position = -Gm*[(M1/R1) + (M2/R2) ]

= -6.67*10^-11*100*[ (1,988,500 / 152,507,112,300) + (5.9726/264,282,940) ]*10^24

= - 8.712*10^10 J

change in potential energy from 1st position to 2nd position = 0.184*10^10 J

so the work required to move it from from position a to position c = 0.184*10^10 J

= 1.84*10^9 J

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