(a) At what point in space should a mass m=100kg be placed so that the Sun\'s gr
ID: 2250753 • Letter: #
Question
(a) 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?
(b) 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.
(c) 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?
(d) How much work would be required to move the mass m from the
location in (a) to the location in (c)?
Explanation / Answer
a)
mS = mass of the sun
mE = mass of the earth
m = 100 kg
rS = distance from the mass to the sun
rE = distance from the mass to the earth
r = distance from the sun to the earth
we have F1 = F2, so
==> G m mS/(r - rS)^2 = G m mE/(r - rS)^2
==> mS/(rS)^2 = mE/(rE)^2
rE = r - rS = 1.496e11 - rS
==> mS/(rS)^2 = mE/(rE)^2
==> 1.989e30/(rS)^2 = 5.972e24/(1.496e11 - rS)^2
==> rS = 1.49341 x 10^11 m
the distance of the mass m from the sun is "1.49 x 10^11 m"
the distance of the mass m from the earth is "2.59 x 10^8 m"
b)
U = G m mS/rS + G m mE/rE
==> U = 6.673e-11 * 100 * 1.989e30/1.49341 + 6.673e-11 * 100 * 5.972e24/2.59e8
==> U = 8.90 x 10^10 J
c)
we have F1 = F2, so
==> G m mS/(rS)^2 = G m mE/(rE)^2
==> mS/(rS)^2 = mE/(rE)^2
rS = r + rE = 1.496e11 + rE
==> mS/(1.496e11+rE)^2 = mE/(rE)^2
==> 1.989e30/(1.496e11+rE)^2 = 5.972e24/(rE)^2
==> rE = 2.5967 x 10^8 m
the distance of the mass m from the sun is "1.50 x 10^11 m"
the distance of the mass m from the earth is "2.60 x 10^8 m"
d)
U = G m mS/rS + G m mE/rE
==> U = 6.673e-11 * 100 * 1.989e30/1.4986e11 + 6.673e-11 * 100 * 5.972e24/2.5967e8
==> U = 8.87 x 10^10 J
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