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Homework Statement
13) (II) At what distance from the Earth will a spacecraft on the way to the Moon experience zero net force due to these two bodies becasue the Earth and Moon pull with equal and opposite forces?
Homework Equations
NET F = ma
G = 6.67 E-11 (Nm^2)/kg^2
Fg = (GmM)/r^2
Mass Moon = 7.35 E 22 kg
Mass Earth = 5.98 E 24 Kg
r Earth to Moon = 384,403,000 m
The Attempt at a Solution
Apply Newton's s second law in the radial direction
NET F = m_craft( a_radial) = Fg moon = Fg Earth = 0
= (G m_craft m_moon)/(384,403,000 m - r)^2 = (G m_craft m_Earth)/r^2
m_craft cancels
G cancels
m_moon/(384,403,000 m - r)^2 = m_Earth/r^2
simplify
m_moon/((384,403,000 m)^2- r^2) = m_Earth/r^2
raise both sides to negative one power
((384,403,000 m)^2- r^2)/m_moon = r^2/m_Earth
multiply both sides by m_Earth
m_Earth( (384,403,000 m)^2 - r^2 )/m_moon = r^2
simplify
( m_Earth(384,403,000 m)^2 - m_Earth(r^2) )/m_moon = r^2
simplify
( m_Earth(384,403,00 m)^2 - 2(m_Earth)(384,403,000 m)m_Earth(r^2) - m_Earth(r^2) )/m_moon = r^2
simplify further
( m_Earth(384,403,00 m)^2 - 2(m_Earth)^2(384,403,000 m)(r^2) - m_Earth(r^2) )/m_moon = r^2
multiply both sides by 1/m_Earth and m_moon to clean up
(384,403,00 m)^2 - 2(m_Earth)(384,403,000 m)(r^2) - (r^2) = (m_moon (r^2))/m_Earth
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