Question Continuation
Derive an expression for x in terms of m, M, and D. b) If the net force is zero a distance ⅔D from the planet, what is the ratio R of the mass of the planet to the mass of the moon, M/m?
Answer:
a. x = (D√M/m)/(√M/m + 1)
b. The ratio R of the mass of the planet to the mass of the moon=4:1
Explanation:
Given
m = Mass of moon
M = Mass of the planet
D = Distance between the centre of the planet and the moon
Net force = 0
Let Y be a point at distance x from the planet
Let mo = mass at point Y
a.
Derive an expression for x in terms of m, M and D.
Formula for Gravitational Force is
F = Gm1m2/r²
Y = D - x
Force on rest mass due to mass M (FM) =Force applied on rest mass due to m (Fm)
FM = G * mo * M/x²
Fm = G * mo * m/Y²
Fm = G * mo * m/(D - x)²
FM = Fm = 0 ------ from the question
So,
G * mo * M/x² = G * mo * m/(D - x)² ----- divide both sides by G * mo
M/x² = m/(D - x)² --- Cross Multiply
M * (D - x)² = m * x²
M/m = x²/(D - x)² ---_ Find square roots of both sides
√(M/m) = x/(D - x) ----- Multiply both sides by (D - x)
(D - x)√(M/m) = x
D√(M/m) - x√(M/m) = x
D√(M/m) = x√(M/m) + x
D√(M/m) = x(√(M/m) + 1) ------- Divide both sides by √M/m + 1
x = (D√M/m)/(√M/m + 1)
b. Here x = ⅔D
FM = G * mo * M/x²
Fm = G * mo * m/(D - x)²
FM = Fm
G * mo * M/x² = G * mo * m/(D - x)² ----- divide both sides by G * mo
M/x² = m/(D - x)² --- (Substitute ⅔D for x)
M/(⅔D)² = m/(D - ⅔D)²
M/(4D/9) = m/(⅓D)²
9M/4D = m/(D/9)
9M/4D = 9m/D ---- Divide both side by 9/D
M/4 = m
M = 4m
M/m = 4
M:m = 4:1
So, the ratio R of the mass of the planet to the mass of the moon=4:1
