A rocket is fired from the earth towards the sun. At what distance from the earth’s centre is the gravitational force on the rocket zero? Mass of the sun = 2 ×1030 kg, mass of the earth = 6 × 1024 kg. Neglect the effect of other planets etc. (orbital radius = 1.5 × 1011 m). Question 8.12 Gravitation
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Mass of the sun (Ms) = 2 × 10^30kg
Mass of the earth (Me) = 6×10²⁴kg
Orbital radius (r) = 1.5 × 10¹¹ m
Let mass of rocket = m
Let Gravitational force is zero at a distance x from the center of the earth .
at point x
______________
Gravitational force between rocket and earth = Gravitational force between sun and rocket .
GmMe/x² = GmMs/(r - x)²
Me/x² = Ms/(r - x)²
Ms/Me = (r - x)²/x²
2× 10^30/6×10²⁴ = (r-x)²/x²
10^6/3 = (r - x)²/x²
Taking square root both sides,
(r - x)/x = 10³/√3
r/x -1 = 1000/√3
r/x = (1000+√3)/√3
x = 1.5×10¹¹×1.732/(1001.732)
= 2.594 × 10^8 m
≈ 2.6 × 10^8 m
Mass of the earth (Me) = 6×10²⁴kg
Orbital radius (r) = 1.5 × 10¹¹ m
Let mass of rocket = m
Let Gravitational force is zero at a distance x from the center of the earth .
at point x
______________
Gravitational force between rocket and earth = Gravitational force between sun and rocket .
GmMe/x² = GmMs/(r - x)²
Me/x² = Ms/(r - x)²
Ms/Me = (r - x)²/x²
2× 10^30/6×10²⁴ = (r-x)²/x²
10^6/3 = (r - x)²/x²
Taking square root both sides,
(r - x)/x = 10³/√3
r/x -1 = 1000/√3
r/x = (1000+√3)/√3
x = 1.5×10¹¹×1.732/(1001.732)
= 2.594 × 10^8 m
≈ 2.6 × 10^8 m
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