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Difficulty: MediumGravitational Field and Orbits

An astronaut has a weight of 720 N720\text{ N} on the surface of the Earth. Calculate the weight of the astronaut, in Newtons (N\text{N}), at an altitude equal to twice the radius of the Earth (h=2Rh = 2R).

Answer: 80 N

Answer

The weight of the astronaut at an altitude of 2R2R is 80 N80\text{ N}.
At an altitude of 2R2R, the total distance from the center of the Earth is r=R+2R=3Rr = R + 2R = 3R. Because gravitational force follows the inverse-square law (W∝1/r2W \propto 1/r^2), tripling the distance reduces the gravitational force and weight by a factor of 32=93^2 = 9. Dividing the surface weight of 720 N720\text{ N} by 99 yields 80 N80\text{ N}.

Step-by-Step Solution

1
Determine the total distance from the center of the Earth
r=R+2R=3Rr = R + 2R = 3R
Gravitational force depends on the distance measured from the center of mass of the Earth, which is the sum of Earth's radius RR and altitude hh.
2
Apply the inverse-square law of gravitation to find field strength at altitude
g′=g32=g9g' = \frac{g}{3^2} = \frac{g}{9}
Acceleration due to gravity is inversely proportional to the square of the distance from the planet's center (g∝1r2g \propto \frac{1}{r^2}).
3
Calculate the astronaut's weight at altitude
W′=7209=80 NW' = \frac{720}{9} = 80\text{ N}
Weight is directly proportional to gravitational field strength (W=mgW = mg).

Key Concept

Variation of Acceleration due to Gravity with Altitude (Inverse Square Law)
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Topics

Gravitational Field and Orbits

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