Many people mistakenly believe that the astronauts who orbit the Earth are "above gravity." Calculate g for space shuttle territory, 290 kilometers above the Earth's surface (dashed line in sketch). Earth's mass is 6 1024 kg, and its radius is 6.38 106 m (6380 km).
1.)in m/s^2?
2.) Your answer is what percentage of 9.8 m/s2?
To calculate the acceleration due to gravity (g) at a distance of 290 kilometers above the Earth's surface, we can use the formula for gravitational acceleration:
g = (G * M) / (R + h)^2
Where:
- G is the gravitational constant (approximately 6.67430 × 10^-11 N m^2/kg^2)
- M is the mass of the Earth (6 × 10^24 kg)
- R is the radius of the Earth (6.38 × 10^6 m)
- h is the height above the Earth's surface (290 kilometers = 290,000 meters)
Let's plug in the values and calculate g:
g = (6.67430 × 10^-11 N m^2/kg^2 * 6 × 10^24 kg) / (6.38 × 10^6 m + 290,000 m)^2
Simplifying the equation:
g = (3.98 × 10^14 N) / (6.67 × 10^6 m)^2
g = (3.98 × 10^14 N) / 4.45 × 10^13 m^2
g ≈ 8.94 m/s^2 (rounded to two decimal places)
Therefore, the acceleration due to gravity at an altitude of 290 kilometers above the Earth's surface is approximately 8.94 m/s^2.
To calculate the percentage of 8.94 m/s^2 compared to 9.8 m/s^2, use the following formula:
Percentage = (8.94 m/s^2 / 9.8 m/s^2) * 100
Percentage ≈ 91.22% (rounded to two decimal places)
Therefore, the acceleration due to gravity at an altitude of 290 kilometers above the Earth's surface is approximately 91.22% of the gravitational acceleration at the Earth's surface.
To calculate the acceleration due to gravity (g) at a specific height above the Earth's surface, you can use the formula:
g = (G * M) / r^2
Where:
G = Universal gravitational constant = 6.67 × 10^-11 N m^2/kg^2
M = Mass of the Earth = 6 × 10^24 kg
r = Distance from the center of the Earth to the point of interest
1.) To find the value of g for space shuttle territory at a height of 290 kilometers (or 290,000 meters) above the Earth's surface, substitute the given values into the formula:
g = (6.67 × 10^-11 N m^2/kg^2 * 6 × 10^24 kg) / (6380000 m + 290000 m)^2
Simplifying the equation:
g = (40.02 × 10^13 N m^2) / (6673200 m)^2
g = 40.02 × 10^13 N m^2 / 4.46 × 10^13 m^2
g ≈ 9.0 m/s^2
Therefore, the acceleration due to gravity at a height of 290 kilometers above the Earth's surface is approximately 9.0 m/s^2.
2.) To find the percentage of 9.8 m/s^2 that the value of g represents, we can calculate the percentage using the formula:
Percentage = (g / 9.8 m/s^2) * 100
Substituting the values:
Percentage = (9.0 m/s^2 / 9.8 m/s^2) * 100
Percentage ≈ 91.8%
Therefore, the value of g for space shuttle territory is approximately 91.8% of the value of 9.8 m/s^2.