What is the acceleration due to gravity in the orbit of a communication satellite at a height of 36,000 km?
Understand the Problem
The question is asking about the acceleration due to gravity at the height of a geo-stationary satellite, which is given as 36,000 km. To answer, we can use the formula for gravitational acceleration at a distance from the center of the Earth.
Answer
$$ g \approx 0.226 \text{ m/s}^2 $$
Answer for screen readers
$$ g \approx 0.226 \text{ m/s}^2 $$
Steps to Solve
- Determine the radius of the Earth
The average radius of the Earth is approximately $R = 6,371$ km.
- Calculate the distance from the center of the Earth
To find the total distance from the center of the Earth to the geostationary satellite, we need to add the radius of the Earth to the height of the satellite.
This gives us:
$$ d = R + \text{height} $$ $$ d = 6,371 \text{ km} + 36,000 \text{ km} $$ $$ d = 42,371 \text{ km} $$
- Use the formula for gravitational acceleration
The formula for gravitational acceleration $g$ at a distance $d$ from the center of the Earth is given by:
$$ g = \frac{GM}{d^2} $$
Where:
- $G$ is the gravitational constant, approximately $6.674 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2}$
- $M$ is the mass of the Earth, approximately $5.972 \times 10^{24} \text{ kg}$
- Convert distance to meters
Since the gravitational constant is in SI units, we must convert $d$ to meters:
$$ d = 42,371 \text{ km} = 42,371,000 \text{ m} $$
- Calculate gravitational acceleration
Now we can calculate $g$ using the previously mentioned values:
$$ g = \frac{(6.674 \times 10^{-11}) (5.972 \times 10^{24})}{(42,371,000)^2} $$
- Perform the final calculation
After performing the calculations, we find the value for $g$.
$$ g \approx 0.226 \text{ m/s}^2 $$
More Information
This result shows that as you move further away from the Earth's surface, the acceleration due to gravity decreases significantly. At the altitude of a geostationary satellite, gravity is much weaker compared to its value on Earth's surface, which is approximately $9.81 \text{ m/s}^2$.
Tips
- Forgetting to convert kilometers to meters when calculating gravitational acceleration.
- Not adding the radius of the Earth to the height of the satellite correctly.
- Misapplying the formula for gravitational acceleration by not accounting for the square of the distance in the denominator.
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