A satellite that is 4175 miles from the center of the earth, orbits with a period of 90 minutes. What is its centripetal acceleration?
To find the centripetal acceleration of the satellite, we can use the following formula:
a = (4π²r) / T²
Where:
a = centripetal acceleration
r = distance from the center of the earth
T = period of the orbit
Using the given information:
r = 4175 miles
T = 90 minutes
First, we need to convert the period from minutes to seconds:
T = 90 minutes × 60 seconds/minute = 5400 seconds
Substituting the values into the formula, we get:
a = (4π² × 4175) / (5400²)
Calculating this expression, we find:
a ≈ 0.04495 miles/second²
Therefore, the centripetal acceleration of the satellite is approximately 0.04495 miles/second².
To find the centripetal acceleration of the satellite, we can use the formula:
a = (4π²r) / T²
Where:
a is the centripetal acceleration
π is a mathematical constant approximately equal to 3.14159
r is the distance of the satellite from the center of the Earth
T is the period of the satellite's orbit
In this case, we are given that the satellite is 4175 miles from the center of the Earth and it has a period of 90 minutes.
First, let's convert the distance from miles to meters, as the standard SI unit for distance is meters. We know that 1 mile is approximately 1609.34 meters. Therefore, the distance from the center of the Earth can be converted as follows:
r = 4175 miles * 1609.34 meters/mile
r = 6703125 meters
Next, let's convert the period from minutes to seconds, as the standard SI unit for time is seconds. We know that there are 60 seconds in 1 minute. Therefore, the period can be converted as follows:
T = 90 minutes * 60 seconds/minute
T = 5400 seconds
Now we can substitute the values into the formula:
a = (4π² * 6703125 meters) / (5400 seconds)²
Simplifying the expression:
a = (4π² * 6703125 meters) / (5400 seconds)²
a = (4 * 3.14159 * 3.14159 * 6703125 meters) / (5400 seconds)²
a = (4 * 3.14159 * 3.14159 * 6703125 meters) / (5400 seconds * 5400 seconds)
a = (12.5664 * 6703125 meters) / (29160000 seconds)
a = 84006351.41875 meters / 29160000 seconds
a ≈ 2.8827 meters/second²
Therefore, the centripetal acceleration of the satellite is approximately 2.8827 meters/second².