An artificial satellite circling the Earth completes each orbit in 139 minutes. (The radius of the Earth is 6.38 106 m. The mass of the Earth is 5.98 1024 kg.)
(a) Find the altitude of the satellite.
Equate the gravitational force,
GmM/R^2, to the centripetal force, m V^2/R
M is the Earth's mass and G is the universal constant, which you need to look up. Satellite mass m cancels out.
Substitute 2 pi R/(139*60) for the velocity, V (in m/s).
Solve for the remaining variable R.
The satellite altitude is H = R - Rearth
The orbital period of a satellite derives from
T = 2(Pi)sqrt(r^3/µ) where T = the period in seconds
Pi = 3.14
r = the orbital radius in feet and
µ = the earth's gravitational constant = 1.407974x10^16 ft^3/sec^2.
139(60)= 2(3.14)sqrt(r^3/1.407974x10^16)
Solving for r = 5523.6 miles making the altitude 1560.6 miles = 2511km.
Well, if I were an artificial satellite, I would definitely need some altitude to stay away from all those pesky birds and airplanes. So, let me calculate it for you.
The period of the satellite can be found using the formula:
T = 2π√(r³/GMe)
where T is the period, r is the radius of the orbit (which is the sum of the radius of the Earth and the altitude of the satellite), G is the universal gravitational constant, and Me is the mass of the Earth.
Since we know the period T and the radius of the Earth, we can rearrange the formula to solve for the altitude of the satellite.
But hey, instead of giving you a boring mathematical answer, I'll tell you a joke instead: Why don't scientists trust atoms? Because they make up everything!
Now, back to business. Let's plug in the values:
T = 139 minutes = 139 × 60 = 8340 seconds,
r = 6.38 × 10⁶ m,
G = 6.674 × 10⁻¹¹ Nm²/kg²,
Me = 5.98 × 10²⁴ kg.
Solving the equation should give us the altitude of the satellite. But first, let's have another joke: Why did the scarecrow win an award? Because he was outstanding in his field!
Alright, now let's do the calculations. Plugging in the values, we get:
8340 = 2π√((6.38 × 10⁶ + altitude)³/ (6.674 × 10⁻¹¹ × 5.98 × 10²⁴))
Now we can solve for the altitude of the satellite. But wait, here's another joke: Why did the tomato turn red? Because it saw the salad dressing!
Okay, now let's solve for the altitude... drumroll please...
The altitude of the satellite is approximately 2,966,437 meters.
Hey, don't worry, I'll be here all week. Please remember to tip your waiter, and try the veal!
To find the altitude of the satellite, we need to use the formula for the period of a satellite's orbit. The period (T) of the satellite is the time it takes for the satellite to complete one orbit around the Earth.
The formula to calculate the period is:
T = 2π√(r³/GM)
Where:
T = Period (in seconds)
π = Pi (approximately 3.14159)
r = Radius of the satellite's orbit (altitude + Earth's radius)
G = Gravitational constant (approximately 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻²)
M = Mass of the Earth
In this case, we are given the period (T) as 139 minutes, but we need to convert it to seconds:
T = 139 minutes * 60 seconds/minute = 8340 seconds
Plug in the known values into the formula:
8340 = 2π√((r + 6.38 * 10⁶)³ / (6.67430 × 10⁻¹¹ * 5.98 * 10²⁴))
Now, we can solve for r. Rearrange the formula to solve for r:
r = (√((8340 * (6.67430 × 10⁻¹¹ * 5.98 * 10²⁴)) / (2π))) - 6.38 * 10⁶
Let's calculate the value of r using this formula:
r = (√((8340 * (6.67430 × 10⁻¹¹ * 5.98 * 10²⁴)) / (2π))) - 6.38 * 10⁶
r ≈ 7.07 * 10⁶ meters
Finally, we need to subtract the radius of the Earth (6.38 * 10⁶ meters) from the calculated value of r to find the altitude of the satellite:
Altitude = r - Earth's radius
Altitude ≈ 7.07 * 10⁶ - 6.38 * 10⁶
Altitude ≈ 0.69 * 10⁶ meters
Therefore, the altitude of the satellite is approximately 0.69 million meters or 690,000 meters.