A satellite has a mass of 6445 kg and is in a circular orbit 4.14 × 105 m above the surface of a planet. The period of the orbit is 2.1 hours. The radius of the planet is 4.52 × 106 m. What would be the true weight of the satellite if it were at rest on the planet’s surface?
To find the true weight of the satellite if it were at rest on the planet's surface, we need to calculate it based on the gravitational force between the satellite and the planet.
First, let's find the gravitational force between the satellite and the planet in its circular orbit using Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2
Where:
F is the gravitational force
G is the gravitational constant (6.67 × 10^-11 N m^2/kg^2)
m1 is the mass of the satellite
m2 is the mass of the planet
r is the distance from the center of the planet to the satellite in its orbit
We can assume that the mass of the planet is much greater than the mass of the satellite, so we can neglect the mass of the satellite compared to the mass of the planet.
The weight of an object is defined as the gravitational force acting on it. Therefore, the true weight of the satellite in its circular orbit is the same as the gravitational force between the satellite and the planet.
In this case, the distance between the satellite and the center of the planet is the sum of the radius of the planet and the altitude of the satellite:
r_total = r_planet + altitude
r_total = 4.52 × 10^6 m + 4.14 × 10^5 m
Now we can substitute the values into the equation to find the gravitational force:
F = G * (m1 * m2) / r_total^2
Note: We can simplify the equation by neglecting the mass of the satellite compared to the mass of the planet, as mentioned earlier.
Finally, the true weight of the satellite on the planet's surface is equal to the gravitational force:
Weight = F
Let's plug in the numbers and calculate the true weight of the satellite:
Weight = G * (m1 * m2) / r_total^2
Weight = (6.67 × 10^-11 N m^2/kg^2) * (6445 kg * m2) / (4.52 × 10^6 m + 4.14 × 10^5 m)^2
Weight = (6.67 × 10^-11 N m^2/kg^2) * (6445 kg * m2) / (5.12 × 10^6 m)^2
Now, we need to know the mass of the planet in order to complete the calculation.