A satellite has a mass of 5381 kg and is in a circular orbit 4.30 105 m above the surface of a planet. The period of the orbit is 2.06 hours. The radius of the planet is 4.09 106 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 understand the concept of gravitational force and calculate the force exerted on the satellite.
The gravitational force acting on an object can be calculated using the formula:
F = (G * (m₁ * m₂)) / r²
Where:
F is the gravitational force,
G is the universal gravitational constant (approximately 6.67430 × 10^-11 m³ kg^(-1) s^(-2)),
m₁ and m₂ are the masses of the two objects, and
r is the distance between the centers of the two objects.
In this case, the mass of the satellite doesn't affect the gravitational force because weight is a measure of the force of gravity on an object. So, we can use the mass of the planet as m₁, the mass of the satellite as m₂, and the radius of the planet as r.
To find the mass of the planet, we need the satellite's period and radius:
T = 2π * sqrt(r³ / (G * M_earth))
Where:
T is the period of the orbit,
r is the distance between the center of the planet and the satellite (radius of the planet + height of the satellite),
G is the universal gravitational constant, and
M_earth is the mass of the planet (unknown).
Rearranging the above formula, we can solve for M_earth:
M_earth = (4π² * r³) / (G * T²)
Once we find M_earth, we can calculate the gravitational force on the satellite using the formula mentioned earlier.
Finally, the true weight of the satellite if it were at rest on the planet's surface will be equal to the gravitational force calculated in the previous step.
Now, let's perform the calculations:
Given:
Mass of the satellite (m₂) = 5381 kg
Orbit radius (r) = 4.30 * 10^5 m
Period of the orbit (T) = 2.06 hours = 2.06 * 3600 seconds
Planet radius = 4.09 * 10^6 m
Universal gravitational constant (G) = 6.67430 × 10^-11 m³ kg^(-1) s^(-2)
Step 1: Calculate the mass of the planet (M_earth):
M_earth = (4π² * r³) / (G * T²)
Substituting the given values:
M_earth = (4 * 3.14² * (4.09 * 10^6 + 4.30 * 10^5)³) / (6.67430 × 10^-11 * (2.06 * 3600)²)
Calculating the above expression will give us the mass of the planet.
Step 2: Calculate the gravitational force on the satellite:
F = (G * (m₁ * m₂)) / r²
Substituting the values:
F = (6.67430 × 10^-11 * M_earth * m₂) / r²
Step 3: Calculate the true weight of the satellite:
The true weight of the satellite would be equal to the gravitational force calculated in Step 2.
Performing the calculations will give you the final result.
Remember to use proper units throughout the calculations, and if any conversions are required, make sure to account for them.