Six artificial satellites complete one circular orbit around a space station in the same amount of time. Each satellite has mass m and radius of orbit L. The satellites fire rockets that provide the force needed to maintain a circular orbit around the space station. The gravitational force is negligible. Rank from largest to smallest.
1-m=400kg
L=2500m
2-m=800kg
L=5000m
3-m=100kg
L=2500m
4-m=100kg
L=10000m
5-m=200kg
L=5000m
6-m=300kg
L=7500m
To rank the satellites from largest to smallest, we need to calculate the centripetal force acting on each satellite. The centripetal force required to maintain a circular orbit is given by the equation:
F = (m * v^2) / r
where F is the centripetal force, m is the mass of the satellite, v is the velocity of the satellite, and r is the radius of the orbit.
Let's calculate the centripetal forces for each satellite:
1. Satellite with m = 400 kg and L = 2500 m:
F₁ = (400 kg * (v₁)^2) / 2500 m
2. Satellite with m = 800 kg and L = 5000 m:
F₂ = (800 kg * (v₂)^2) / 5000 m
3. Satellite with m = 100 kg and L = 2500 m:
F₃ = (100 kg * (v₃)^2) / 2500 m
4. Satellite with m = 100 kg and L = 10000 m:
F₄ = (100 kg * (v₄)^2) / 10000 m
5. Satellite with m = 200 kg and L = 5000 m:
F₅ = (200 kg * (v₅)^2) / 5000 m
6. Satellite with m = 300 kg and L = 7500 m:
F₆ = (300 kg * (v₆)^2) / 7500 m
Since all the satellites complete one circular orbit around the space station in the same amount of time, the velocities will be the same for all the satellites:
v₁ = v₂ = v₃ = v₄ = v₅ = v₆
Therefore, we can compare the centripetal forces using the mass and radius of orbit values.
Now, let's compare the centripetal forces:
F₁ = (400 kg * (v₁)^2) / 2500 m
F₂ = (800 kg * (v₁)^2) / 5000 m
F₃ = (100 kg * (v₁)^2) / 2500 m
F₄ = (100 kg * (v₁)^2) / 10000 m
F₅ = (200 kg * (v₁)^2) / 5000 m
F₆ = (300 kg * (v₁)^2) / 7500 m
Simplifying the equations:
F₁ = 0.16 kg * (v₁)^2
F₂ = 0.16 kg * (v₁)^2
F₃ = 0.04 kg * (v₁)^2
F₄ = 0.01 kg * (v₁)^2
F₅ = 0.04 kg * (v₁)^2
F₆ = 0.04 kg * (v₁)^2
From the equations, we can see that F₁ = F₂, F₃ = F₅ = F₆, and F₄ is smaller than all the other forces. Therefore, ranking the satellites from largest to smallest:
1. Satellite with m = 100 kg and L = 10000 m
2. Satellites with m = 400 kg and L = 2500 m, and m = 800 kg and L = 5000 m
3. Satellites with m = 100 kg and L = 2500 m, m = 200 kg and L = 5000 m, and m = 300 kg and L = 7500 m
To rank the artificial satellites from largest to smallest, we need to look at the centripetal force required to maintain a circular orbit. The centripetal force is provided by the rockets on the satellites.
The centripetal force (Fc) is given by the formula:
Fc = m * v^2 / r
Where:
- m is the mass of the satellite
- v is the velocity of the satellite in the circular orbit
- r is the radius of the orbit
Since all the satellites complete one circular orbit in the same amount of time, and the force required is the same for all of them, we can calculate the velocity (v) using the formula:
T = 2πr / v
Where:
- T is the time period of the orbit (given as the same for all satellites)
- r is the radius of the orbit
Let's calculate the velocities and centripetal forces for each satellite:
1. For satellite 1:
m = 400 kg
L = 2500 m
v1 = (2π * L) / T
v1 = (2 * 3.14 * 2500) / T
Fc1 = m * v1^2 / L
Fc1 = 400 * (v1^2) / 2500
Similarly, we can calculate the centripetal forces for the remaining satellites:
2. For satellite 2:
m = 800 kg
L = 5000 m
Fc2 = m * v2^2 / L
3. For satellite 3:
m = 100 kg
L = 2500 m
Fc3 = m * v3^2 / L
4. For satellite 4:
m = 100 kg
L = 10000 m
Fc4 = m * v4^2 / L
5. For satellite 5:
m = 200 kg
L = 5000 m
Fc5 = m * v5^2 / L
6. For satellite 6:
m = 300 kg
L = 7500 m
Fc6 = m * v6^2 / L
Now, we can calculate the centripetal forces for each satellite and compare them to rank them from largest to smallest.