A sattelite of mass 1000kg moves in circular orbit of radius 7000km ar...
Total Energy required to place a satellite in orbit from Earth's surface
Given:
- Mass of the satellite (m) = 1000 kg
- Radius of the orbit (r) = 7000 km = 7,000,000 meters
- Mass of the Earth (M) = 5.972 × 10^24 kg
- Gravitational constant (G) = 6.674 × 10^-11 Nm^2/kg^2
Explanation:
Step 1: Calculate the gravitational force between the satellite and the Earth
The gravitational force between two objects can be calculated using the formula:
F = (G * M * m) / r^2
Where:
- F is the gravitational force
- G is the gravitational constant
- M is the mass of the Earth
- m is the mass of the satellite
- r is the distance between the centers of the Earth and the satellite
Plugging in the values, we get:
F = (6.674 × 10^-11 Nm^2/kg^2 * 5.972 × 10^24 kg * 1000 kg) / (7,000,000 meters)^2
Simplifying the equation, we get:
F = 8.974 × 10^15 N
Step 2: Calculate the work done to move the satellite to the orbit
The work done to move an object against a force is given by the formula:
Work = Force * Distance
In this case, the force is the gravitational force calculated in the previous step and the distance is the radius of the orbit.
Work = 8.974 × 10^15 N * 7,000,000 meters
Simplifying the equation, we get:
Work = 6.282 × 10^22 J
Step 3: Calculate the kinetic energy of the satellite in orbit
The kinetic energy of an object in orbit can be calculated using the formula:
Kinetic Energy = (1/2) * m * v^2
In this case, since the satellite is in a circular orbit, the velocity can be calculated using the formula:
v = √(G * M / r)
Plugging in the values, we get:
v = √(6.674 × 10^-11 Nm^2/kg^2 * 5.972 × 10^24 kg / 7,000,000 meters)
Simplifying the equation, we get:
v = 7456.35 m/s
Plugging the velocity into the kinetic energy formula, we get:
Kinetic Energy = (1/2) * 1000 kg * (7456.35 m/s)^2
Simplifying the equation, we get:
Kinetic Energy = 2.778 × 10^10 J
Step 4: Calculate the total energy required
The total energy required to place the satellite in orbit from Earth's surface is the sum of the work done and the kinetic energy:
Total Energy = Work + Kinetic Energy
Plugging in the values, we get:
Total Energy = 6.282 × 10^22 J + 2.778 × 10^10 J
Simplifying the equation, we get:
Total Energy =
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