What's the total mechanical energy of a 300kg satellite in circular orbit 1400km above Earth's surface?
To calculate the total mechanical energy of a satellite in a circular orbit around Earth, we need to add the gravitational potential energy and the kinetic energy.
First, let's calculate the gravitational potential energy. The gravitational potential energy of an object near the Earth's surface is given by the formula:
PE = m * g * h
Where PE is the potential energy, m is the mass of the object, g is the acceleration due to gravity near the Earth's surface, and h is the height above the Earth's surface.
We are given that the mass of the satellite, m, is 300 kg, and the height above the Earth's surface, h, is 1400 km. We need to convert the height into meters, as the gravitational constant, g, is measured in m/s^2.
1 km = 1000 m, so 1400 km = 1400 * 1000 m = 1,400,000 m
The acceleration due to gravity near the Earth's surface, g, is approximately 9.8 m/s^2.
Now we can calculate the gravitational potential energy:
PE = m * g * h
= 300 kg * 9.8 m/s^2 * 1,400,000 m
= 4,116,000,000 joules
Next, let's calculate the kinetic energy of the satellite. The kinetic energy of an object in motion is given by the formula:
KE = (1/2) * m * v^2
Where KE is the kinetic energy, m is the mass of the object, and v is the velocity of the object.
To find the velocity of the satellite, we need to use the concept of centripetal force in circular motion. The centripetal force is given by the formula:
F = m * a
Where F is the force, m is the mass, and a is the acceleration.
In the case of circular motion, the centripetal force is provided by the gravitational force between the satellite and the Earth. So, we can equate these forces:
F = G * (m * M) / r^2
Where G is the gravitational constant, M is the mass of the Earth, and r is the radius of the orbit.
The centripetal force is also given by:
F = m * (v^2 / r)
Equating the two expressions for F, we find:
G * (m * M) / r^2 = m * (v^2 / r)
Canceling the mass m on both sides and rearranging the equation, we get:
v^2 = G * M / r
Now we can calculate the velocity of the satellite:
v^2 = G * M / r
v^2 = (6.6743 × 10^-11 N(m/kg)^2) * (5.972 × 10^24 kg) / (6,371,000 m + 1,400,000 m)
Simplifying the equation and calculating v, we find:
v = 10,831 m/s
Now, let's calculate the kinetic energy of the satellite:
KE = (1/2) * m * v^2
= (1/2) * 300 kg * (10,831 m/s)^2
= 4,427,393,500 joules
Finally, the total mechanical energy is the sum of the gravitational potential energy and the kinetic energy:
Total Mechanical Energy = PE + KE
= 4,116,000,000 joules + 4,427,393,500 joules
= 8,543,393,500 joules
Therefore, the total mechanical energy of the 300 kg satellite in circular orbit 1400 km above Earth's surface is approximately 8,543,393,500 joules.