How much work is required to lift a 700kg satellite to an altitude of 1x10^6m above the surface of the earth? The gravitation force is F=GMm/r^2, where M is the mass of the earth, m is the mass of the satellite, and r is the distance between them. The radius of the earth is 6.4x10^6m, its mass is 6x10^24 kg, and in these units the gravitational constant, G, is 6.67x10^-11.
To lift the mass a distance of dr, the work involved is
F dr = 700GM/r^2 dr
= (700 * 6.67*10^-11 * 6*10^24)/r^2
= 2.8*10^17/r^2 dr
So, the work is
∫[6.4*10^6,7.4*10^6] 2.8*10^17/r^2 dr = 5.9*10^9 Joules
To calculate the work required to lift the satellite to an altitude of 1x10^6m, we need to consider the change in potential energy of the satellite as it moves away from the surface of the Earth.
The potential energy of an object near the surface of the Earth is given by the equation PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height above the ground.
In this case, the potential energy of the satellite is given by PE = mgh', where h' is the altitude above the surface of the Earth. We need to calculate the difference in potential energy between the initial and final altitudes.
Let's go step by step to find the work required:
Step 1: Calculate the gravitational force between the Earth and the satellite.
Using the equation F = GMm / r^2, where G is the gravitational constant, M is the mass of the Earth, m is the mass of the satellite, and r is the distance between them.
F = (6.67x10^-11 N*m^2/kg^2) * (6x10^24 kg) * (700 kg) / (6.4x10^6 m)^2
Step 2: Calculate the initial potential energy of the satellite.
The initial height, h_initial, is 0 because the satellite is at the surface of the Earth.
PE_initial = m * g * h_initial
PE_initial = (700 kg) * (9.8 m/s^2) * (0 m) = 0 J
Step 3: Calculate the final potential energy of the satellite.
The final height, h_final, is 1x10^6 m.
PE_final = m * g * h_final
PE_final = (700 kg) * (9.8 m/s^2) * (1x10^6 m)
Step 4: Calculate the difference in potential energy.
ΔPE = PE_final - PE_initial
ΔPE = (700 kg) * (9.8 m/s^2) * (1x10^6 m)
Step 5: Calculate the work required.
The work done on the satellite is equal to the change in potential energy.
Work = ΔPE = (700 kg) * (9.8 m/s^2) * (1x10^6 m)
By evaluating this expression, you will get the work required to lift the satellite to an altitude of 1x10^6m above the surface of the Earth.