Use Newton's Law of Gravitation to compute the work W required to propel a 1100 kg satellite out of the earth's gravitational field. You may assume that the earth's mass is 5.981024 kg and is concentrated at its center. Take the radius of the earth to be 6.37106 m and G = 6.6710-11 Nm2/kg2
To compute the work required to propel a satellite out of the Earth's gravitational field, we can use Newton's Law of Gravitation. The formula for the gravitational force between two objects is given by:
F = (G * m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant (6.67 * 10^-11 Nm^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between their centers of mass.
In this case, we need to calculate the work done to move the satellite out of the Earth's gravitational field, which is equal to the change in gravitational potential energy. The work done is given by:
W = U_final - U_initial
where U_final is the gravitational potential energy of the satellite at the final distance from the center of the Earth, and U_initial is the gravitational potential energy of the satellite at its initial position on the Earth's surface.
The gravitational potential energy of an object near the surface of the Earth is given by:
U = (-G * m1 * m2) / r
where negative sign indicates that the potential energy is zero at infinity.
Now let's calculate the work required:
1. Calculate the initial gravitational potential energy of the satellite on the Earth's surface:
U_initial = (-G * m_satellite * m_earth) / r_earth
where m_satellite is the mass of the satellite (1100 kg), m_earth is the mass of the Earth (5.98 * 10^24 kg), and r_earth is the radius of the Earth (6.37 * 10^6 m).
2. Calculate the final gravitational potential energy of the satellite when it is far away from the Earth's surface:
U_final = (-G * m_satellite * m_earth) / r_final
where r_final is the final distance between the center of the Earth and the satellite. Since the satellite is being propelled out of the Earth's gravitational field, we can assume that r_final approaches infinity. Therefore, r_final can be considered very large compared to the radius of the Earth.
3. Substitute the values into the equation for work:
W = U_final - U_initial
Simplify the expression to get the final answer.
By following these steps, you should be able to compute the work required to propel the satellite out of the Earth's gravitational field using Newton's Law of Gravitation.