In certain cases, using both the momentum principle and energy principle to anal

Question

In certain cases, using both the momentum principle and energy principle to analyze a system is useful, as they each can reveal different information. You will use the both momentum principle and the energy principle in this problem.

A satellite of mass 5000 kg orbits the Earth in a circular orbit of radius of 7.9 106 m (this is above the Earth's atmosphere).The mass of the Earth is 6.0 1024 kg.

What is the magnitude of the gravitational force on the satellite due to the earth? F = N

Using the momentum principle, find the speed of the satellite in orbit. (HINT: Think about the components of (dp^^->)/(dt) parallel and perpendicular to p^^->.)

v = m/s

Using the energy principle, find the minimum amount of work needed to move the satellite from this orbit to a location very far away from the Earth. (You can think of this energy as being supplied by work due to something outside of the system of the Earth and the satellite.)

work = J

Explanation / Answer

Magnitude of gravitational force on the satellite due to the earth, Fg

Fg = GMm/r^2

where, G is gravitational constant = 6.67*10^-11 m^3/kg s^2

M, mass of earth

m, mass of satellite

r , distance

Fg = 6.67*10^-11*6*10^24*5000/(7.9*10^6)^2

= 32062 N answer

The two forces, gravity and centripetal are equal,

GMm/r^2 = mV^2/r

V = sqrt(GM/r)

V = sqrt(6.67*10^-11*6*10^24/(7.9*10^6)

= 7117 m/s answera

The energy of satellite at that orbit; note that gravitational energy has minus sign as its attraction energy

E = - GMm/r + 0.5mV^2

So,

E = (- 6.67*10^-11*6*10^24*5000/(7.9*10^6)) + (0.5*5000*(7117)^2)

= -1.27 * 10^11 J

To take it very far away, in another words to infinity

delta E = E final - E initial

E final = 0,  energy at infinity is zero as distance is infinite

deltaE = 0 - (-1.27*10^11)

so the minimum amount of energy to move it very far away is 1.27*10^11 J

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