Gravitation-Variations in g- Binding energy

Binding Energy is the smallest quantity of energy that must be provided to the satellite in order for it to be free of the planet’s gravitational pull or influence.

So, we can state the binding energy definition as – The energy that links the satellite to the Earth is referred to as “binding energy.” This binding energy will overcome the pull of gravity between the satellite and the Earth.

Consider a spacecraft orbiting the earth in a circular pattern. The centripetal force required to maintain the satellite spinning in a stable circular orbit is provided by the gravitational pull between the earth and the satellite. As the satellite circles the earth, it generates two types of mechanical energy. Now, let us learn in detail what is binding energy?

What is binding energy?

The energy of gravitational binding is the amount of energy it requires to separate two objects connected by gravity. If you decide to launch an object into space from the Earth. You must give it some positive energy in the kinetic field before, and when the bullet gets closer to infinity, it slowly slows.

Therefore, we can claim that the binding energy was negative and was neutralised by the positive kinetic energy and the total energy needed to get away from the ground that had to be identical to the energy it has at the point of infinity, which is zero energy since energy is stored.

How is binding energy calculated?

Let’s say the earth’s mass is M, the satellite’s mass is m, the earth’s radius is R, the radius of the satellite’s circular orbit is r or R+h, the satellite’s height above the earth’s surface is h, and the satellite’s critical velocity is v.

Because it supplies the essential centripetal force for circular motion, a centripetal force equal to gravitational force is equivalent to gravitational force.

mv2r = GMmr2

mv2 = GMmr2

12mv2 = GMm2r

where the universal gravitational constant is G.

With this binding energy definition and formula, we can write the kinetic energy as:

Kinetic Energy = GMm2r

Because the force acting between the earth and the satellite is attracting, the potential energy between the satellite and the earth may now be expressed as the inverse of kinetic energy.

The magnitude of the K.E., on the other hand, is half that of the P.E, therefore the total E will be the sum of kinetic and potential energy.

E = Kinetic Energy + Potential Energy

E = GMm2r+ (- GMmr)

E = GMm2r – GMmr

E = – GMmr

The binding energy of a satellite circling around the earth in a stable circular orbit is given by the formula above.

The binding energy of a satellite on the earth’s surface is expressed as an expression.

Given that the mass of the stationary satellite is m, the kinetic energy is zero.

The potential energy may now be stated as follows:

P.E. = – GMmr

The total E  will be the sum of kinetic energy and potential energy,

E = Kinetic energy + Potential energy

E = 0 + (- GMmr)

E = 0 – GMmr

E = – GMmr

It’s sometimes easier to think about orbital motion in terms of a celestial body’s energy rather than the forces that are acting on it. We’ll focus on the energy that the body possesses because of the motion, and (gravitational) potential energy, which is the energy owned by a system as a result of the relative positions of its component elements. Imagine a satellite of mass m travelling at v at a distance r from the centre of its planet of mass M (M>>m), then, by relating Newton’s law of gravitation to the centripetal force, we get:

GMm/r2 = mv2/r

As a result, the satellite’s total energy is negative, indicating that it will never be able to break free of the planet’s gravitational attraction.

If other than gravitational forces, there are lack of forces, the total of the mutual potential energies and  kinetic energies of the objects in an isolated system remains constant, i.e.

E = constant

Conclusion

While the satellite’s kinetic and potential energy fluctuates as it goes around the globe, the overall energy of its orbit remains unchanged. The kinetic energy component of the total energy is at its highest at periapsis, when the satellite is closest to the planet and hence travelling at its quickest, and the gravitational potential energy is at its lowest (i.e. its most negative value). The satellite is slowing down and hence losing kinetic energy as it climbs out of the planet’s gravitational potential well towards apoapsis, acquiring potential energy in the process.