Does the gravitational force act on gases

Planetary system

Repetition from the intermediate level

The following applies to lifting work in the laboratory: \ [W = F _ {\ rm g} \ cdot h \ qquad \ text {or} \ qquad E _ {\ rm {pot, above}} - E _ {\ rm {pot, below}} \] This is a special case of the equation \ (W = F \ cdot s \), so work equals force times displacement.

This only applies under the following 2 conditions:

1) The force \ (F \) always acts in the direction of travel

2) The force is constant along the way

Generalizations for the concept of work:

1.
If the force is not parallel to the path, one takesthe Force component in the direction of the road times the path or

the Path component in the direction of force times the force.

In the gravitational field, the work is only dependent on the start and end point, but not on the path!

One therefore looks for the most favorable path, the first part of which is a section of the path in which the path and force are parallel, and the second part of which is a section of the path where the path and force are perpendicular to each other. So only the first part contributes to the work. \ [W = E _ {\ rm {pot, end point}} - E _ {\ rm {pot, start point}} \]

2.

If the force along the path is not constant, the work consists of a sum of small amounts of work along small parts of the path, along which the force does not change significantly.
This leads to the integral \ (W = \ int \ limits _ {{r_ {Anf}}} ^ {{r_ {End}}} {F \ cdot dr} \)

This leads to work in the gravitational field: \ (W = \ int \ limits _ {{r_ {Anf}}} ^ {{r_ {End}}} {G \ cdot m \ cdot M \ cdot \ frac {1} {{{{ r ^ 2}}} \ cdot dr} = G \ cdot m \ cdot M \ cdot \ left [{- \ frac {1} {r}} \ right] _ {{r_A}} ^ {{r_E}} \ )
\ (W = - G \ cdot m \ cdot M \ cdot \ frac {1} {{{r_E}}} + G \ cdot m \ cdot M \ cdot \ frac {1} {{{r_A}}} \) = EPot, the end - E.Pot, start

Determination: \ (E _ {\ rm {pot,} \ infty} = 0 \). In the free space a body has no potential energy.

Gravitational field near bodies

In the vicinity of a heavy body, the potential energy decreases as the body approaches the body, in inverse proportion to the distance.

One can imagine the gravitational field (reduced by one spatial dimension) in the vicinity of heavy bodies as sketched opposite.
In the vicinity of the body, as in a stretching rubber skin, pits form due to the mass of the body, the walls of which drop by 1 / r² and other bodies, such as moons or satellites, can circle along the walls without being able to leave the area of ​​attraction.

Calculation of the escape speed:

The lifting work from the body surface to infinity (negative potential energy) must be given to the body in the form of kinetic energy: \ [\ frac {1} {2} m \ cdot {v ^ 2} = \ frac {{G \ cdot {\ rm {m}} \ cdot {\ rm {M}}}} {r} \ Rightarrow v = \ sqrt {\ frac {{2 \ cdot G \ cdot M}} {r}} {\ rm {} } \] This speed is called the escape speed.

For the earth the escape speed is \ (v _ {\ rm Escape} = 11 {,} 2 \, \ rm {\ frac {km} {s}} \)

Significance of the escape speed:

The lower the escape speed, the more difficult it is for the body to hold gases. The formation or maintenance of an atmosphere is therefore dependent on a minimum size of the escape speed.