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Equation of the orbit
Storyboard 
In dem Fall, dass ein Körper um eine bestimmte Höhe gegen die Schwerkraft angehoben wird, wird die potentielle Schwerkraftenergie erhalten, die proportional zu Masse, Schwerkraftbeschleunigung und Höhe ist.
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Equation of the orbit
Storyboard
When a body is lifted against the gravitational force to a given height, it gains gravitational potential energy, which is proportional to its mass, the gravitational acceleration, and the height reached.
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The energy required for an object to change its angular velocity from $\omega_1$ to $\omega_2$ can be calculated using the definition
Applying Newton's second law, this expression can be rewritten as
$\Delta W=I \alpha \Delta\theta=I\displaystyle\frac{\Delta\omega}{\Delta t}\Delta\theta$
Using the definition of angular velocity
we get
$\Delta W=I\displaystyle\frac{\Delta\omega}{\Delta t}\Delta\theta=I \omega \Delta\omega$
The difference in angular velocities is
$\Delta\omega=\omega_2-\omega_1$
On the other hand, angular velocity itself can be approximated with the average angular velocity
$\omega=\displaystyle\frac{\omega_1+\omega_2}{2}$
Using both expressions, we obtain the equation
$\Delta W=I \omega \Delta \omega=I(\omega_2-\omega_1)\displaystyle\frac{(\omega_1+\omega_2)}{2}=\displaystyle\frac{I}{2}(\omega_2^2-\omega_1^2)$
Thus, the change in energy is given by
$\Delta W=\displaystyle\frac{I}{2}\omega_2^2-\displaystyle\frac{I}{2}\omega_1^2$
This allows us to define kinetic energy as
The work variance ($\Delta W$) required for an object to change from the initial Angular Speed ($\omega_0$) to the angular Speed ($\omega$) is obtained by applying a the torque ($T$) that produces an angular displacement the difference of Angles ($\Delta\theta$), according to:
Applying Newton's second law for rotation, in terms of the moment of inertia for axis that does not pass through the CM ($I$) and the mean Angular Acceleration ($\bar{\alpha}$):
this expression can be rewritten as:
$\Delta W = I \alpha \Delta\theta$
or, using the difference in Angular Speeds ($\Delta\omega$) and the time elapsed ($\Delta t$):
we get:
$\Delta W = I\displaystyle\frac{\Delta\omega}{\Delta t} \Delta\theta$
Using the definition of the mean angular velocity ($\bar{\omega}$) and the time elapsed ($\Delta t$):
results in:
$\Delta W = I\displaystyle\frac{\Delta\omega}{\Delta t} \Delta\theta = I\omega \Delta\omega$
where the difference in Angular Speeds ($\Delta\omega$) is expressed as:
On the other hand, the angular velocity can be approximated by the average angular velocity:
$\bar{\omega}=\displaystyle\frac{\omega_1 + \oemga_2}{2}$
By combining both expressions, we obtain the equation:
$\Delta W = I \omega \Delta\omega = I(\omega_2 - \omega_1) \displaystyle\frac{(\omega_1 + \omega_2)}{2} = \displaystyle\frac{I}{2}(\omega_2^2 - \omega_1^2)$
Therefore, the change in energy is expressed as:
$\Delta W = \displaystyle\frac{I}{2}\omega_2^2 - \displaystyle\frac{I}{2}\omega_1^2$
This allows us to define the rotational kinetic energy as:
The relationship between the angular Momentum ($L$) and the moment ($p$) is expressed as:
Using the radius ($r$), this expression can be equated with the moment of Inertia ($I$) and the angular Speed ($\omega$) as follows:
Then, substituting with the inertial Mass ($m_i$) and the speed ($v$):
and
it can be concluded that the moment of inertia of a particle rotating in an orbit is:
Just as the relationship between the speed ($v$) and the angular Speed ($\omega$) with the radius ($r$) is expressed by the equation:
we can establish a relationship between the angular Momentum ($L$) and the moment ($p$) in the context of translation. However, in this case, the multiplicative factor is not the arm ($r$), but rather the moment ($p$). This relationship is expressed as:
The total Kinetic Energy ($K$) corresponds to the sum of the translational Kinetic Energy ($K_t$) and the kinetic energy of rotation ($K_r$):
Since the translational Kinetic Energy ($K_t$) is expressed as a function of the inertial Mass ($m_i$) and the speed ($v$):
and the kinetic energy of rotation ($K_r$), as a function of the moment of inertia for axis that does not pass through the CM ($I$) and the angular Speed ($\omega$), is defined as:
the final expression is:
As the gravitational force is
To move a mass $m$ from a distance $r_1$ to a distance $r_2$ from the center of the planet, a potential energy is required
resulting in the gravitational potential energy being
$W_2-W_1=\displaystyle\int_{r_1}^{r_2}\displaystyle\frac{GmM}{r^2}dr=\displaystyle\frac{GmM}{r_1}-\displaystyle\frac{GmM}{r_2}$
thus yielding
The total Energy ($E$) depends on the total Kinetic Energy ($K$) and the general gravitational potential energy ($V$), according to:
When the object is in orbit, the total Kinetic Energy ($K$) consists of a translational part and a rotational part. Considering the inertial Mass ($m_i$), the speed ($v$), the moment of Inertia ($I$) and the angular Speed ($\omega$), we have:
Since the angular Momentum ($L$) is:
and using the distance to the center of the celestial body ($r$), we obtain:
On the other hand, the gravitational potential, expressed in terms of the mass of the celestial body ($M$), the gravitational mass ($m_g$) and the gravitational constant ($G$), is:
Therefore, it follows that:
Examples
The translational Kinetic Energy ($K_t$) is determined based on the speed ($v$) and the inertial Mass ($m_i$), according to:
5288 is associated with 6290 and not with 8762, even though they are numerically equal. The energy that an object possesses is a direct consequence of the inertia that had to be overcome to set it in motion.
The kinetic energy of rotation ($K_r$) is a function of the angular Speed ($\omega$) and of a measure of inertia represented by the moment of inertia for axis that does not pass through the CM ($I$):
The total Kinetic Energy ($K$) can have translational and/or rotational components. Therefore, it is expressed as the sum of the translational Kinetic Energy ($K_t$) and the kinetic energy of rotation ($K_r$):
The total kinetic energy is calculated by adding the kinetic energies of translation and rotation
so we have:
The gravitational force in general is expressed as
while the energy
can be shown to be
The total Energy ($E$) corresponds to the sum of the total Kinetic Energy ($K$) and the potential Energy ($V$):
For a particle of mass the point Mass ($m$) orbiting around an axis at a distance the radius ($r$), the relationship can be established by comparing the angular Momentum ($L$), expressed in terms of the moment of Inertia ($I$) and the moment ($p$), which results in:
The angular Momentum ($L$) is the analogue of the moment ($p$). Therefore, just as in translational motion it corresponds to the product of the inertial Mass ($m_i$) and the speed ($v$), in rotational motion it is obtained from the moment of Inertia ($I$) and the angular Speed ($\omega$), according to the relation:
The masses that Newton used in his principles are related to the inertia of bodies, which leads to the concept of the inertial Mass ($m_i$).
Newton's law, which is linked to the force between bodies due to their masses, is related to gravity, hence known as the gravitational mass ($m_g$).
Empirically, it has been concluded that both masses are equivalent, and therefore, we define
Einstein was the one who questioned this equality and, from that doubt, understood why both 'appear' equal in his theory of gravity. In his argument, Einstein explained that masses deform space, and this deformation of space causes a change in the behavior of bodies. Thus, masses turn out to be equivalent. The revolutionary concept of space curvature implies that even light, which lacks mass, is affected by celestial bodies, contradicting Newton's theory of gravitation. This was experimentally demonstrated by studying the behavior of light during a solar eclipse. In this situation, light beams are deflected due to the presence of the sun, allowing stars behind it to be observed.
The total Energy ($E$) depends on the kinetic energy of radial motion, determined by the inertial Mass ($m_i$) and the speed ($v$); the kinetic energy of rotation, which depends on the angular Momentum ($L$) and the distance to the center of the celestial body ($r$); and the potential energy, which depends on the mass of the celestial body ($M$), the gravitational mass ($m_g$), and the gravitational constant ($G$):
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