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Newton's Principles for Rotation
Storyboard 
Newton's principles are stated for what is translation, however, because of the analogy between translation and rotation, they can also be formulated for what is rotation.
In that case the role of the moment is assumed by the angular momentum, that of the mass, the moment of inertia and that of the force, the so-called torque.
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Rotation Generation
Definition
Up until now, we have explored how force results in translation, but we haven't yet delved into how rotation is generated.
From the previous discussion, it can be concluded that any force $\vec{F}$ can be decomposed into two components. The first component, $\vec{F}{\parallel}$, lies along the line connecting the point of application (PA) to the center of mass (CM) of the object. The second component is $\vec{F}{\perp}$, which is perpendicular to the line connecting the point of application to the center of mass.
The first component causes the translation of the object, while the second component gives rise to its rotation.
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Newton's Laws for the Rotation
Image
Due to the relationship between force and torque, it becomes possible to formulate the laws of rotation based on Newton's principles. Therefore, a connection should exist between the following concepts:
Principle 1
A constant moment > corresponds to a constant angular momentum.
Principle 2
A force: Change in momentum over time > corresponds to a torque: Change in angular momentum over time.
Principle 3
A reaction force > corresponds to a reaction torque.
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Newton's Principles for Rotation
Storyboard
Newton's principles are stated for what is translation, however, because of the analogy between translation and rotation, they can also be formulated for what is rotation. In that case the role of the moment is assumed by the angular momentum, that of the mass, the moment of inertia and that of the force, the so-called torque.
Variables
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Calculations
Variable
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Equation
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Equations
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:
Examples
So far, we have examined how force generates translation, but we have not yet analyzed how rotation occurs.
From the previous discussion, it follows that any force $\vec{F}$ can be decomposed into two components. The first, $\vec{F}{\parallel}$, lies along the line that connects the point of application (PA) to the center of mass (CM) of the body. The second component, $\vec{F}{\perp}$, is perpendicular to the line joining the point of application to the center of mass.
The first component generates the translation of the body, while the second component is responsible for its rotation.
Due to the relationship between force and torque, it becomes possible to formulate the laws of rotation based on Newton's principles. Therefore, a connection should exist between the following concepts:
Principle 1
A constant moment > corresponds to a constant angular momentum.
Principle 2
A force: Change in momentum over time > corresponds to a torque: Change in angular momentum over time.
Principle 3
A reaction force > corresponds to a reaction torque.
The moment ($p$) was defined as the product of the inertial Mass ($m_i$) and the speed ($v$), which is equal to:
The analogue of the speed ($v$) in the case of rotation is the instantaneous Angular Speed ($\omega$), therefore, the equivalent of the moment ($p$) should be a the angular Momentum ($L$) of the form:
the inertial Mass ($m_i$) is associated with the inertia in the translation of a body, so the moment of Inertia ($I$) corresponds to the inertia in the rotation of a body.
Similar to the relationship that exists between linear velocity and angular velocity, represented by the equation:
we can establish a relationship between angular momentum and translational momentum. However, in this instance, the multiplying factor is not the radius, but rather the moment. The relationship is expressed as:
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:
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