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Newton's Principles for Rotation
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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.
ID:(756, 0)
Newton's Laws for the Rotation
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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.
ID:(1073, 0)
Newton's Principles for Rotation
Description
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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(ID 1072)
(ID 3251)
The relationship between the angular Momentum ($L$) and the moment ($p$) is expressed as:
| $ L = r p $ |
Using the radius ($r$), this expression can be equated with the moment of Inertia ($I$) and the angular Speed ($\omega$) as follows:
| $ L = I \omega $ |
Then, substituting with the inertial Mass ($m_i$) and the speed ($v$):
| $ p = m_i v $ |
and
| $ v = r \omega $ |
it can be concluded that the moment of inertia of a particle rotating in an orbit is:
| $ I = m_i r ^2$ |
(ID 3602)
Just as the relationship between the speed ($v$) and the angular Speed ($\omega$) with the radius ($r$) is expressed by the equation:
| $ v = r \omega $ |
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:
| $ L = I \omega $ |
(ID 9874)
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.
(ID 322)
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.
(ID 1073)
ID:(756, 0)