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Concept
Euler's laws of motion
Résumé
In classical mechanics, Euler's laws of motion are equations of motion which extend Newton's laws of motion for point particle to rigid body motion. They were formulated by Leonhard Euler about 50 years after Isaac Newton formulated his laws.
Euler's first law states that the rate of change of linear momentum p of a rigid body is equal to the resultant of all the external forces Fext acting on the body:
Internal forces between the particles that make up a body do not contribute to changing the momentum of the body as there is an equal and opposite force resulting in no net effect.
The linear momentum of a rigid body is the product of the mass of the body and the velocity of its center of mass vcm.
Balance of angular momentum
Euler's second law states that the rate of change of angular momentum L about a point that is fixed in an inertial reference frame (often the center of mass of the body), is equal to the sum of the external moments of force (torques) acting on that body M about that point:
Note that the above formula holds only if both M and L are computed with respect to a fixed inertial frame or a frame parallel to the inertial frame but fixed on the center of mass.
For rigid bodies translating and rotating in only two dimensions, this can be expressed as:
where:
rcm is the position vector of the center of mass of the body with respect to the point about which moments are summed,
acm is the linear acceleration of the center of mass of the body,
m is the mass of the body,
α is the angular acceleration of the body, and
I is the moment of inertia of the body about its center of mass.
See also Euler's equations (rigid body dynamics).
The distribution of internal forces in a deformable body are not necessarily equal throughout, i.e. the stresses vary from one point to the next. This variation of internal forces throughout the body is governed by Newton's second law of motion of conservation of linear momentum and angular momentum, which for their simplest use are applied to a mass particle but are extended in continuum mechanics to a body of continuously distributed mass.
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