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The moment of force, or torque, is a first moment: =, or, more generally, .; Similarly, angular momentum is the 1st moment of momentum: =.Momentum itself is not a moment.; The electric dipole moment is also a 1st moment: = for two opposite point charges or () for a distributed charge with charge density ().
Moment arm diagram. A very useful special case, often given as the definition of torque in fields other than physics, is as follows: = (). The construction of the "moment arm" is shown in the figure to the right, along with the vectors r and F mentioned above. The problem with this definition is that it does not give the direction of the torque ...
The trivial case of the angular momentum of a body in an orbit is given by = where is the mass of the orbiting object, is the orbit's frequency and is the orbit's radius.. The angular momentum of a uniform rigid sphere rotating around its axis, instead, is given by = where is the sphere's mass, is the frequency of rotation and is the sphere's radius.
Momentum is also conserved in special relativity (with a modified formula) and, in a modified form, in electrodynamics, quantum mechanics, quantum field theory, and general relativity. It is an expression of one of the fundamental symmetries of space and time: translational symmetry.
In 1744, Euler was the first to use the principles of momentum and of angular momentum to state the equations of motion of a system. In 1750, in his treatise "Discovery of a new principle of mechanics" [ 3 ] he published the Euler's equations of rigid body dynamics , which today are derived from the balance of angular momentum, which Euler ...
Note on second moment of area: The moment of inertia of a body moving in a plane and the second moment of area of a beam's cross-section are often confused. The moment of inertia of a body with the shape of the cross-section is the second moment of this area about the z {\displaystyle z} -axis perpendicular to the cross-section, weighted by its ...
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 acting on that body M about that point: [1] [4] [5]
where M k are the components of the applied torques, I k are the principal moments of inertia and ω k are the components of the angular velocity. In the absence of applied torques, one obtains the Euler top. When the torques are due to gravity, there are special cases when the motion of the top is integrable.