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In Newtonian mechanics, momentum (pl.: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction.
Defining equation SI units Dimension Flow velocity vector field u = (,) m s −1 [L][T] −1: Velocity pseudovector field ω = s −1 [T] −1 ... Momentum current ...
The Navier–Stokes momentum equation can be derived as a particular form of the Cauchy momentum equation, whose general convective form is: = +. By setting the Cauchy stress tensor σ {\textstyle {\boldsymbol {\sigma }}} to be the sum of a viscosity term τ {\textstyle {\boldsymbol {\tau }}} (the deviatoric stress ) and a pressure term − p I ...
This equation holds for a body or system, such as one or more particles, with total energy E, invariant mass m 0, and momentum of magnitude p; the constant c is the speed of light. It assumes the special relativity case of flat spacetime [ 1 ] [ 2 ] [ 3 ] and that the particles are free.
The angular momentum of m is proportional to the perpendicular component v ⊥ of the velocity, or equivalently, to the perpendicular distance r ⊥ from the origin. Angular momentum is a vector quantity (more precisely, a pseudovector) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about a ...
In the general case a conservation equation can be also a system of this kind of equations (a vector equation) in the form: [10]: 43 + = where y is called the conserved (vector) quantity, ∇y is its gradient, 0 is the zero vector, and A(y) is called the Jacobian of the current density.
This is what is expected, so the division by 2 in the spin contribution to the momentum density is necessary. The absence of a division by 2 in the formula for the current reflects the = gyromagnetic ratio of the electron. In other words, a spin-density gradient is twice as effective at making an electric current as it is at contributing to the ...
Viewing a magnetic dipole as current loop brings out the close connection between magnetic moment and angular momentum. Since the particles creating the current (by rotating around the loop) have charge and mass, both the magnetic moment and the angular momentum increase with the rate of rotation.