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Thin, solid disk of radius r and mass m. ... The given formula is for the ... the object is a solid ball (above).
Here, the function gives the mass density at each point (,,), is a vector perpendicular to the axis of rotation and extending from a point on the rotation axis to a point (,,) in the solid, and the integration is evaluated over the volume of the body . The moment of inertia of a flat surface is similar with the mass density being replaced by ...
The fundamental equation describing the behavior of a rotating solid body is Euler's equation of motion: = = + = + = + where the pseudovectors τ and L are, respectively, the torques on the body and its angular momentum, the scalar I is its moment of inertia, the vector ω is its angular velocity, the vector α is its angular acceleration, D is ...
This is done by integrating an infinitesimally thin spherical shell with mass of , and we can obtain the total gravity contribution of a solid ball to the object outside the ball F t o t a l = ∫ d F r = G m r 2 ∫ d M . {\displaystyle F_{total}=\int dF_{r}={\frac {Gm}{r^{2}}}\int dM.}
In particular, a ball (open or closed) always includes p itself, since the definition requires r > 0. A unit ball (open or closed) is a ball of radius 1. A ball in a general metric space need not be round. For example, a ball in real coordinate space under the Chebyshev distance is a hypercube, and a ball under the taxicab distance is a cross ...
The Euler equations can be generalized to any simple Lie algebra. [1] The original Euler equations come from fixing the Lie algebra to be s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} , with generators t 1 , t 2 , t 3 {\displaystyle {t_{1},t_{2},t_{3}}} satisfying the relation [ t a , t b ] = ϵ a b c t c {\displaystyle [t_{a},t_{b}]=\epsilon ...
Stated formally, in general, an equation of motion M is a function of the position r of the object, its velocity (the first time derivative of r, v = dr / dt ), and its acceleration (the second derivative of r, a = d 2 r / dt 2 ), and time t. Euclidean vectors in 3D are denoted throughout in bold.
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]