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The parallel axis theorem, also known as Huygens–Steiner theorem, or just as Steiner's theorem, [1] named after Christiaan Huygens and Jakob Steiner, can be used to determine the moment of inertia or the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of gravity and the perpendicular distance between ...
The parallel axis theorem can be used to determine the second moment of area of a rigid body about any axis, given the body's second moment of area about a parallel axis through the body's centroid, the area of the cross section, and the perpendicular distance (d) between the axes. ′ = +
The result is the parallel axis theorem, = [], where is the vector from the center of mass to the reference point . Note on the minus sign : By using the skew symmetric matrix of position vectors relative to the reference point, the inertia matrix of each particle has the form − m [ r ] 2 {\displaystyle -m\left[\mathbf {r} \right]^{2 ...
An arbitrary shape. ρ is the distance to the element dA, with projections x and y on the x and y axes.. The second moment of area for an arbitrary shape R with respect to an arbitrary axis ′ (′ axis is not drawn in the adjacent image; is an axis coplanar with x and y axes and is perpendicular to the line segment) is defined as ′ = where
Parovicenko's theorem ; Parallel axis theorem ; Parseval's theorem (Fourier analysis) Parthasarathy's theorem (game theory) Pascal's theorem ; Pasch's theorem (order theory) Peano existence theorem (ordinary differential equations) Peeling theorem ; Peetre theorem (functional analysis) Peixoto's theorem (dynamical systems)
In classical mechanics, the stretch rule (sometimes referred to as Routh's rule) states that the moment of inertia of a rigid object is unchanged when the object is stretched parallel to an axis of rotation that is a principal axis, provided that the distribution of mass remains unchanged except in the direction parallel to the axis. [1]
Another elementary proof of Mozzi–Chasles' theorem was given by E. T. Whittaker in 1904. [7] Suppose A is to be transformed into B. Whittaker suggests that line AK be selected parallel to the axis of the given rotation, with K the foot of a perpendicular from B. The appropriate screw displacement is about an axis parallel to AK such that K is ...
The perpendicular axis theorem (or plane figure theorem) states that for a planar lamina with a uniform mass distribution, the moment of inertia about an axis perpendicular to the plane of the lamina is equal to the sum of the moments of inertia about two mutually perpendicular axes in the plane of the lamina, which intersect at the point where the perpendicular axis passes through.