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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 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 ...
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]
Let Q be the semicircle with diameter on the x-axis that passes through the points (1,0) and (0,y), where y > 1. Since Q is tangent to the unit semicircle centered at the origin, the two semicircles represent parallel hyperbolic lines. The y-axis crosses both semicircles, making a right angle with the unit semicircle and a variable angle Φ with Q.
The proofs given in this article use these definitions, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles , see Trigonometric functions . Other definitions, and therefore other proofs are based on the Taylor series of sine and cosine , or on the differential equation f ″ + f = 0 ...
The presumption that the axis is parallel to the y axis allows one to consider a parabola as the graph of a polynomial of degree 2, and conversely: the graph of an arbitrary polynomial of degree 2 is a parabola (see next section).
To show that a system S is required to prove a theorem T, two proofs are required. The first proof shows T is provable from S; this is an ordinary mathematical proof along with a justification that it can be carried out in the system S. The second proof, known as a reversal, shows that T itself implies S; this proof is carried out in the base ...