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The Holditch-Type Theorem for the Polar Moment of Inertia of the Orbit Curve in the Generalized Complex Plane

Research paper by Tülay Erişir, Mehmet Ali Güngör; Murat Tosun

Indexed on: 09 Nov '16Published on: 01 Dec '16Published in: Advances in Applied Clifford Algebras



Abstract

Abstract In this study, we first calculate the polar moment of inertia of orbit curves under one-parameter planar motion in the generalized complex plane \({{\mathbb{C}_p}}\) and then give the Holditch-type theorem for \({{\mathbb{C}_p}}\) : When the fixed points \({X}\) and \({Y}\) on the moving plane \({{\mathbb{K}_p} \subset {\mathbb{C}_p}}\) trace the same curve \({k}\) with the polar moment of inertia \({{T_X}}\) , the different point \({Z}\) on this line segment \({XY}\) traces another curve \({{k_Z}}\) with the polar moment of inertia \({{T_Z}}\) during the one-parameter planar motion in the fixed plane \({{\mathbb{K}'_p} \subset {\mathbb{C}_p}}\) . Thus, we obtain that the difference between the polar moments of inertia of these curves \({( {{T_Z} - {T_X}} )}\) depends on the only the \({p}\) -distances of this points and \({p}\) -rotation angle of the motion, \({{T_X} - {T_Z} = {\delta _p}ab.}\) AbstractIn this study, we first calculate the polar moment of inertia of orbit curves under one-parameter planar motion in the generalized complex plane \({{\mathbb{C}_p}}\) and then give the Holditch-type theorem for \({{\mathbb{C}_p}}\) : When the fixed points \({X}\) and \({Y}\) on the moving plane \({{\mathbb{K}_p} \subset {\mathbb{C}_p}}\) trace the same curve \({k}\) with the polar moment of inertia \({{T_X}}\) , the different point \({Z}\) on this line segment \({XY}\) traces another curve \({{k_Z}}\) with the polar moment of inertia \({{T_Z}}\) during the one-parameter planar motion in the fixed plane \({{\mathbb{K}'_p} \subset {\mathbb{C}_p}}\) . Thus, we obtain that the difference between the polar moments of inertia of these curves \({( {{T_Z} - {T_X}} )}\) depends on the only the \({p}\) -distances of this points and \({p}\) -rotation angle of the motion, \({{T_X} - {T_Z} = {\delta _p}ab.}\) \({{\mathbb{C}_p}}\) \({{\mathbb{C}_p}}\) \({{\mathbb{C}_p}}\) \({{\mathbb{C}_p}}\) \({X}\) \({X}\) \({Y}\) \({Y}\) \({{\mathbb{K}_p} \subset {\mathbb{C}_p}}\) \({{\mathbb{K}_p} \subset {\mathbb{C}_p}}\) \({k}\) \({k}\) \({{T_X}}\) \({{T_X}}\) \({Z}\) \({Z}\) \({XY}\) \({XY}\) \({{k_Z}}\) \({{k_Z}}\) \({{T_Z}}\) \({{T_Z}}\) \({{\mathbb{K}'_p} \subset {\mathbb{C}_p}}\) \({{\mathbb{K}'_p} \subset {\mathbb{C}_p}}\) \({( {{T_Z} - {T_X}} )}\) \({( {{T_Z} - {T_X}} )}\) \({p}\) \({p}\) \({p}\) \({p}\) \({{T_X} - {T_Z} = {\delta _p}ab.}\) \({{T_X} - {T_Z} = {\delta _p}ab.}\)