2D Kinetic Energy - 2 P G/P p 2 p I 2 1 ) r v m v m 2 1 T +...

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2D Kinetic Energy of a Rigid-Body Kinetic energy of the i th particle: i i i 2 i i i v v m 2 1 v m 2 1 T = = Note : r ω v v p i × + = (relative motion analysis equation) where k ω ω , j y i x r = + = ) r ω ( ) r ω m 2 1 ) r ω v m v m 2 1 ) r ω v ) r ω v m 2 1 T i p i 2 p i p p i i × × + × + = × + × + = ( ( ( ( For planar problems: ) y (x ω ) r ω ( ) r ω ( 2 2 2 + = × × Therefore, ) y (x ω m 2 1 ) r ω v m v m 2 1 T 2 2 2 i p i 2 p i i + + × + = ( This is for the i th particle. Now integrate over the entire body: Letting dm m i + + × + = m m m )dm y (x ω 2 1 ) dm r ω v v dm 2 1 T 2 2 2 p 2 p ( -axis) bout the z inertia a (moment of t P) of mass wr of center (location s of body) (total mas note: m P m 2 2 G/P m I dm ) y (x r m dm r m dm = + = = 2 P G/P p 2 p ω I 2 1 ) r ω v m v m 2 1 T + × + = ( General expression for the kinetic energy of a rigid-body in plane motion
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General Expression:
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Unformatted text preview: 2 P G/P p 2 p I 2 1 ) r v m v m 2 1 T + + = ( Special conditions and cases for point P: 1) Rigid-body translations ( = 0 ) 2 G v m 2 1 T = 2) Fixed axis rotation about point P ( = P v ) 2 P I 2 1 T = 3) Point P = center-of-mass ( r G/P = ) 2 G 2 G I 2 1 v m 2 1 T + = Translational Kinetic Energy Rotational Kinetic Energy v G is the same for any point on the body including point P Also true if point P is the IC of zero velocity...
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2D Kinetic Energy - 2 P G/P p 2 p I 2 1 ) r v m v m 2 1 T +...

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