phys documents (dragged) 3

phys documents (dragged) 3 - to each other. For a principal...

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6 Physics Formulary by ir. J.C.A. Wevers 1.6 Dynamics of rigid bodies 1.6.1 Moment of Inertia The angular momentum in a moving coordinate system is given by: ± L ± = I ±ω + ± L ± n where I is the moment of inertia with respect to a central axis, which is given by: I = ± i m i ± r i 2 ; T ± = W rot = 1 2 ω I ij ± e i ± e j = 1 2 I ω 2 or, in the continuous case: I = m V ² r ± 2 n dV = ² r ± 2 n dm Further holds: L i = I ij ω j ; I ii = I i ; I ij = I ji = - ± k m k x ± i x ± j Steiner’s theorem is: I w . r . t . D = I w . r . t . C + m ( DM ) 2 if axis C ± axis D. Object I Object I Cavern cylinder I = mR 2 Massive cylinder I = 1 2 mR 2 Disc, axis in plane disc through m I = 1 4 mR 2 Halter I = 1 2 μR 2 Cavern sphere I = 2 3 mR 2 Massive sphere I = 2 5 mR 2 Bar, axis through c.o.m. I = 1 12 ml 2 Bar, axis through end I = 1 3 ml 2 Rectangle, axis plane thr. c.o.m. I = 1 12 m ( a 2 + b 2 ) Rectangle, axis ± b thr. m I = ma 2 1.6.2 Principal axes Each rigid body has (at least) 3 principal axes which stand
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Unformatted text preview: to each other. For a principal axis holds: I x = I y = I z = 0 so L n = 0 The following holds: k =-a ijk i j with a ijk = I i-I j I k if I 1 I 2 I 3 . 1.6.3 Time dependence For torque of force holds: = I ; d L dt = - L The torque T is dened by: T = F d . 1.7 Variational Calculus, Hamilton and Lagrange mechanics 1.7.1 Variational Calculus Starting with: b a L ( q, q,t ) dt = 0 with ( a ) = ( b ) = 0 and du dx = d dx ( u )...
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