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phys documents (dragged) 13

phys documents (dragged) 13 - ⊥ to each other For a...

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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 r 2 n dV = 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 ∂ω x = ∂ I ∂ω y = ∂ I ∂ω z = 0 so L ± n = 0 The following holds: ˙ ω k =-a ijk ω i ω 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 ± τ holds: ± τ ± = I ¨ θ ; d ±± ± L ± dt = ± τ ±-±ω × ± L ± The torque ± T is de±ned by: ± T = ± F × ± d . 1.7 Variational Calculus, Hamilton and Lagrange mechanics 1.7.1 Variational Calculus Starting with: δ b ² a L ( q, ˙ q,t ) dt = 0 with δ ( a ) = δ ( b ) = 0 and δ ³ du dx ´ = d dx ( δ u )...
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