Lecture 11 - Rigid Rotations

Lecture 11 - Rigid Rotations - Rotation of Rigid Bodies u...

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Rotation of Rigid Bodies
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1 or better, 1 0 R rdm M rdm M = = ur r r
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x θ dx v dt = d dt ϖ= dv a dt = d dt ϖ α= Displacement Rotational
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2 2 2 2 2 2 2 vector from origin distance to axis r l r x y z l x y v r v l ϖ = = = + + = + = × = r r ur r
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2 2 2 2 Kinetic energy 1 2 1 2 i i i i i i i i i i i K m v v l K m l I m l ϖ = = = =
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Displacement Rotational m I 2 1 2 mv 2 1 2 I ϖ
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r F L r p τ= × = × r r ur ur r ur
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( 29 d L d r p dt dt dr d p p r dt dt mv v r F τ = × = × + × = × + × = ur r ur r ur ur r r r r ur r
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Displacement Rotational F ur r F τ= × r r ur p ur L r p = × ur r ur d p F dt = ur ur d L dt τ= ur r
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2 2 i i i I l m l dm = = 2 2 Bicycle wheel Same hollow cyclinder I R dm MR = =
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2 Spin up bicycle wheel ˆ ˆ d dt dv r ma r adm RM k dt dv d R R dt dt MR k I ϖ α τ × = = × = × = = = = = r r r r r r ur
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Displacement Rotational I τ α = r ur F ma = ur r
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2 2 2 2 2 Parallel Axis Theorem 1 1 1 2 2 2 c c I I MR K I I MR ϖ = + = = +
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2 2 4 2 Solid disc 2 2 4 1 2 Same solid cylinder I l dm M l ldl A M R A I MR π = = = =
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2 3 2 2 2 2 2 2 Long stick uniform mass
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Lecture 11 - Rigid Rotations - Rotation of Rigid Bodies u...

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