PHYS 422 Lecture 13 Free Vibrations II

PHYS 422 Lecture 13 Free Vibrations II - Free Vibrations of...

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Free Vibrations f Physical Systems II of Physical Systems II
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Pendula O Simple Pendulum θ – 2D problem though expressible rough the single parameter l-y l through the single parameter θ – exchange of KE and PE during the motion P m N – consider θ small y << x 2 x y x y – conservation of energy 2 l 2 1 v mgy E 22 2 mv dx dy + = ⎛⎞ + 2 v dt dt = ⎜⎟ ⎝⎠
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Pendula Simple Pendulum O – use the fact y << x 2 11 dx mg θ l-y l 2 22 g mx E dt l g ⎛⎞ ⇒+ = ⎜⎟ ⎝⎠ – use the angular displacement instead l ω= P y m N d vl dt θ = x 2 2 1 (1 cos ) 2 yl l θθ =− d ml mg lE dt =
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Pendula Arbitrary Object O – center-of-mass (C) a distance h from the suspension point θ h 2 for deflection 2 mgh PE θ Δ= C – KE is the rotation of the whole body about the suspension point O ote each point has angular speed t note, each point has angular speed d θ /dt 2 1 I moment of inertia about d KE I ⎛⎞ ⇒= 2 horizontal axis @ O dt ⎜⎟ ⎝⎠ 2 2 11 22 d I mgh E dt +=
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Pendula Arbitrary Object O θ h – utilize the parallel axis theorem –let±mk 2 = I about center-of-mass C – KE with respect to the center-of-mass is 2 2 1 d mk t θ ⎛⎞ ⎜⎟ – but we must include the KE associated with the linear speed of the center-of-mass 2 dt ⎝⎠ 2 2 1 2 d mh dt 22 2 2 11 d 1 2 111 222 dd mk mh mg hE dt dt h h k θθ ⇒++ = 2 2 gh T hk g h ω π + ⇒= = +
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U – Tube – Potential Energy of moving column of water is Um g y
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This note was uploaded on 03/23/2011 for the course PHYS 422 taught by Professor Staff during the Spring '08 term at Purdue.

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PHYS 422 Lecture 13 Free Vibrations II - Free Vibrations of...

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