51 lines
2 KiB
Text
51 lines
2 KiB
Text
Angular velocity
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97
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relative to the centre of mass resulting from a rotation through an infinitesimal
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angle do (see (9.1)): dr = dR + do xr. Dividing this equation by the time
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dt during which the displacement occurs, and putting
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dr/dt = V,
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dR/dt =
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do/dt = La
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(31.1)
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we obtain the relation
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V = V+Sxr.
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(31.2)
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Z
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X3
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P
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X2
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r
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o
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R
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X1
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Y
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X
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FIG. 35
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The vector V is the velocity of the centre of mass of the body, and is also
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the translational velocity of the body. The vector S is called the angular
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velocity of the rotation of the body; its direction, like that of do, is along the
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axis of rotation. Thus the velocity V of any point in the body relative to the
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fixed system of co-ordinates can be expressed in terms of the translational
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velocity of the body and its angular velocity of rotation.
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It should be emphasised that, in deriving formula (31.2), no use has been
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made of the fact that the origin is located at the centre of mass. The advan-
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tages of this choice of origin will become evident when we come to calculate
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the energy of the moving body.
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Let us now assume that the system of co-ordinates fixed in the body is
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such that its origin is not at the centre of mass O, but at some point O' at
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a distance a from O. Let the velocity of O' be V', and the angular velocity
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of the new system of co-ordinates be S'. We again consider some point P
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in the body, and denote by r' its radius vector with respect to O'. Then
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= r'+a, and substitution in (31.2) gives V = V+2xa+2xr'. The
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definition of V' and S' shows that V = Hence it follows that
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(31.3)
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The second of these equations is very important. We see that the angular
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velocity of rotation, at any instant, of a system of co-ordinates fixed in
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the body is independent of the particular system chosen. All such systems
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t
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To avoid any misunderstanding, it should be noted that this way of expressing the angular
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velocity is somewhat arbitrary: the vector so exists only for an infinitesimal rotation, and not
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for all finite rotations.
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4*
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98
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Motion of a Rigid Body
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