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By Brian H. Chirgwin, Charles Plumpton

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Additional resources for A Course of Mathematics for Engineers and Scientists. Volume 6: Advanced Theoretical Mechanics

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14, where we use alphabetical notation. In this frame j sincc), w2= co, (i cosoc — j sinoc). rA = a k, co, = co, (i cos oc The resultant angular velocity of the body is = Wl + 632 = (col + w2) i Cosa + (co, — w2) j sin ce. co2 = co + w2 + 2co1co2cos2oc. The velocity of the point of the body instantaneously coincident with 0 is vo = w2 x ( — ak) = — aco2 (i cosoc — j sing) x k = aco2 (i sing j cosoc). Hence the invariant v . w of this motion is vo . (co, + 032) raw, (i since j cosoc) • [(co, + w2) i cosoc = aco, co, sin2oc.

46) The scalar product v . w is an invariant for the change of base point. This result concerning the equivalent screw motion is the threedimensional counterpart of the theorem of the existence of the instantaneous centre in plane motion. As the body moves the change of position of the central axis generates one surface in space and another surface in the body; the motion of the body is given by the combination of rolling of the "body-surface" on the space-surface with sliding along the direction of the central axis.

The instantaneous velocities of particles at the points (a, 0, 0), (0, a/y3, 0), (0, 0, 2a) of a rigid body are (u, 0, 0), (u, 0, v), (u + v, —v 1/3, v/2) respectively, referred to a rectangular cartesian frame. Find the magnitude and direction of the spin of the body and the point at which the central axis cuts the xz-plane. 3. A body is pivoted about an axis AB and makes n/27r revolutions per second relative to its bearing. The latter is itself made to revolve with spin co about a fixed axis OZ inclined at an angle 0 to AB, the shortest distance OX between OZ and AB being a.

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A Course of Mathematics for Engineers and Scientists. Volume 6: Advanced Theoretical Mechanics by Brian H. Chirgwin, Charles Plumpton


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