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Additional info for A Course of Mathematics for Engineers and Scientists. Volume 6: Advanced Theoretical Mechanics
12) by the unit vector P. ) x , vB = b (co' x t). Hence the velocity of C is vc = EVA + vB) = ia(co x t) + ( a co + b co' \ x ( 1 a+b ) 2 (a + b)i) = x t) x rc, where rcis the position vector of C. Hence C moves as if it were a point of a rigid body rotating about 0 with angular velocity St - aw + bw' a+b § 1: 10] KINEMATICS IN THREE DIMENSIONS 41 Since S2 is constant, w and co' being constants, C moves in a circle. ) = a(w x [bco' — a co — (b — a) 0] x = 0. Since i is variable in direction, 0— bco' — ao, b— a Exercises 1:9 1.
The term (5 (114) is a small rotation which can be written as 60 x (r — rA), using eqn. 32). 44a) Sr = SrA + 60 x (r rA). — [This result may also be obtained from eqn. ] Examples. (i) Two wheels, of radii a and b and centres A and B respectively, are freely mounted at the ends of an axle AB with their planes at right angles to the axle. They are placed with their rims resting on a rough horizontal plane on which they roll without slipping, the distance apart of their points of contact being c. If the spins of the wheels about the axle are w, w' respectively, find the angular velocity of the vertical plane containing the axle.
Co x v, (02 aw2 (w2 + a), costa) k o4. 55) This gives a point on the z-axis and is shown at C in Fig. 14. Therefore the central axis intersects the common perpendicular 0A. From eqn. 56) CA w1 (w1 w2cos2a) When a = 0, n/2 this result is equivalent to eqn. 53) above. Miscellaneous Exercises 1 1. Prove that the angle of rotation a of the resultant rotation defined by Euler's angles, eqn. 17), is given by cos ia = coslO cos i(q) p). 2. A right circular cone of vertical angle 2a rolls without slipping on a rough plane, its vertex being at a fixed point in the plane.