tailieunhanh - Classical Mechanics Joel phần 6

Các locus có thể cho một điểm trên trục đối xứng của đầu. Trục nutates giữa θmin = 50 ◦ và θmax = 60 ◦ ˙ trục, φ tỷ lệ đó không phải là không đổi, nhưng chức năng của θ (Eq. 4,36). Chất lượng, chúng tôi có thể phân biệt ba loại chuyển động, có phụ thuộc vào ˙ giá trị của φ tại các điểm quay trong θ. | . DYNAMICS 119 ỡ 52 e 44 nỉ n V Vmin Figure Possible loci for a point on the symmetry axis of the top. The axis nutates between Vmin 50 and Vmax 60 axis at a rate Ộ which is not constant but a function of V Eq. . Qualitatively we may distinguish three kinds of motion depending on the values of Ộ at the turning points in V. These in turn depend on the initial conditions and the parameters of the top expressed in a b and Vmin Vmax. If the value of u0 cos V at which Ộ vanishes is within the range of nutation then the precession will be in different directions at Vmin and Vmax and the motion is as in Fig. . On the other hand if V cos-1 b a 2 ớmin ớmax the precession will always be in the same direction although it will speed up and slow down. We then get a motion as in Fig. . Finally it is possible that cos Vmin b a so that the precession stops at the top as in Fig. . This special case is of interest because if the top s axis is held still at an angle to the vertical and then released this is the motion we will get. Exercises Prove the following properties of matrix algebra a Matrix multiplication is associative A B C A B C. b A-BB T Bt At where AT is the transpose of A that is AT ij Aji. c If A-1 and B-1 exist A B -1 B-1 A-1. d The complex conjugate of a matrix A ij A j is the matrix with every element complex conjugated. The hermitean conjugate Ay is the 120 CHAPTER 4. RIGID BODY MOTION transpose of that Ay A T AT with Ay ij A ị. Show that A B A B and A B y By Ay. In section we considered reexpressing a vector V ypi Vịềị in terms of new orthogonal basis vectors. If the new vectors are ei 55j Ajêj we can also write ei 52j Ajịẽ j because AT A-1 for an orthogonal transformation. Consider now using a new basis ei which are not orthonormal. Then we must choose which of the two above expressions to generalize. Let èí 52j Ajid j and find the expressions for a ej in terms of ei b V in terms of Vj and c Vi in terms of Vj. Then show d that .