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Unformatted text preview: UNIT 10 COMPONENTS OF ACCELERATION AND CURVATURE INTRO: In this Unit we will continue our study of the trajectory-aligned coordinate system T, N, B , in particular how it may be used to decompose acceleration along a trajectory into its components along and perpendicular to the trajectory. The curvature of the trajectory will play a role in all of this. 1. Orthogonal Decomposition of a Vector In the previous Unit we defined a complete orthonormal set of vectors T , N and B for R 3 , determined by a trajectory in R 3 , and a complete orthonormal set of vectors T, N in R 2 . By orthonormal we mean that these vectors are mutually orthogonal to each other and that they are unit vectors: they have norm 1. By complete we mean there are 3 of them for R 3 and two of them for R 2 . Completeness guarantees that all directions in the vector space will be accounted for. For example, we know i, j, k also form a complete orthonormal set of vectors in R 3 and i, j a complete orthonormal set in R 2 . The vector T is the unit tangent vector , which is tangent to the trajectory at each point along the curve. For any particle traveling along the curve, at any speed, T will be pointing in the direction of its velocity. The vector N is the unit normal vector , perpendicular to T and pointing to the center of the circle which the trajectory is instantaneously tracing, therefore in the direction of its centripetal acceleration. This is a complete set of orthonormal vectors in R 2 . In R 3 the vector B is the unit binormal vector , which is perpendicular to both T and N and defined so as to make a right-handed coordinate system, thus B = T N . Since the set is mutually orthogonal, we know T N = T B = N B = 0 and since the vectors are unit vectors, || T || = || N || = || B || = 1 In two dimensions T and N form a complete moving coordinate system as one travels along a trajectory. In three dimensions T , N and B form a complete moving ccordinate system as one travels along a trajectory. As long as the travel is along the trajectory, T will always be pointing in the direction of the velocity vector ~v , and the unit normal vector N will always be perpendicular to the velocity vector. We know that any vector ~w = < w x ,w y ,w z > can be decomposed into its components along i, j, k , namely, ~w = w x i + w y j + w z k , where the coefficients are just the x,y,z components of ~w . We also know from Unit 5 that these coefficients are also the scalar projections of ~w onto the unit vectors i, j, k , w x = ~w i w y = ~w j w z = ~w k, More generally, any vector in R 3 can be decomposed in the same fashion in terms of any three orthogonal unit vectors using the scalar projections onto the unit vectors (and any vector in R 2 in terms of any two orthogonal unit vectors). For example, the accelerationin terms of any two orthogonal unit vectors)....
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- Fall '08