Unit_10 - UNIT 10 COMPONENTS OF ACCELERATION AND CURVATURE...

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UNIT 10 COMPONENTS OF ACCELERATION AND CURVATURE INTRO: In this Unit we will continue our study of the trajectory-aligned coordinate system ˆ T, ˆ N, ˆ 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 κ 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 ˆ T , ˆ N and ˆ B for R 3 , determined by a trajectory in R 3 , and a complete orthonormal set of vectors ˆ T, ˆ 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 ˆ i, ˆ j, ˆ k also form a complete orthonormal set of vectors in R 3 and ˆ i, ˆ j a complete orthonormal set in R 2 . The vector ˆ 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, ˆ T will be pointing in the direction of its velocity. The vector ˆ N is the unit normal vector , perpendicular to ˆ 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 ˆ B is the unit binormal vector , which is perpendicular to both ˆ T and ˆ N and defined so as to make a right-handed coordinate system, thus ˆ B = ˆ T × ˆ N . Since the set is mutually orthogonal, we know ˆ T · ˆ N = ˆ T · ˆ B = ˆ N · ˆ B = 0 and since the vectors are unit vectors, || ˆ T || = || ˆ N || = || ˆ B || = 1 In two dimensions ˆ T and ˆ N form a complete “moving” coordinate system as one travels along a trajectory. In three dimensions ˆ T , ˆ N and ˆ B form a complete “moving” ccordinate system as one travels along a trajectory. As long as the travel is along the trajectory, ˆ T will always be pointing in the direction of the velocity vector ~v , and the unit normal vector ˆ 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 ˆ i, ˆ j, ˆ k , namely, ~w = w x ˆ i + w y ˆ j + w z ˆ 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 ˆ i, ˆ j, ˆ k , w x = ~w · ˆ i w y = ~w · ˆ j w z = ~w · ˆ 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
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in terms of any two orthogonal unit vectors). For example, the acceleration ~a = < a x , a y , a z > of any object moving along a trajectory in three dimensions may be written ~a = a x ˆ i + a y ˆ j + a z ˆ k .
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