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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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)....
View Full Document

Page1 / 10

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online