Home / Math and Physics Files / Physics / Mechanics / frames doc stuff / Reviewed Docs

Passive View of V with Translation and Translation+Rotation

DOCX · 79.5 KB

A brief note by Phil (dated 7.1.12) that serves as a precursor to the equation r' = r + b used in his frames document. Part 1 treats pure translation between frames S and S' with identical basis vectors, giving V' = V + b and v' = v + vb. Part 2 adds a rotation R of the basis vectors and writes V' = R(V + b). It stresses that V and V' live in different spaces, so the prime is needed.

AI-written summary; may contain errors.

Extracted text (machine-read; may contain errors)

Passive View of V with Translation and Translation+Rotation PhL 7.1.12 This doc provides a precursor to the equation r' = r + b used in frames doc. In both main pictures below, V' = V + b . I ended up never drawing or plotting velocity vectors in frames doc, maybe I should have.! This is a very early doc in this whole effort! Part 1: Translation Only 1. We start with a vector sitting in x-space like so 2. We now imagine some other frame S' related to S by a simple 2D translation: The origin x=0 of S is located at position x' = b in S'. The basis unit vectors in either system are the same: = ' and = ' We have these two ways to look at vector V V = Vx + Vy = (Vx,Vy) = the vector V as seen in no-prime space V' = V'x' + V'y' = (V'x,V'y)' = the same vector V as seen in primed space Notice that we do not have V = V' . So the prime is needed to tell which frame we are in! The vector V' exists in x'-space whereas V exists in x-space, so it makes no sense to say V = V'. It certainly seems that this is true V' = V + b V'x = Vx + bx and similarly for y Suppose V is the vector x. Then we have x' = x + b If b is changing in time, we then have dx'/dt = dx/dt + db/dt or v' = v + vb Here v is a vector in x-space, v' is the corresponding vector in x'-space, and vb is in neither space. Part 2: Translation and Rotation Replace the earlier two-frame picture now with this picture The origin x=0 of S is located at position x' = b in S'. (same) The basis unit vectors in either system are different: = R' and = R' (different) We have these two ways to look at vector V (same) V = Vx + Vy = (Vx,Vy) = the vector V as seen in no-prime space V' = V'x' + V'y' = (V'x,V'y)' = the same vector V as seen in primed space Notice that we do not have V = V' . So the prime is needed to tell which frame we are in! The vector V' exists in x'-space whereas V exists in x-space, so it makes no sense to say V = V'. It certainly seems that this is true V' = R(V + b) or V'x' + V'y' = R(Vx + Vy + b) or V'x' + V'y' = R(Vx + Vy + b)