university physics young and freedman 13th edition-Chapter Summary Chapter 9

## university physics young and freedman 13th edition-Chapter Summary Chapter 9

Section 9.1 “Angular velocity,” of an object is its instantaneous … Read More...

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Problem 3a: (6pts) As shown below, a two-dimensional vector can be defined by its x and y coordinates in an x-y Cartesian coordinate system. The vector can also be defined by its x ‘ and y ‘ coordinates in an x ‘− y’ Cartesian coordinate system that is rotated by a positive angle θ with respect to the x-y Cartesian coordinate system The relation between the two sets of coordinates, (x, y) and ( x ‘ , y ‘), is defined by the following transformation: A x y ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = x ‘ y ‘ ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ where the transformation matrix is given by: A = cosθ sinθ −sinθ cosθ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ (i) Determine the inverse of the matrix A. (4 pts) (ii) Show that the inverse determined in Part (a) is the transformation matrix corresponding to rotation by a negative angle (i.e., rotation by –θ). (2 pts) Hint: cos(θ) = cos (−θ) and sin (θ) = −sin (−θ)

## Problem 3a: (6pts) As shown below, a two-dimensional vector can be defined by its x and y coordinates in an x-y Cartesian coordinate system. The vector can also be defined by its x ‘ and y ‘ coordinates in an x ‘− y’ Cartesian coordinate system that is rotated by a positive angle θ with respect to the x-y Cartesian coordinate system The relation between the two sets of coordinates, (x, y) and ( x ‘ , y ‘), is defined by the following transformation: A x y ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = x ‘ y ‘ ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ where the transformation matrix is given by: A = cosθ sinθ −sinθ cosθ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ (i) Determine the inverse of the matrix A. (4 pts) (ii) Show that the inverse determined in Part (a) is the transformation matrix corresponding to rotation by a negative angle (i.e., rotation by –θ). (2 pts) Hint: cos(θ) = cos (−θ) and sin (θ) = −sin (−θ)

Solution ( ) ( ) A 1 1 adj A … Read More...

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