Question

(10 marks) Consider rotation around the z-axis, X = Q(t)A where Q is the rotation matrix $Q(t) = \begin{pmatrix} \cos(\omega t) & -\sin(\omega t) & 0\\ \sin(\omega t) & \cos(\omega t) & 0\\ 0 & 0 & 1 \end{pmatrix}$ Determine the kinematic variables in material coordinates U, V, as well as in spatial coordinates u, v. Sketch the vector field $\mathbf{v}$ in the x-y plane.

          (10 marks) Consider rotation around the z-axis, X = Q(t)A where Q is the rotation matrix
$Q(t) = \begin{pmatrix} \cos(\omega t) & -\sin(\omega t) & 0\\ \sin(\omega t) & \cos(\omega t) & 0\\ 0 & 0 & 1 \end{pmatrix}$
Determine the kinematic variables in material coordinates U, V, as well as in spatial coordinates u, v. Sketch the vector field $\mathbf{v}$ in the x-y plane.

(10 marks) Consider rotation around the z-axis, X = Q(t)A where Q is the rotation matrix
Q(t) =
< p m a t r i x >
Determine the kinematic variables in material coordinates U, V, as well as in spatial coordinates u, v. Sketch the vector field 𝐯 in the x-y plane.

Added by Christian V.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Chasen Shaw

Step 1

The material coordinates U, V are related to the spatial coordinates u, v by the rotation matrix Q. We can write the relationship as: [U; V] = Q * [u; v; 0] Expanding this equation, we have: U = cos(wt) * u - sin(wt) * v V = sin(wt) * u + cos(wt) * v Show more…

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Consider rotation around the z-axis, X = QtA where Q is the rotation matrix: [cos(wt) -sin(wt) 0] [sin(wt) cos(wt) 0] [0 0 1] Determine the kinematic variables in material coordinates U, V, as well as in spatial coordinates u, v. Sketch the vector field v in the x-y plane.