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Concept# Rotation matrix

Summary

In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin{bmatrix}
\cos \theta & -\sin \theta \

\sin \theta & \cos \theta \end{bmatrix} rotates points in the xy plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates v = (x, y), it should be written as a column vector, and multiplied by the matrix R:# :
R\mathbf{v} = \begin{bmatrix}
\cos \theta & -\sin \theta \

\sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}

\begin{bmatrix}
x\cos\theta-y\sin\theta \
x\sin\theta+y\cos\theta
\end{bmatrix}.
If x and y are the endpoint coordinates of a vector, where x is cosine and y is sine, then the

\sin \theta & \cos \theta \end{bmatrix} rotates points in the xy plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates v = (x, y), it should be written as a column vector, and multiplied by the matrix R:

\sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}

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PHYS-100: Advanced physics I (mechanics)

La Physique Générale I (avancée) couvre la mécanique du point et du solide indéformable. Apprendre la mécanique, c'est apprendre à mettre sous forme mathématique un phénomène physique, en modélisant la situation et appliquant les lois de la physique.

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The presence of aerodynamic vortices is widespread in nature. They can be found at small scales near the wing tip of flying insects or at bigger scale in the form of hurricanes, cyclones or even galaxies. They are identified as coherent regions of high vorticity where the flow is locally dominated by rotation over strain. A better comprehension of vortex dynamics has a great potential to increase aerodynamic performances of moving vehicles, such as drones or autonomous underwater vehicles. An accelerated flat plate, a pitching airfoil or a jet flow ejected from a nozzle give rise to the formation of a primary vortex, followed by the shedding of smaller secondary vortices. We experimentally study the growth, timing and trajectory of primary and secondary vortices generated from a rectangular flat plate that is rotated around its centre location in a quiescent fluid. We systematically vary the rotational speed of the plate to get a chord based Reynolds number \Rey that ranges from 800 to 12000. We identify the critical \Rey for the occurrence of secondary vortices to be at 2500. The timing of the formation of the primary vortex is \Rey independent but is affected by the plate's dimensions. The circulation of the primary vortex increases with the angular position $\alpha$ of the plate, until the plate reaches 30°. Increasing the thickness and decreasing the chord lead to a longer growth of the primary vortex. Therefore, the primary vortex reaches a higher dimensionless limit strength. We define a new dimensionless time $T^*$ based on the thickness of the plate to scale the age of the primary vortex. The primary vortex stops growing when $T^* \approx 10$, regardless of the dimensions of the plate. We consider this value to be the vortex formation number of the primary vortex generated from a rotating rectangular flat plate in a Reynolds number range that goes from 800 to 12000. When $\alpha$ > 30°, the circulation released in the flow is entrained into secondary vortices for $\Rey > 2500$. The circulation of all secondary vortices is approximately 4 to 5 times smaller than the circulation of the primary vortex. We present a modified version of the Kaden spiral that accurately predicts the shear layer evolution and the trajectory of primary and secondary vortices during the entire rotation of the plate.We model the timing dynamics of secondary vortices with a power law equation that depends on two distinct parameter: $\chi$ and $\alpha_{0}$.The parameter $\chi$ indicates the relative increase in the time interval between the release of successive secondary vortices.The parameter $\alpha_{0}$ indicates the angular position at which the primary vortex stops growing and pinches-off from the plate.We also observe that the total circulation released in the flow is proportional to $\alpha^{1/3}$, as predicted by the inviscid theory.The combination of the power law equation with the total circulation computed from inviscid theory predict the strength of primary and secondary vortices, based purely on the plate's geometry and kinematics.The strength prediction is confirmed by experimental measurements.In this thesis we provided a valuable insight into the growth, timing and trajectory of primary and secondary vortices generated by a rotating flat plate. Future work should be directed towards more complex object geometries and kinematics, to confirm the validity of the modified Kaden spiral and explore the influence on the formation number.

We report a state-prepared, state-resolved study of rotational scattering of a diatomic molecule from a solid surface. Specifically, H-2 molecules with 80 meV kinetic energy are rotationally aligned in the j = 3 rotational state via stimulated Raman pumping and then scattered from a Si(100) surface at normal incidence. The rotational alignment of the scattered molecules is determined by measuring, for both the incident and scattered molecules, the ionization yield of a probe laser, tuned to selectively ionize molecules in the j = 3 rotation level, as the probe laser polarization is rotated. The measurement is performed for two initial rotational alignments: a "helicoptering " alignment with the bonds constrained to lie primarily parallel to the surface and a "cartwheeling " alignment with the bonds lying primarily normal to the surface. For both initial alignments, the modulation of the probe ionization yield with laser polarization for the scattered molecules is pronounced, although significantly weaker than the modulation measured for the incident molecules. This indicates a significant modification but not a complete elimination of the initial alignment. The modulation is found to be stronger for the scattered molecules originating in the cartwheeling alignment than for the helicoptering alignment. These results contribute toward an improved understanding of the role of rotational motion in molecule-surface dynamics.

Paul Karoly Otto Ery, Carlotta Guiducci, Kevin Keim, Samuel Kilchenmann, Mohamed Zakarya Alsaid Ali Rashed

This paper reports a method for label-free single-cell biophysical analysis of multiple cells trapped in suspension by electrokinetic forces. Tri-dimensional pillar electrodes arranged along the width of a microfluidic chamber define actuators for single cell trapping and selective release by electrokinetic force. Moreover, a rotation can be induced on the cell in combination with a negative DEP force to retain the cell against the flow. The measurement of the rotation speed of the cell as a function of the electric field frequency define an electrorotation spectrum that allows to study the dielectric properties of the cell. The system presented here shows for the first time the simultaneous electrorotation analysis of multiple single cells in separate micro cages that can be selectively addressed to trap and/or release the cells. Chips with 39 micro-actuators of different interelectrode distance were fabricated to study cells with different sizes. The extracted dielectric properties of HeLa, HEK 293, and human immortalized T lymphocytes cells were found in agreements with previous findings. Moreover, the membrane capacitance of M17 neuroblastoma cells was investigated and found to fall in in the range of 7.49 ± 0.39 mF/m2.

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