A Kirkpatrick-Baez X-ray microscope has been developed for use on the Titan laser facility at the Lawrence Livermore National Laboratory in Fast Ignition experiments. It was developed as a broadband alternative to narrow band Bragg crystal imagers for imaging Kα emission from tracer layers. A re-entrant design is employed which allows for alignment from outside the chamber. The mirrors are coated with Pt and operate at a grazing incident angle of 0.5° providing higher resolution than an equal brightness pinhole and sufficient bandwidth to image thermally shifted characteristic Kα emission from heated Cu tracer layers in Fast Ignition experiments. The superpolished substrates (<1 Å rms roughness) had a final visible wavelength roughness of 1.7 Å after coating, and exhibited a reflectivity corresponding to an X-ray wavelength roughness of 7 ± 1 Å. A unique feature of this design is that during experiments, the unfiltered direct signal along with the one-dimensional reflections are retained on the detector in order to enable a live indication of alignment and incident angle. The broad spectral window from 4 to 9 keV enables simultaneous observation of emission from several spectral regions of interest, which has been demonstrated to be particularly useful for cone-wire targets. An experimentally measured resolution of 15 μm has been obtained at the center of the field of view.
We present a pedagogical treatment of a circular cylinder moving over a smooth surface with one point of the cylinder making contact with the surface. We derive the equations of motion using Newton’s laws. The simplicity of this approach makes the results easily within reach of undergraduate students. A careful derivation of the equations is presented first, and then we show how easily one could make an error. We illustrate how instructors could use this calculation to teach students how to detect errors and critically examine assumptions, including those that seem beyond question. Circular motion with no slipping is examined and we demonstrate the extent of possible motions for static friction. Some calculations are listed that instructors can assign to students to teach the points made in this paper.
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