Rocking and rolling: a can that appears to rock might actually roll

A beer bottle or soda can on a table, when slightly tipped and released, falls to an upright position and then rocks up to a somewhat opposite tilt. Superficially this rocking motion involves a collision when the flat circular base of the container slaps the table before rocking up to the opposite tilt. A keen eye notices that the after-slap rising tilt is not generally just diametrically opposite the initial tilt but is veered to one side or the other. Cushman and Duistermaat (2006) recently noticed such veering when a flat disk with rolling boundary conditions is dropped nearly flat. Here, we generalize these rolling disk results to arbitrary axi-symmetric bodies and to frictionless sliding. More specifically, we study motions that almost but do not quite involve a face-down collision of the round container's bottom with the table-top. These motions involve a sudden rapid motion of the contact point around the circular base. Surprisingly, like for the rolling disk, the net angle of motion of this contact point is nearly independent of initial conditions. This angle of turn depends simply on the geometry and mass distribution but not on the moment of inertia about the symmetry axis. We derive simple asymptotic formulas for this "angle of turn" of the contact point and check the result with numerics and with simple experiments. For tall containers (height much bigger than radius) the angle of turn is just over $\pi$ and the sudden rolling motion superficially appears as a nearly symmetric collision leading to leaning on an almost diametrically opposite point on the bottom rim.Comment: 10 pages, 5 figure

Bounds for the Multiplicity of Gorenstein algebras

We prove upper bounds for the Hilbert-Samuel multiplicity of standard graded Gorenstein algebras. The main tool that we use is Boij-S\"oderberg theory to obtain a decomposition of the Betti table of a Gorenstein algebra as the sum of rational multiples of symmetrized pure tables. Our bound agrees with the one in the quasi-pure case obtained by Srinivasan [J. Algebra, vol.~208, no.~2, (1998)]

Walking, Running, and Resting Under Time, Distance and Average Speed Constraints: Optimality of Walk-Run-Rest Mixtures

On a treadmill, humans switch from walking to running beyond a characteristic transition speed. Here, we study human choice between walking and running in a more ecological (non-treadmill) setting. We asked subjects to travel a given distance overground in a given allowed time duration. During this task, the subjects carried, and could look at, a stop-watch that counted down to zero. As expected, if the total time available were large, humans walk the whole distance. If the time available were small, humans mostly run. For an intermediate total time, humans often use a mixture of walking at a slow speed and running at a higher speed. With analytical and computational optimization, we show that using a walk-run mixture at intermediate speeds and a walk-rest mixture at the lowest average speeds is predicted by metabolic energy minimization, even with costs for transients { a consequence of non-convex energy curves. Thus, sometimes, steady locomotion may not be energy optimal, and not preferred, even in the absence of fatigue. Assuming similar non-convex energy curves, we conjecture that similar walk-run mixtures may be energetically beneficial to children following a parent and animals on long leashes. Humans and other animals might also benefit energetically from alternating between moving forward and standing still on a slow and sufficiently long treadmill

Synthesis of Magnesium Based Nano-composites

Magnesium based nanocomposites are new lightweight and high-performance materials for potential applications in automotive, aerospace, space, electronics, sports and biomedical sectors primarily due to their lower density when compared to aluminum-based materials and steels. Synthesis of magnesium-based materials is relatively challenging and accordingly this chapter explicitly provides an insight into various techniques hitherto devised/adopted by various researcher for synthesizing magnesium based nano-composites (MMNCs). Overall processing of MMNCs often includes combination of primary and secondary processing. Primary processing fundamentally leads to the initial formulation and creation of MMNC ingots by solid, semi-solid or liquid state processing routes. This is followed by secondary processing that includes plastic deformation or severe plastic deformation to alleviate inhomogeneity, clustering of particles and fabrication defects to enhance the properties of the MMNCs. This chapter provides an insight into different fabrication methodologies, their benefits and limitations for MMNCs

Enhancing Fodder Oat (\u3cem\u3eAvena sativa\u3c/em\u3e) Production in Problem Soils Using Phosphate Solubilizing Fungi Isolated from Acid and Salt Affected Soils of India

Tropical and subtropical soils are predominantly acidic, and often extremely phosphorus-deficient with high phosphorus sorption (fixation) capacities. Phosphorus (P) is one of the major essential macronutrients for plants, present at levels of 400–1200 mg/kg of soil. One of the drawbacks of fertilization is that only a fraction of the P added is eventually assimilated by plants, due to high reactivity of soluble phosphate with other elements the rest becomes unavailable to plants by forming complexes with either, Al, Fe, Ca or Mn depending on soil type (Rodríguez and Fraga, 1999). Even though some soils may have high levels of total P, they can still be P-deficient due to low levels of soluble phosphate available to plants. Available P concentrations for maximum pasture production are estimated to be between 20 and 50 µg/g. Because of the spiraling cost of phosphatic fertilizers coupled with low recovery (10- 30%) of phosphorous applied in the field, the developing tropical countries are attempting to utilize their indigenous reactive ground phosphate rock as a cheap alternative (Sabannavar and Lakshman, 2009). Many soil bacteria, Pseudomonas, Bacillus, Burkholderia, Arthrobacter, Alcaligenes, Serratia, Enterobacter, Acinetobacter and Flavobacterium and fungi especially Aspergillus, Penicillium, Trichoderma have the ability to solubilize elemental phosphate (Pi) and make it available to plants. They are used as biofertilizers for supplying the P requirement of the plants. However, scanty information is available on the occurrence of PSMicroorganisms (PSMs) in acid and salt affected soils. The present investigation was aimed to isolate PSMs from acid and salt affected soils that could survive and solubilize insoluble phosphate efficiently in the presence of higher salt concentration and acidic medium so as to obtain efficient isolates for application as a potential biofertilizer in acid and saline or problematic soils

Design and Analysis of Bus Body Right Side Frame

Conventional designs of bus bodies have more weight and which will affect the carrying capacity, performance and resulting higher fuel consumption. This project describes a new approach and specific design procedure for more lightweight bus body design. The purpose of this project is to redesign the parameters like gauges, materials, geometry/shapes, topology variables, weld pitch, joints for improving the structure so that the total weight of the bus is reduced. A light weight design of bus body structure is designed and structural analysis of the body structure under different load conditions is done. The modeling will be done on solidworks 2014 and analysis part will be done on ANSYS 14