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Register a new userTeaching an Old Ball New Tricks: Another Look at Energetics, Motion Detectors and a Bouncing Rubber Ball
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David Marasco
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A bouncing rubber ball under a motion sensor is a classic of introductory physics labs. It is often used to measure the acceleration due to gravity, and can also demonstrate conservation of energy. By observing that the ball rises to a lower height upon each bounce, posing the question what is the main source of energy loss? and requiring students to construct their own measured values for velocity from position data, a rich lab experience can be created that results in good student discussions of proper analysis of data, and implementation of models. The payoff is student understanding that seemingly small differences in definitions can lead to very different conclusions.
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Reversible debuggers and process replay have been developed at least since 1970. This vision enables one to execute backwards in time under a debugger. Two important problems in practice are that, first, current reversible debuggers are slow when reversing over long time periods, and, second, after building one reversible debugger, it is difficult to transfer that achievement to a new programming environment. The user observes a bug when arriving at an error. Searching backwards for the corresponding fault may require many reverse steps. Ultimately, the user prefers to write an expression that will transition to false upon arriving at the fault. The solution is an expression-transition watchpoint facility based on top of snapshots and record/replay. Expression-transition watch- points are implemented as binary search through the timeline of a program execution, while using the snapshots as landmarks within that timeline. This allows for debugging of subtle bugs that appear only after minutes or more of program execution. When a bug occurs within seconds of program startup, repeated debugging sessions suffice. Reversible debugging is preferred for bugs seen only after minutes. This architecture allows for an efficient and easy-to-write snapshot-based reversibe debugger on top of a conventional debugger. The validity of this approach was tested by developing four personalities (for GDB, MATLAB, Perl, and Python), with each personality typically requiring just 100 lines of code.
Nonlinear dynamics of a bouncing ball moving vertically in a gravitational field and colliding with a moving limiter is considered and the Poincare map, describing evolution from an impact to the next impact, is described. Displacement of the table is approximated in one period by four cubic polynomials. Results obtained for this model are used to elucidate dynamics of the standard model of bouncing ball with sinusoidal motion of the limiter.
Some dynamical properties of a bouncing ball model under the presence of an external force modeled by two nonlinear terms are studied. The description of the model is made by use of a two dimensional nonlinear measure preserving map on the variables velocity of the particle and time. We show that raising the straight of a control parameter which controls one of the nonlinearities, the positive Lyapunov exponent decreases in the average and suffers abrupt changes. We also show that for a specific range of control parameters, the model exhibits the phenomenon of Fermi acceleration. The explanation of both behaviours is given in terms of the shape of the external force and due to a discontinuity of the moving walls velocity.
We investigate that the two types of the Q balls explain the baryon asymmetry and the dark matter of the universe in the gauge-mediated supersymmetry breaking. The gauge-mediation type Q balls of one flat direction produce baryon asymmetry, while the new type Q balls of another flat direction become the dark matter. We show that the dark matter new type Q balls are free from the neutron star constraint. n=5 gauge mediation type and n=6 new type Q balls are displayed as an example, where the potential is lifted by the superpotential Phi^n. These dark matter Q balls may be detected by future observations, such as in advanced IceCube-like observations.
In this paper, based on a matrix norm, we first present a ball of separable unnormalized states around the identity matrix for the bipartite quantum system, which is larger than the separable ball in Frobenius norm. Then the proposed ball is used to get not only simple sufficient conditions for the separability of pseudopure states and the states with strong positive partial transposes, but also a separable ball centered at the identity matrix for the multipartite quantum system.
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