This HTML version of the book is provided as a convenience, but some math equations are not translated correctly. The PDF version is more reliable. Chapter 14 Final Thoughts14.1 Dot and cross productsMultiplying a vector by a scalar is a straightforward operation; so is adding two vectors. But multiplying two vectors is more subtle. It turns out that there are two vector operations that resemble multiplication: the dot product and the cross product. The dot product of vectors u and v is a scalar:
where θ is the smallest angle between u and v. We already know how to compute magnitudes, and you could probably figure out how to compute θ, but you don’t have to. MATLAB provides a function, dot, that computes dot products: d = dot(U, V) dot works in any number of dimensions, as long as U and V have the same number of elements. If one of the operands is a unit vector, you can use the dot product to compute the component of a vector u that is in the direction of a unit vector, r: s = dot(U, Rhat) In this example, s is the scalar projection of u onto r. The vector projection is the vector that has magnitude s in the direction of r: V = dot(U, Rhat) * Rhat The cross product of vectors u and v is a vector whose direction is perpendicular to u and v (with orientation determined by the right-hand rule^{1}) and whose magnitude is
where (again) θ is the smallest angle between u and v. MATLAB provides a function, cross, that computes cross products. C = cross(U, V) cross calculates the cross product of corresponding vectors along the first array dimension whose size equals 3. A common use of cross is to compute torques. If you represent a moment arm R and a force F as vectors with size equal to 3, then the torque is just Tau = cross(R, F) If the components of R are in meters and the components of F are in Newtons, then the torques in Tau are in Newton-meters. 14.2 What is a model for?In Section 8.2 I defined a “model” as a simplified description of a physical system, and said that a good model lends itself to analysis and simulation, and makes predictions that are good enough for the intended purpose. Since then, we have seen a number of examples; now we can say more about what models are for. The goals of a model tend to fall into three categories.
The exercises at the end of this chapter include one model of each type. 14.3 Glossary
14.4 ExercisesExercise 1
If you put two identical bowls of water into a freezer, one at
room temperature and one boiling, which one freezes first? Hint: you might want to do some research on the Mpemba effect. Exercise 2
You have been asked to design a new skateboard ramp; unlike a typical
skateboard ramp, this one is free to pivot about a support point.
Skateboarders approach the ramp on a flat surface and then coast up
the ramp; they are not allowed to put their feet down while on the
ramp. If they go fast enough, the ramp will rotate and they will
gracefully ride down the rotating ramp. Technical and artistic
display will be assessed by the usual panel of talented judges. Your job is to design a ramp that will allow a rider to accomplish this feat, and to create a physical model of the system, a simulation that computes the behavior of a rider on the ramp, and an animation of the result. Exercise 3
A binary star system contains two stars orbiting each other and sometimes planets that orbit one or both stars.^{2} In a binary system, some orbits are “stable” in the sense that a planet can stay in orbit without crashing into one of the stars or flying off into space. Simulation is a useful tool for investigating the nature of these orbits, as in Holman, M.J. and P.A. Wiegert, 1999, “Long-Term Stability of Planets in Binary Systems,” Astronomical Journal 117, available from http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.255.4314. Read this paper and then modify your planetary simulation to replicate or extend the results. |
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