dimensionalreasoning：空間推理(存儲版)
【正文】 ing f: c2 = a2 + b2, . Proof of the Pythagorean Theorem Dimensions Scaling, modeling, similarity ? Types of “similarity” between two objects/processes. – Geometric similarity – linear dimensions are proportional。 R。s law of universal gravitation. We could add the mass m to the list, but if we assume that the density is constant, then m = ρ(4πR3/3) and the mass is redundant. Therefore, ω is the governed parameter, with dimensions [ω] = T1, and (ρ。 2c = 1 Solving, we find a = c = 1/2, b = 0, so that ω = C(Gσ)1/2, with C a constant. We see that the frequency of oscillation is proportional to the square root of the density, and independent of the radius. Buckingham Pi Theorem (cont.) Oscillations of a star Dimensions “Dimensionless” Quantities ? Dimensional quantities can be made “dimensionless” by “normalizing” them with respect to another dimensional quantity of the same dimensionality. Example: speed V (m/s) can be made dimensionless“ by dividing by the velocity of sound c (m/s) to obtain M = V/c, a dimensionless speed known as the Mach number. M1 is faster than the speed of sound。G) have independent dimensions。Dimensions Dimensional Reasoning Dimensions Dimensions and Measurements ? “Dimension” is characteristic of the object, condition, or event and is described quantitatively in terms of defined “units”. ? A physical quantity is equal to the product of two elements: – A quality or dimension – A quantity expressed in terms of “units” ? Dimensions – Physical things are measurable in terms of three primitive qualities (Maxwell 1871) ? Mass (M) ? Length (L) ? Time (T) Note: (Temperature, electrical charge, chemical quantity, and luminosity were added as “primitives” some years later.)
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