Scaling Relations Illustrated by a Simple Harmonic Oscillator

Consider a simple harmonic oscillator (SHO) such as a mass on a spring, as described above, and imagine shrinking the system in three dimensions. As stated, mal3and KaL, so a)=(K/m)y2al"1.

It is easy to see that the spring constant scales as I if the "spring" is taken a (massless) rod of cross section A and length I described by Young's modulus Y = (F/A)I(AL/L). Under a compressive force F, AL=-(LY/A)F, so that the spring constant K = LY/A , aL.

A more detailed analysis of the familiar coiled spring gives a spring constant K = (ji/32R2)ßsd4/j!, where R and d, respectively, are the radii of the coil and of the wire, ßs is the shear modulus and i the total length of the wire.  This spring constant K scales in three dimensions as I1.

Insight into the typical scaling of other kinetic parameters such as velocity, acceleration, energy density, and power density can be understood by further consideration of a SHO, in operation, as it is scaled to smaller size. A reasonable quantity to hold constant under scaling is the strain, %max/I, where %max is the amplitude of the motion and I is length of the spring. So the peak velocity of the mass vmax= coxmax which is then constant under scaling: vaL°, since coaLT1. Similarly, the maximum acceleration is amax= co2xmsLX, which then scales as aaL'1. (The same conclusion can be reached by thinking of a mass in circular motion. The centripetal acceleration is a = v2lr, where r is the radius of the circular motion of constant speed v.) Thus for the oscillator under isotropic scaling the total energy U = 1¡2 Kxmax2 scales as I3.

In simple harmonic motion, the energy resides entirely in the spring when % = xmax, but has completely turned into kinetic energy at x = 0, a time T/4 later. The spring then has done work Uin a time l/co, so the power P = dU/dt produced by the spring is a w U, which thus scales as I2. Finally, the power per unit volume (power density) scales as IT1. The power density strongly increases at small sizes. These conclusions are generally valid as scaling relations.

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