FYKOS Year 28 Series 5 Problem 1 — stiffness of Mr. Planck · FYKOS 28 #28.5.1

Physics · mechanics · optics

Problem

Maybe you have heard about the so called Planck's units ie. units expressed in the form of fundamental physics constants – speed of light

c≈3.00\cdot 10^{8}\;\mathrm{m}\cdot \mathrm{s}^{−1}

, gravitational&nbspconstant

G=6.67\cdot 10^{-11}\;\mathrm{m}\cdot \mathrm{kg}^{−1}\cdot \mathrm{s}^{−2}

and the reduced Planck's constant

h=1.05\cdot 10&nbsp^{-34}\;\mathrm{kg}\cdot \mathrm{m}\cdot \mathrm{s}^{−1}

. This way Planck's time, Planck's length and Planck's weight are often mentioned. What if we were interested in „Planck's spring constant“? Using dimensional analysis with

c

G

and

h

the equation of the unit relating to the unit of a spring constant [

k]=\;\mathrm{kg}\cdot \mathrm{s}^{−2}

. To determine the equation assume that the unknown and from dimensional analysis undeterminable dimensionless constant is equal to 1.

Intuitive explanation

Planck's spring constant asks: what combination of the three fundamental constants $c$ , $G$ , and $\hbar$ has the dimensions of stiffness $[\mathrm{kg·s^{-2}}]$ ? Use dimensional analysis by writing $k_P = c^a G^b \hbar^c$ and matching powers of mass, length, and time on both sides to find the exponents $a$ , $b$ , and $c$ . This technique—treating dimensions as algebraic quantities—is your key tool for building any physical quantity from fundamental constants.

Intuitive explanation

Related theory

optics

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