A very short ripple on the surface of water is controlled by surface tension rather…

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Oscillations, Waves & Acoustics · Quantities, units and the practice of measurement

Measurement and Method, Ch What dimensions decide before you measure anything

Problem

A very short ripple on the surface of water is controlled by surface tension rather than gravity. The relevant quantities are judged to be the surface tension

\sigma

(unit

\mathrm{N\,m^{-1}}

), the density of water

\rho_{\rm m}

(unit

\mathrm{kg\,m^{-3}}

), and the wavelength

\lambda

(unit

\mathrm{m}

). ((a)) Write the ripple speed as

v = C\sigma^{\alpha}\rho_{\rm m}^{\beta}\lambda^{\gamma}

with

C

dimensionless, and use dimensional homogeneity to find

\alpha

\beta

and

\gamma

. ((b)) For water at

20\ ^\circ\mathrm{C}

, with

\sigma = 0.0728\ \mathrm{N\,m^{-1}}

and

\rho_{\rm m} = 998\ \mathrm{kg\,m^{-3}}

, and for a ripple of wavelength

\lambda = 2.0\ \mathrm{mm}

, evaluate

v

in terms of

C

. ((c)) Wave theory (not this method) gives

C = \sqrt{2\pi}

for this ripple. State the numerical speed to two significant figures.

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