A flat rectangular steel shim of length

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Fluid statics and the continuum picture · Solid continuum mechanics and complex fluids

Fluids and Continuum Physics, Ch Surface tension, capillarity and wetting

Problem

L = 50\ \mathrm{mm}

, width

b = 6.0\ \mathrm{mm}

and thickness

t = 0.15\ \mathrm{mm}

is laid gently on a clean water surface and floats, resting in a dimple of the surface. Take steel density

\rho_s = 7800\ \mathrm{kg\,m^{-3}}

and Young modulus

E = 200\ \mathrm{GPa}

; water density

\rho_w = 1000\ \mathrm{kg\,m^{-3}}

, clean-water surface tension

\gamma = 0.072\ \mathrm{N\,m^{-1}}

, soapy-water surface tension

\gamma_s = 0.025\ \mathrm{N\,m^{-1}}

, and

g = 9.8\ \mathrm{m\,s^{-2}}

. The shim stays flat and horizontal, the water surface meets it along its whole perimeter, and where it does so the surface pulls vertically upward with

\gamma

per unit length of contact line. ((a)) Form the Bond number for the shim using its thickness as the length,

Bo = \rho_w g t^{2}/\gamma

, and say which of gravity and surface tension governs the water surface on that scale. ((b)) If the shim were held just below the surface, fully wetted, what upward force would the displaced water provide? Compare it with the shim's weight. ((c)) Find the largest upward force the clean surface can exert along the shim's perimeter and say whether the shim floats. Repeat for soapy water. ((d)) A drop of detergent is placed on the water at one end of the shim only, so that the tension is

\gamma_s

at that end and still

\gamma

at the other. Find the resulting horizontal force on the shim, and say which way it moves and why. ((e)) The shim is fished out, dried, and hung vertically in air from one short end. Find the tensile strain in the steel at the top of the shim.

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