An aluminium rod of circular cross-section

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

Bending, Ringing, Breaking and Flowing, Ch Elastic waves in a solid and at its surface

Problem

An aluminium rod of circular cross-section, radius

a = 6.00\,\mathrm{mm}

, length

L = 1.20\,\mathrm{m}

and density

\rho = 2700\,\mathrm{kg\,m^{-3}}

, is to be used as a pin-ended column (effective-length factor

K_e = 1

). Its Young's modulus is found by sending a longitudinal ultrasonic pulse along the rod's axis and timing the echo from the far end: the round-trip time is

t = 471\,\mathrm{\mu s}

. Because the rod's diameter is far smaller than the pulse wavelength, the measured speed is the thin-bar speed

c_{\rm bar} = \sqrt{E/\rho}

. ((a)) Find

E

from the echo time. ((b)) Find the Euler buckling load

P_{\rm cr} = \pi^2EI/(K_eL)^2

of the rod, with

I = \pi a^4/4

the second moment of area of the circular cross-section.

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