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A 1 m long metallic rod is clamped at its left end. A longitudinal standing wave is set up in it, described by y = (10⁻⁶ m) sin(pi*x/2), where x is in meters. Find the maximum tensile stress at the midpoint of the rod. (Young's modulus of the rod material Y = 10¹² N/m²)
- 10⁶ * pi/(2*sqrt(2))
- 10⁶ * pi/2
- 10⁶ * pi/4
- 10⁶ * pi/(4*sqrt(2))
Correct answer: 10⁶ * pi/(2*sqrt(2))
Solution
The longitudinal strain at any point x is dy/dx = (10⁻⁶)*(pi/2)*cos(pi*x/2). At the midpoint x = 0.5 m: strain = (10⁻⁶)*(pi/2)*cos(pi/4) = (10⁻⁶)*(pi/2)*(1/sqrt(2)). Stress = Y * strain = 10¹² * 10⁻⁶ * pi/(2*sqrt(2)) = 10⁶ * pi/(2*sqrt(2)).
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