Questions

SMAT011 Week 9 Practice Quiz

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The following second-order linear differential equations model the vibrations of a spring-mass system: Equation (1): 49 𝑦 ″ + 42 𝑦 ′ + 9 𝑦 = 0 Equation (2): 16 𝑦 ″ + 9 𝑦 = 0 Equation (3): 10 𝑦 ″ + 37 𝑦 ′ + 7 𝑦 = 0 Equation (4): 𝑦 ″ + 6 𝑦 ′ + 58 𝑦 = 0 The motion of the spring modeled by Equation (1) is classified as [ Select ] an overdamped motion a simple harmonic motion. an underdamped motion. a critically damped motion . The motion of the spring modeled by Equation (2) is classified as [ Select ] a critically damped motion an overdamped motion a simple harmonic motion an underdamped motion . The motion of the spring modeled by Equation (3) is classified as [ Select ] a simple harmonic motion an overdamped motion a critically damped motion an underdamped motion . The motion of the spring modeled by Equation (4) is classified as [ Select ] an underdamped motion an overdamped motion a simple harmonic motion a critically damped motion .

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When we classify a second-order linear differential equation of the form ay'' + by' + cy = 0, we compare the damping characteristics using the discriminant b^2 - 4ac or, equivalently, by inspecting the roots of the characteristic equation ar^2 + br + c = 0. Real, distinct roots correspond to overdamped motion, a repeated root to critical damping, and complex conjugate roots to underdamped (oscillatory) motion. Oscillatory behavior only occurs if the roots are complex, which happens when b^2 - 4ac < 0......Login to view full explanation

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