11. Consider a spring-mass system as discussed in class, modeled by the equation

Question

11. Consider a spring-mass system as discussed in class, modeled by the equation mu+bu'+ku - 0mbkO A) (3 pts) What additional condition(s) on the parameters produces simple harmonic motion? 19 pts. s es simple harmonic motion? B)C pto) Underwthat conditionlo would the sprig mas ysem e riialy ampet f-Hala B) @ pts) Under what condition(s) would the spring-mass system be critically damped? if vk C) (3 pts) If the mass is pulled down 3 cm and then released, write the initial conditions that would complete the IVP. Assume forces are measured in Newtons (N). 3 D) (5 pts) What value for > 0 produces resonance in the system u" + 16" = 5 cos(pt) ? math! -. Explain why with E) (2 pts) Is there another, linearly independent, forcing function that would also cause resonance in the system?If so, give an example of such a function. F (2 pts) A mass weighing 80 kg stretches a spring 2 m. Assuming Hooke's law, give the spring constant with units G) (2 pts) Which letter represents the damping coefficient?G sa ing2m Assun ingHookeslaw, give the spring constant wit"nits o e po) Wich literrets aine tsuns ifhe far measuement s Give its units if the force measurement is N.

Explanation / Answer

(a) No damping coefficient , b = 0

(b) b2 - 4mk = 0

(c) Pulled down 3 cm down and assuming downward motion as positive

=> u(0) = 3 ; u'(0) = 0

(d) Resonance Frequence = 4

u'' + 16 u = 0

=> u = c1cos(4t) + c2sin(4t)

w0 = w produce resonance => 4

(e) 5sin(4t) might produce similar resonance

(f) mg = kx

=> 80(9.8) = k(2)

=> k = 392 N/m

(g) b , bu' = Force units

u' = Velocity units (m/s)

=> ( N ) = b( m/s )

=> Units of b = Ns/m

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