11 Consider a springmass system as discussed in class modele

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.

Solution

(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