Question

A mass weighing 8 lb is attached to a spring hanging from the ceiling, and comes to rest at its equilibrium position. The spring constant is 9 lb/ft and there is no damping.

A. How far (in feet) does the mass stretch the spring from its natural length? L=________ (do not include units).

B. What is the resonance frequency for the system? ?₀= _________(do not include units).

C. At time t=0 seconds, an external force F(t)=2cos(?₀t) is applied to the system (where ?₀ is the resonance frequency from part B). Find the equation of motion of the mass. u(t)=___________

D.The spring will break if it is extended by 5L feet beyond its natural length (where L is the answer in part A). How many times does the mass pass through the equilibrium position traveling downward before the spring breaks? (Count t=0 as the first such time. Remember that the spring is already extended L ft when the mass is at equilibrium. Make the simplifying assumption that the local maxima of u(t) occur at the maxima of its trigonometric part.)_____ times.

Answer 1

Given,

mass weighing, m = 8 lb

spring constant, k = 9 lb/ft

Part (a)

Natural length, L = 8/9

Natural length, L = 0.89 ft

Part (b)

Resonance frequency, w_o = sqrt[k/m] = sqrt[9/8]

Resonance frequency, w_o = 1.061 rad/sec

Part (c)

the equation of motion is given by

8 u" + 9 u = F(t)

F(t) = 2*cos(w_o t)

8 u" + 9 u = 2*cos(w_o t)

solve the homogeneous equation,

(8 D² + 9) u = 0

D = 1.125 i

u(t) = A Cos (1.061 t) + B Sin (1.061 t)

Equation of motion,

u(t) = A Cos (1.061 t) + B Sin (1.061 t) + 1.78*Sin (1.061 t)