Exercises

(1)
A large undamped (vertical) spring-mass system begins (at time \(t=0\)) at rest at equilibrium. Then the mass is constantly forced upward by a force of magnitude \(12\) Newtons. Suppose the mass is \(2\) kg and the spring stiffness is \(8\) Newtons per meter. Find the vertical displacement of the mass (in meters) at time \(t=10\) seconds. Round to the nearest hundredth of a meter.
(2)
Which CANNOT be displacement of the mass above equilibrium for a free damped spring-mass system?
(a)
\(y=(1+t)e^{-t/2}\)
(b)
\(y=2e^{-t}-e^{-10t}\)
(c)
\(y=e^{-0.2t}\cos {(\pi t)}\)
(d)
\(y=e^{-t/10}-10e^{-t/100}\)
(e)
\(y=\dfrac {\cos {(2\pi t)}-\sin {(2\pi t)}}{e^{t}}\)
(f)
\(y=\dfrac {1+t}{e^{1-t/5}}\)
(3)
Each curve below is the graph of the displacement above equilibrium of a mass in a spring-mass system. Match each curve to the types of spring-mass systems below.
Some solution curves to spring-mass systems. [Picture]
Figure 1: Some solution curves to spring-mass systems
(a)
A simple harmonic oscillator
(b)
A free underdamped spring-mass system
(c)
A free critically damped spring-mass system
(d)
A forced undamped spring-mass system in resonance
(e)
A forced and underdamped spring-mass system
(4)
A small object of mass 1 kg is attached to an elastic spring of stiffness \(1\) N/m and is immersed in a viscous medium that resists the motion of the mass with 2 Newtons of force per m/s of speed. Suppose that at a certain instant the spring is \(0.25\) meters below equilibrium and moving upward at \(1\) meter per second. How many times does the mass cross through equilibrium thereafter?
(a)
Zero times
(b)
One time
(c)
Two times
(d)
Three times
(e)
Infinitely-many times
(5)
The gun of a U.S. M60 tank is attached to spring-mass system with a \(100\) kg mass and a dashpot damper. The spring stiffness is engineered to equal \(\dfrac {c^2}{400}\) where \(c\) is the damping constant of the dashpot damper. Assume that when the gun is fired, the mass is at equilibrium with a velocity of \(100\) m/s. It is desired that one second later, the quantity \(y^2+(y\,^{\prime })^2\) be no greater than \(0.01\). What is the minimum possible value of \(c\) in \(N\,s/m\)? Report the answer with 2 significant figures (e.g. \(890\) for \(893\)).

Hint: use a numerical inequality solver (e.g. Wolfram Alpha)