Review Guide for Exam 3, Page 1 of 6

ST UD Y G UID E F O R E X A M 3 M ASS-SPRIN G-D A MPE R M O D E LS, L APL A C E

1. Suppose a 4-lb weight stretches a spring 6 inches beyond its natural length. An

external force equal to t8cos21 is acting on the spring. If the weight is started in

motion from its equilibrium position with an upward velocity of 4 ft/sec and there is no damping, set up a initial value problem whose solution will give an equation of the motion of the mass + spring.

2. (#9, Sec. 3.8) A mass of 20 g stretches a spring 5 cm. Suppose that the mass is also

attached to a viscous damper with a damping constant of 400 dyne-sec/cm. If the mass is pulled down an additional 2 cm and then released, set up an initial value problem to model the motion of the mass + spring.

Review Guide for Exam 3, Page 2 of 6

3. Suppose the solution of a vibrating model (unforced) is tttu 4sin24cos2)( . a) Describe the type of motion of the system. b) Put the solution in the amplitude-phase angle form. c) Identify the period and (circular) frequency of the motion.

d) If the mass attached to the spring is 4 kilograms, what is the spring constant of the spring?

4. Classify the type of motion of each equation below. a) tectectu tt 4sin4cos)( 2

22

1 _____________________________ b) tt tecectu 3

23

1)( __________________________________ c) tt ecectu 2

23

1)( __________________________________ d) ttctctu 4sin2sin2cos)( 2

121 ______________________________

5. Suppose a mass-spring-damper system is modeled by tuuu sin14'9'' 2

1 , u(0) = 0, u (0) = 1. If the specific solution is tteetu tt cos02.sin03.2.18.)( 72 , identify the steady-state portion of the solution and rewrite it in amplitude-phase angle form.

Review Guide for Exam 3, Page 3 of 6

6. a) Convert )()()1()()1(2)( 321 tututtutttf to a piece-wise function and sketch.

b) Find L(f(t))

7. The current flow of a circuit can be modeled by the equation )(tVdtdILRI . If R =

2, L = 1, and V(t) = te 3 , use Laplace transforms to find the zero-state output of the circuit.

Review Guide for Exam 3, Page 4 of 6

8. Use Laplace transforms to find the zero-state output of the mass-damper system modeled by tetuuu 3'2'' .

9. Find the Laplace or inverse Laplace transforms as indicated. a) )2cos3( ttL b) )2cos( 3 te tL

c) sss 23

1

22

L

Review Guide for Exam 3, Page 5 of 6

9. continued

d) t

d0

2cosL

e) t

t de0

3 2cosL

f) )(2

3 tue tL

g) 4

12

11

ssL Evaluate as a convolution.

h) 542

1

sss

L

i) 542

21

sse s

L

Review Guide for Exam 3, Page 6 of 6

10. Find the Laplace transform of the following periodic function.