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3. For the following, choose the best description of the system from the following: Simple Harmonic Motion (SHM) Overdamped (OD) Underdamped (UD) Critically Damped (CD) Beating (B) Resonant (R) Steady-State plus Transient (SST) a. y'' + 4y = 0 b. y'' + (1.8)^2y = cos(2t) c. y'' + 4y = cos(2t) d. y'' + y' + y = 0 e. y'' + y' + y = cos(t) f. y'' + 2y' + y = 0 4. The motion of a force mass-spring system is described by the following IVP: u'' + 9u = cos(3t), u(0) = 0, u'(0) = 0 (a) Explain why you expect resonance to occur. (b) Solve this IVP and sketch the graph of the solution. 5. The motion of a force mass-spring system is described by the following IVP: u'' + (2.8)^2u = cos(3t), u(0) = 0, u'(0) = 0 Explain why you expect the beats phenomenon to occur. (a) Explain why you expect resonance to occur. (b) Solve this IVP and write your solution in the form A sin(?t) sin(?t). (c) Determine the length of the beats and the period of the oscillation. 6. A mass m = 1 is attached to a spring with constant k = 2 and damping constant ?. Determine the value of ? so that the motion is critically damped. 7. The position function of a mass-spring system satisfies the differential equation mx'' + ?x' + kx = cos(?t), x(0) = 0, x'(0) = 0. Assume m = 1 and k = 9. If ? ? 0, the amplitude of the forced oscillation is given by C = 1 / sqrt((9-?^2)^2 + ?^2?^2) Assume ? = 1. Differentiate C to find the value of ? at which practical resonance occurs. Determine the corresponding value of C. V. Laplace Transform 1. Use the definition of the Laplace transform to find F(s) = L{f(t)} for the following functions. (a) f(t) = {3, 0 ? t < 4; 0, 4 ? t < ? (b) f(t) = {0, t < 2; 6, 2 ? t (c) f(t) = {0, t < 2; 5e^-3t, 2 ? t (d) f(t) = {2e^t, t < 1; 2e, 1 ? t

          3. For the following, choose the best description of the system from the following: Simple Harmonic Motion (SHM) Overdamped (OD) Underdamped (UD) Critically Damped (CD) Beating (B) Resonant (R) Steady-State plus Transient (SST)

a. y'' + 4y = 0
b. y'' + (1.8)^2y = cos(2t)
c. y'' + 4y = cos(2t)
d. y'' + y' + y = 0
e. y'' + y' + y = cos(t)
f. y'' + 2y' + y = 0

4. The motion of a force mass-spring system is described by the following IVP:
u'' + 9u = cos(3t), u(0) = 0, u'(0) = 0
(a) Explain why you expect resonance to occur.
(b) Solve this IVP and sketch the graph of the solution.

5. The motion of a force mass-spring system is described by the following IVP:
u'' + (2.8)^2u = cos(3t), u(0) = 0, u'(0) = 0
Explain why you expect the beats phenomenon to occur.
(a) Explain why you expect resonance to occur.
(b) Solve this IVP and write your solution in the form A sin(?t) sin(?t).
(c) Determine the length of the beats and the period of the oscillation.

6. A mass m = 1 is attached to a spring with constant k = 2 and damping constant ?. Determine the value of ? so that the motion is critically damped.

7. The position function of a mass-spring system satisfies the differential equation
mx'' + ?x' + kx = cos(?t), x(0) = 0, x'(0) = 0.
Assume m = 1 and k = 9.
If ? ? 0, the amplitude of the forced oscillation is given by C = 1 / sqrt((9-?^2)^2 + ?^2?^2)
Assume ? = 1. Differentiate C to find the value of ? at which practical resonance occurs.
Determine the corresponding value of C.

V. Laplace Transform
1. Use the definition of the Laplace transform to find F(s) = L{f(t)} for the following functions.
(a) f(t) = {3, 0 ? t < 4; 0, 4 ? t < ?
(b) f(t) = {0, t < 2; 6, 2 ? t
(c) f(t) = {0, t < 2; 5e^-3t, 2 ? t
(d) f(t) = {2e^t, t < 1; 2e, 1 ? t

3. For the following, choose the best description of the system from the following: Simple Harmonic Motion (SHM) Overdamped (OD) Underdamped (UD) Critically Damped (CD) Beating (B) Resonant (R) Steady-State plus Transient (SST)

a. y” + 4y = 0
b. y” + (1.8)^2y = cos(2t)
c. y” + 4y = cos(2t)
d. y” + y' + y = 0
e. y” + y' + y = cos(t)
f. y” + 2y' + y = 0

4. The motion of a force mass-spring system is described by the following IVP:
u” + 9u = cos(3t), u(0) = 0, u'(0) = 0
(a) Explain why you expect resonance to occur.
(b) Solve this IVP and sketch the graph of the solution.

5. The motion of a force mass-spring system is described by the following IVP:
u” + (2.8)^2u = cos(3t), u(0) = 0, u'(0) = 0
Explain why you expect the beats phenomenon to occur.
(a) Explain why you expect resonance to occur.
(b) Solve this IVP and write your solution in the form A sin(?t) sin(?t).
(c) Determine the length of the beats and the period of the oscillation.

6. A mass m = 1 is attached to a spring with constant k = 2 and damping constant ?. Determine the value of ? so that the motion is critically damped.

7. The position function of a mass-spring system satisfies the differential equation
mx” + ?x' + kx = cos(?t), x(0) = 0, x'(0) = 0.
Assume m = 1 and k = 9.
If ? ? 0, the amplitude of the forced oscillation is given by C = 1 / sqrt((9-?^2)^2 + ?^2?^2)
Assume ? = 1. Differentiate C to find the value of ? at which practical resonance occurs.
Determine the corresponding value of C.

V. Laplace Transform
1. Use the definition of the Laplace transform to find F(s) = Lf(t) for the following functions.
(a) f(t) = 3, 0 ? t < 4; 0, 4 ? t < ?
(b) f(t) = 0, t < 2; 6, 2 ? t
(c) f(t) = 0, t < 2; 5e^-3t, 2 ? t
(d) f(t) = 2e^t, t < 1; 2e, 1 ? t

Added by Megan D.

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