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Diff Eq 3.4

 

<== Differential Equations
Number Question Answer
See the work
1. A mass m = 4 is attached to both a spring with spring constant k = 485 and a dash-pot with damping constant c = 4.

The ball is started in motion with initial position x_0 = 3 and initial velocity v_0 = 3 .

Determine the position function x(t).

x(t)=

Note that, in this problem, the motion of the spring is underdamped, therefore the solution can be written in the form x(t) = C_1 e^{-pt} cos(\omega_1 t -  \alpha_1). Determine C_1, \omega_1 ,\alpha_1and p.

C_1 =
\omega_1 =
\alpha_1 =
p =

Graph the function x(t) together with the "amplitude envelope" curves x = - C_1 e^{- p t} and x = C_1 e^{- p t}.

Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected ( so c = 0). Solve the resulting differential equation to find the position function u(t).
In this case the position function u(t) can be written as u(t) = C_0 cos ( \omega_0 t - \alpha_0). Determine C_0, \omega_0 and \alpha_0.

C_0 =
\omega_0 =
\alpha_0 =

Finally, graph both function x(t) and u(t) in the same window to illustrate the effect of damping.

x(t)=[e^(-0.5*t)]*[3*cos(11*t)+(9/22)*sin(11*t)]

C_1 =3.02776

\omega_1 = 11.011

\alpha_1 = 0.135528

p = 1/2

C_0 = 3.0123

\omega_0 = 11.0114

\alpha_0 = .09057
The work

Point Cost: 3
2. This problem is an example of critically damped harmonic motion.
A mass m =8 is attached to both a spring with spring constant k =392 and a dash-pot with damping constant c = 112.

The ball is started in motion with initial position x_0 = 3 and initial velocity v_0 = -25 .

Determine the position function x(t).

  x(t)=

 

 

Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected ( so c = 0). Solve the resulting differential equation to find the position function u(t).
In this case the position function u(t) can be written as u(t) = C_0 cos ( \omega_0 t - \alpha_0). Determine C_0, \omega_0 and \alpha_0.

C_0 =
\omega_0 =
\alpha_0 =

Finally, graph both function x(t) and u(t) in the same window to illustrate the effect of damping.

x(t)=


C_0 = 4.642

\omega_0 = 7

\alpha_0 = 5.41105
The work

Point Cost: 3
3. This problem is an example of over-damped harmonic motion.
A mass m = 4 is attached to both a spring with spring constant k = 112 and a dash-pot with damping constant c = 44.

The ball is started in motion with initial position x_0 = -8 and initial velocity v_0 = 2 .

Determine the position function x(t).

x(t)=

 

Graph the function x(t).

Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected ( so c = 0). Solve the resulting differential equation to find the position function u(t).
In this case the position function u(t) can be written as u(t) = C_0 cos ( \omega_0 t - \alpha_0). Determine C_0, \omega_0 and \alpha_0.

C_0 =
\omega_0 =
\alpha_0 =

Finally, graph both function x(t) and u(t) in the same window to illustrate the effect of damping.

x(t) = -18*[e^(-4*t)]+10*[e^(-7*t)]

C_0 = 8.00892

\omega_0 = 5.2915

\alpha_0 = 3.09438
The work