query 21

#$&*

course Mth 279

4/12 6

Query 21 Differential Equations

********************************************* Question: A 10 kg mass stretches a spring 9.8 cm beyond its original rest position. A driving force F(t) = 20 N * cos((8 s^-1) * t) begins at t = 0, where the downward direction is regarded as positive. Write down and solve the appropriate differential equation, obtaining the position function for the motion of the mass. Plot your solution, and find the maximum distance of the mass from its equilibrium position. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To find k we use the fact that m*g = k*`ds, so k = m*g/`ds = 98(kg*m/s^2)/.98m = 100 kg/s^2 So we have y’’ + 100y = 20 cos(8*t), y(0) = 0, y’(0) = 0 Complimentry solution is y_c(t) = c1sin(10*t) + c2 cos(10*t) Particular solution is going to be y_p(t) = A cos(8*t) + B sin(8*t) Plugging into our equation we get y’’ + 100y = 20 cos(8*t) ( A cos(8*t) + B sin(8*t) )’’ + 100(A cos(8*t) + B sin(8*t) ) = 20 cos(8s^-1 t) -64Acos(8*t) - 64Bsin(8t) + 100A cos(8t) + 100B sin(8t) = 20 cos(8t) Coefficants give us 36A = 20 36B = 0 A= 5/9, B=0, where A = 5(kg*m)/9 Imposing init cond y(t) = c1sin(10*t) + c2 cos(10*t) + 5/9 cos(8*t), at t=0 y(t) = c2 cos(10*t) + 5/9 cos(8*t) confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating:

@& Good solution.*@

********************************************* Question: The motion of a mass is governed by the equation m y '' + 2 delta y ' + omega_0^2 y = F(t), with m = 2 kg, gamma = 8 kg / s and k = 80 N / m and F(t) = 20 N * e^(- t s^-1). Solve the equation for the function y(t). What is the long-term behavior of this system? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating:

********************************************* Question: Solve the equation y '' + 2 delta y ' + omega_0^2 y = F cos( omega_1 * t), y(0) = 0, y ' (0) = 0. Give an outline of your work. A very similar problem was set up and partially solved in class on 110309, and your text gives the solution but not the steps. Find the limiting function as omega_1 approaches omega_0, and discuss what this means in terms of a real system. Find the limiting function as delta approaches 0, and discuss what this means in terms of a real system. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating:

********************************************* Question: An LC circuit with L = 1 Henry and C = 4 microFarads is driven by voltage V_S(t) = 10 t e^(-t). Write and solve the differential equation for the system. Interpret your result. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating:

query 21

#$&*

course Mth 279

4/12 6

Query 21 Differential Equations

*********************************************

Question: A 10 kg mass stretches a spring 9.8 cm beyond its original rest position.

A driving force F(t) = 20 N * cos((8 s^-1) * t) begins at t = 0, where the downward direction is regarded as positive.

Write down and solve the appropriate differential equation, obtaining the position function for the motion of the mass.

Plot your solution, and find the maximum distance of the mass from its equilibrium position.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

To find k we use the fact that m*g = k*`ds, so k = m*g/`ds = 98(kg*m/s^2)/.98m = 100 kg/s^2

So we have

y’’ + 100y = 20 cos(8*t), y(0) = 0, y’(0) = 0

Complimentry solution is

y_c(t) = c1sin(10*t) + c2 cos(10*t)

Particular solution is going to be

y_p(t) = A cos(8*t) + B sin(8*t)

Plugging into our equation we get

y’’ + 100y = 20 cos(8*t)

( A cos(8*t) + B sin(8*t) )’’ + 100(A cos(8*t) + B sin(8*t) ) = 20 cos(8s^-1 t)

-64Acos(8*t) - 64Bsin(8t) + 100A cos(8t) + 100B sin(8t) = 20 cos(8t)

Coefficants give us

36A = 20

36B = 0

A= 5/9, B=0, where A = 5(kg*m)/9

Imposing init cond

y(t) = c1sin(10*t) + c2 cos(10*t) + 5/9 cos(8*t), at t=0

y(t) = c2 cos(10*t) + 5/9 cos(8*t)

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

@& Good solution.*@

*********************************************

Question: The motion of a mass is governed by the equation

m y '' + 2 delta y ' + omega_0^2 y = F(t),

with m = 2 kg, gamma = 8 kg / s and k = 80 N / m and F(t) = 20 N * e^(- t s^-1).

Solve the equation for the function y(t).

What is the long-term behavior of this system?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

*********************************************

Question:

Solve the equation

y '' + 2 delta y ' + omega_0^2 y = F cos( omega_1 * t), y(0) = 0, y ' (0) = 0.

Give an outline of your work. A very similar problem was set up and partially solved in class on 110309, and your text gives the solution but not the steps.

Find the limiting function as omega_1 approaches omega_0, and discuss what this means in terms of a real system.

Find the limiting function as delta approaches 0, and discuss what this means in terms of a real system.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

*********************************************

Question: An LC circuit with L = 1 Henry and C = 4 microFarads is driven by voltage V_S(t) = 10 t e^(-t).

Write and solve the differential equation for the system.

Interpret your result.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

query 21

#$&*

course Mth 279

4/12 6

Query 21 Differential Equations

*********************************************

Question: A 10 kg mass stretches a spring 9.8 cm beyond its original rest position.

A driving force F(t) = 20 N * cos((8 s^-1) * t) begins at t = 0, where the downward direction is regarded as positive.

Write down and solve the appropriate differential equation, obtaining the position function for the motion of the mass.

Plot your solution, and find the maximum distance of the mass from its equilibrium position.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

To find k we use the fact that m*g = k*`ds, so k = m*g/`ds = 98(kg*m/s^2)/.98m = 100 kg/s^2

So we have

y’’ + 100y = 20 cos(8*t), y(0) = 0, y’(0) = 0

Complimentry solution is

y_c(t) = c1sin(10*t) + c2 cos(10*t)

Particular solution is going to be

y_p(t) = A cos(8*t) + B sin(8*t)

Plugging into our equation we get

y’’ + 100y = 20 cos(8*t)

( A cos(8*t) + B sin(8*t) )’’ + 100(A cos(8*t) + B sin(8*t) ) = 20 cos(8s^-1 t)

-64Acos(8*t) - 64Bsin(8t) + 100A cos(8t) + 100B sin(8t) = 20 cos(8t)

Coefficants give us

36A = 20

36B = 0

A= 5/9, B=0, where A = 5(kg*m)/9

Imposing init cond

y(t) = c1sin(10*t) + c2 cos(10*t) + 5/9 cos(8*t), at t=0

y(t) = c2 cos(10*t) + 5/9 cos(8*t)

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

@& Good solution.*@

*********************************************

Question: The motion of a mass is governed by the equation

m y '' + 2 delta y ' + omega_0^2 y = F(t),

with m = 2 kg, gamma = 8 kg / s and k = 80 N / m and F(t) = 20 N * e^(- t s^-1).

Solve the equation for the function y(t).

What is the long-term behavior of this system?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

*********************************************

Question:

Solve the equation

y '' + 2 delta y ' + omega_0^2 y = F cos( omega_1 * t), y(0) = 0, y ' (0) = 0.

Give an outline of your work. A very similar problem was set up and partially solved in class on 110309, and your text gives the solution but not the steps.

Find the limiting function as omega_1 approaches omega_0, and discuss what this means in terms of a real system.

Find the limiting function as delta approaches 0, and discuss what this means in terms of a real system.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

*********************************************

Question: An LC circuit with L = 1 Henry and C = 4 microFarads is driven by voltage V_S(t) = 10 t e^(-t).

Write and solve the differential equation for the system.

Interpret your result.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique rating: