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Finding a cosine function that models damped harmonic motion

Find and graph a function of the form y = a e c t cos ( ω t ) that models the information given.

  1. a = 20 , c = 0.05 , p = 4
  2. a = 2 , c = 1.5 , f = 3

Substitute the given values into the model. Recall that period is 2 π ω and frequency is ω 2 π .

  1. y = 20 e 0.05 t cos ( π 2 t ) . See [link] .
    Graph of f(t) = 20(e^(-.05t))cos(pi/2 * t), which begins with a high amplitude and slowly decreases.
  2. y = 2 e 1.5 t cos ( 6 π t ) . See [link] .
    Graph of f(t) = 2(e^(-1.5t))cos(6pi * t), which begins with a small amplitude and quickly decreases to almost a straight line.
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The following equation represents a damped harmonic motion model:   f ( t ) = 5 e 6 t cos ( 4 t ) Find the initial displacement, the damping constant, and the frequency.

initial displacement =6, damping constant = -6, frequency = 2 π

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Finding a sine function that models damped harmonic motion

Find and graph a function of the form y = a e c t sin ( ω t ) that models the information given.

  1. a = 7 , c = 10 , p = π 6
  2. a = 0.3 , c = 0.2 , f = 20

Calculate the value of ω and substitute the known values into the model.

  1. As period is 2 π ω , we have
        π 6 = 2 π ω   ω π = 6 ( 2 π )     ω = 12

    The damping factor is given as 10 and the amplitude is 7. Thus, the model is y = 7 e 10 t sin ( 12 t ) . See [link] .

    Graph of f(t) = 7(e^(-10t))sin(12t), which spikes close to t=0 and quickly becomes almost a straight line.
  2. As frequency is ω 2 π , we have
       20 = ω 2 π 40 π = ω

    The damping factor is given as 0.2 and the amplitude is 0.3. The model is y = 0.3 e 0.2 t sin ( 40 π t ) . See [link] .

    Graph of f(t) = .3e^(-.2t)sin(40pi*t), which has a small amplitude but quickly decreases to the appearance of a straight line. The frequency is so high that, in this scaling, the function looks like a solid shape. The zoom in cut out of the graph shows the actual sinusoidal image of the function.
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Write the equation for damped harmonic motion given a = 10 , c = 0.5 , and p = 2.

y = 10 e 0.5 t cos ( π t )

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Modeling the oscillation of a spring

A spring measuring 10 inches in natural length is compressed by 5 inches and released. It oscillates once every 3 seconds, and its amplitude decreases by 30% every second. Find an equation that models the position of the spring t seconds after being released.

The amplitude begins at 5 in. and deceases 30% each second. Because the spring is initially compressed, we will write A as a negative value. We can write the amplitude portion of the function as

A ( t ) = 5 ( 1 0.30 ) t

We put ( 1 0.30 ) t in the form e c t as follows:

0.7 = e c     c = ln .7     c = 0.357

Now let’s address the period. The spring cycles through its positions every 3 seconds, this is the period, and we can use the formula to find omega.

3 = 2 π ω ω = 2 π 3

The natural length of 10 inches is the midline. We will use the cosine function, since the spring starts out at its maximum displacement. This portion of the equation is represented as

y = cos ( 2 π 3 t ) + 10

Finally, we put both functions together. Our the model for the position of the spring at t seconds is given as

y = 5 e 0.357 t cos ( 2 π 3 t ) + 10

See the graph in [link] .

Graph of the function y = -5e^(-.35t)cos(2pi/3 t) + 10 from 0 to 24. It starts out as waves with a high amplitude and decreases to almost a straight line very quickly.
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A mass suspended from a spring is raised a distance of 5 cm above its resting position. The mass is released at time t = 0 and allowed to oscillate. After 1 3 second, it is observed that the mass returns to its highest position. Find a function to model this motion relative to its initial resting position.

y = 5 cos ( 6 π t )

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Finding the value of the damping constant c According to the given criteria

A guitar string is plucked and vibrates in damped harmonic motion. The string is pulled and displaced 2 cm from its resting position. After 3 seconds, the displacement of the string measures 1 cm. Find the damping constant.

The displacement factor represents the amplitude and is determined by the coefficient a e c t in the model for damped harmonic motion. The damping constant is included in the term e c t . It is known that after 3 seconds, the local maximum measures one-half of its original value. Therefore, we have the equation

a e c ( t + 3 ) = 1 2 a e c t

Use algebra and the laws of exponents to solve for c .

  a e c ( t + 3 ) = 1 2 a e c t e c t e 3 c = 1 2 e c t Divide out  a . e 3 c = 1 2   Divide out  e c t . e 3 c = 2 Take reciprocals .

Then use the laws of logarithms.

e 3 c = 2   3 c = ln 2     c = ln 2 3

The damping constant is ln 2 3 .

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Questions & Answers

"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
Karim Reply
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
unknown Reply
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
KARMEL Reply
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
Jhon Reply
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
Practice Key Terms 2

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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