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Simple harmonic motion

A type of motion described as simple harmonic motion    involves a restoring force but assumes that the motion will continue forever. Imagine a weighted object hanging on a spring, When that object is not disturbed, we say that the object is at rest, or in equilibrium. If the object is pulled down and then released, the force of the spring pulls the object back toward equilibrium and harmonic motion begins. The restoring force is directly proportional to the displacement of the object from its equilibrium point. When t = 0 , d = 0.

Simple harmonic motion

We see that simple harmonic motion    equations are given in terms of displacement:

d = a cos ( ω t )   or   d = a sin ( ω t )

where | a | is the amplitude, 2 π ω is the period, and ω 2 π is the frequency, or the number of cycles per unit of time.

Finding the displacement, period, and frequency, and graphing a function

For the given functions,

  1. Find the maximum displacement of an object.
  2. Find the period or the time required for one vibration.
  3. Find the frequency.
  4. Sketch the graph.
    1. y = 5 sin ( 3 t )
    2. y = 6 cos ( π t )
    3. y = 5 cos ( π 2 t )
  1. y = 5 sin ( 3 t )
    1. The maximum displacement is equal to the amplitude, | a | , which is 5.
    2. The period is 2 π ω = 2 π 3 .
    3. The frequency is given as ω 2 π = 3 2 π .
    4. See [link] . The graph indicates the five key points.
      Graph of the function y=5sin(3t) from 0 to 2pi/3. The five key points are (0,0), (pi/6, 5), (pi/3, 0), (pi/2, -5), (2pi/3, 0).
  2. y = 6 cos ( π t )
    1. The maximum displacement is 6.
    2. The period is 2 π ω = 2 π π = 2.
    3. The frequency is ω 2 π = π 2 π = 1 2 .
    4. See [link] .
      Graph of the function y=6cos(pi t) from 0 to 3.
  3. y = 5 cos ( π 2 ) t
    1. The maximum displacement is 5.
    2. The period is 2 π ω = 2 π π 2 = 4.
    3. The frequency is 1 4 .
    4. See [link] .
      Graph of the function y=5cos(pi/2 t) from 0 to 4.
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Damped harmonic motion

In reality, a pendulum does not swing back and forth forever, nor does an object on a spring bounce up and down forever. Eventually, the pendulum stops swinging and the object stops bouncing and both return to equilibrium. Periodic motion in which an energy-dissipating force, or damping factor, acts is known as damped harmonic motion    . Friction is typically the damping factor.

In physics, various formulas are used to account for the damping factor on the moving object. Some of these are calculus-based formulas that involve derivatives. For our purposes, we will use formulas for basic damped harmonic motion models.

Damped harmonic motion

In damped harmonic motion    , the displacement of an oscillating object from its rest position at time t is given as

f ( t ) = a e c t sin ( ω t ) or   f ( t ) = a e c t cos ( ω t )

where c is a damping factor, | a | is the initial displacement and 2 π ω is the period.

Modeling damped harmonic motion

Model the equations that fit the two scenarios and use a graphing utility to graph the functions: Two mass-spring systems exhibit damped harmonic motion at a frequency of 0.5 cycles per second. Both have an initial displacement of 10 cm. The first has a damping factor of 0.5 and the second has a damping factor of 0.1.

At time t = 0 , the displacement is the maximum of 10 cm, which calls for the cosine function. The cosine function will apply to both models.

We are given the frequency f = ω 2 π of 0.5 cycles per second. Thus,

   ω 2 π = 0.5      ω = ( 0.5 ) 2 π         = π

The first spring system has a damping factor of c = 0.5. Following the general model for damped harmonic motion, we have

f ( t ) = 10 e 0.5 t cos ( π t )

[link] models the motion of the first spring system.

Graph of the first spring system, f(t) = 10(e^(-.5t))cos(pi*t), which begins with a high amplitude and quickly decreases.

The second spring system has a damping factor of c = 0.1 and can be modeled as

f ( t ) = 10 e 0.1 t cos ( π t )

[link] models the motion of the second spring system.

Graph of f(t) = 10(e^(-.1t))cos(pi*t), which begins with a high amplitude and slowly decreases (but has a high frequency).
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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
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
strategies to form the general term
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
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
y=x will obviously be a straight line with a zero slope
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
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.
yes, correction on my end, I meant slope of 1 instead of slope of 0
what is f(x)=
Karim Reply
I don't understand
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."
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Now it shows, go figure?
what is this?
unknown Reply
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I've been struggling so much through all of this. my final is in four weeks 😭
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
thank you I have heard of him. I should check him out.
is there any question in particular?
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Sure, are you in high school or college?
Hi, apologies for the delayed response. I'm in college.
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?
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?
I done know
What kind of answer is that😑?
I had just woken up when i got this message
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
i have a question.
how do you find the real and complex roots of a polynomial?
@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
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
@Nare please let me know if you can solve it.
I have a question
hello guys I'm new here? will you happy with me
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
if not then how would I find it from a graph
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.
you could also do it with two consecutive minimum points or x-intercepts
I will try that thank u
Case of Equilateral Hyperbola
Jhon Reply
f(x)=4x+2, find f(3)
f(3)=4(3)+2 f(3)=14
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
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
Practice Key Terms 2

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