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<h2 class="hd hd-2 unit-title">Introduction to Phase, Amplitude, and Period</h2>
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<p> In the previous lesson, we derived the simple harmonic oscillator equation for the ideal spring-block system:</p><p>
\[\dfrac{d^2x}{dt^2} = -\omega_0^2 x\]
</p><p> The general solution of this equation was expressed as a linear combination of \(\cos(\omega_0 t)\) and \(\sin(\omega_0 t)\):</p><p>
\[ x(t) = A\cos(\omega_0 t)+B\sin(\omega_0 t)\,\]
</p><p>where \(\omega_0 = \sqrt{\dfrac{k}{m}}\) is the angular frequency of the oscillations, and \(A\) and \(B\) are constants obtained using the initial position, \(x_0\), and the initial velocity \(v_0\).</p><p> In this lesson we will explore a different set of variables that can be used to describe the same motion that might be more or less useful in different contexts. We can think about angular frequency, \(\omega_0\), or we can think about the period, \(T\), over which the motion repeats itself. Instead of writing coefficients to describe the initial conditions, we can think about the amplitude of the oscillation, and where in the cycle the motion started, the phase shift.</p><p> The solution of the equation will be expressed as:</p><p>
\[ x(t) = C \sin(\omega_0 t+ \phi )\]
</p><p> where \(C\) is the amplitude of the oscillations, and \(\phi\) the phase shift. These two constants are determined using the initial conditions of the problem.</p><p> As a final note before we begin, the videos and problems in this lesson will often use slightly different notation from each other. Sometimes we will use \(A\) and \(B\) as the coefficients of the \(\cos(\omega_0 t)\) and \(\sin(\omega_0 t)\), and sometimes \(C\) and \(D\). When writing in terms of amplitude and phase, sometimes the coefficient will be written as \(A\) and other times as \(C\).</p><p> Reading for this Lesson:</p><ul><li> Simple Harmonic Motion - Amplitude and Phase <a href="/courses/course-v1:MITx+8.01.4x+1T2019/pdfbook/0/chapter/23/8"><i class="it"> example 23.1 and 23.2</i></a></li></ul>
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<h2 class="hd hd-2 unit-title">L40Q1: Solution in Terms of Amplitude and Phase Shift</h2>
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Solution in terms of amplitude and phase shift.
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<p>
A box of mass [mathjaxinline]m[/mathjaxinline] is attached to a horizontal and ideal spring of spring constant [mathjaxinline]k[/mathjaxinline]. The origin of the coordinate system is set at the location of the block when the spring is at its equilibrium length. The equation of motion of the box is: </p>
<table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
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<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \dfrac {d^2x}{dt^2}=-\dfrac {k}{m} x[/mathjaxinline]
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<span>(<span>1</span>)</span>
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Consider the box's position to be given by: </p>
<table id="a0000000004" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000005">
<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle x(t) = C\cos (\omega _0 t +\phi )[/mathjaxinline]
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<span>(<span>2</span>)</span>
</td>
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<p>
where [mathjaxinline]\omega _0[/mathjaxinline], [mathjaxinline]C[/mathjaxinline] and [mathjaxinline]\phi[/mathjaxinline] are constant. In the exercise below you will show that Eq. (2) is a solution of Eq. (1). </p>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]\dfrac {dx}{dt}[/mathjaxinline]. Express your answer in terms of [mathjaxinline]t[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], omega_0 for [mathjaxinline]\_ 0[/mathjaxinline], and phi for [mathjaxinline]\phi[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\dfrac {dx}{dt}=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> Calculate [mathjaxinline]\dfrac {d^2x}{dt^2}[/mathjaxinline]. Express your answer in terms of [mathjaxinline]x[/mathjaxinline] as given by (2), [mathjaxinline]t[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], omega_0 for [mathjaxinline]\omega _0[/mathjaxinline], and phi for [mathjaxinline]\phi[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\dfrac {d^2x}{dt^2}=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part c)</b> What is the value of [mathjaxinline]\omega _0[/mathjaxinline] in Eq. (2) so that Eq. (1) is valid for all times? Express your answer in terms of [mathjaxinline]k[/mathjaxinline], [mathjaxinline]m[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], and phi for [mathjaxinline]\phi[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\omega _0=[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">L40Q2: Displacement and Simple Harmonic Motion</h2>
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Displacement in SHM
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<title>CheckPoint: Displacement in Simple Harmonic Motion</title>
<p>Which of the following equations describe simple harmonic motion? <b>Check all that apply.</b> Symbols \(b\), \(c\), and \(d\) stand for positive constants. Symbols \(x\) and \(t\) stand for position and time, respectively. Note that some of these are written in a somewhat non-standard (or overly complicated) manner so you need to look closely at the time dependence in each case.</p>
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\(x(t)=\cos(bt)+ct\)
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\(x(t)=\cos(bt+ct)\)
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\(x(t)=(\cos(bt+ct))/d\)
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\(x(t)=d\cos(bt+ct)\)
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\(x(t)=d\cos(bt+c)\)
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\(x(t)=d\cos(bt/c)\)
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\(x(t)=(d\cos(bt+c))t\)
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\(x(t)=(d\cos(bt+c))/t\)
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<h2 class="hd hd-2 unit-title">L40v1: Simple Harmonic Motion Solution with Amplitude and Phase</h2>
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<h3 class="hd hd-2">L40v1: Simple Harmonic Motion Solution with Amplitude and Phase</h3>
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<h2 class="hd hd-2 unit-title">L40Q3: Displacement and Simple Harmonic Motion</h2>
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<h3 class="hd hd-3 problem-header" id="SHM-005-1b-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@SHM-005-1b-problem-progress" tabindex="-1">
Constants of Integration, Boundary Conditions, SHM
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<p>
A block of mass [mathjaxinline]m=50\text { g}[/mathjaxinline] rests upon a frictionless surface. It is attached to a horizontal spring of unstretched length [mathjaxinline]10\text { cm}[/mathjaxinline] and spring constant [mathjaxinline]k=20 \text { N/m}[/mathjaxinline], with the other end of the spring attached to a wall. The block is initially pulled such that the spring is stretched [mathjaxinline]1\text { cm}[/mathjaxinline] beyond its unstretched length in the positive [mathjaxinline]\hat{\textbf{i}}[/mathjaxinline]-direction. At time [mathjaxinline]t=0\text { s}[/mathjaxinline], the block is released with initial velocity [mathjaxinline]20\text { cm/s }\hat{\textbf{i}}[/mathjaxinline], and the block-spring system undergoes simple harmonic motion. </p>
<p><b class="bfseries">(Part a)</b> We can describe the position of the block as a function of time with the equation </p>
<table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000003">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle x(t) = A\cos {(\omega _0 t)} + B\sin {(\omega _0 t)} \, .[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p>
Determine the values of the natural frequency [mathjaxinline]\omega _0[/mathjaxinline] and the constants of integration [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline]. Enter your numerical answers such that they agree with the units presented below. </p>
<p>
<p style="display:inline">[mathjaxinline]\omega _0=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_SHM-005-1b_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<span class="trailing_text" id="trailing_text_SHM-005-1b_2_1">[mathjaxinline]\text { rad/s}[/mathjaxinline]</span>
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<div id="display_SHM-005-1b_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_SHM-005-1b_2_1_dynamath" name="input_SHM-005-1b_2_1_dynamath"/>
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<p>
<p style="display:inline">[mathjaxinline]A=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_SHM-005-1b_3_1" class="text-input-dynamath capa_inputtype inline textline">
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<span class="trailing_text" id="trailing_text_SHM-005-1b_3_1">[mathjaxinline]\text { cm}[/mathjaxinline]</span>
<span class="status unanswered" id="status_SHM-005-1b_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<div id="display_SHM-005-1b_3_1" class="equation">`{::}`</div>
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<p>
<p style="display:inline">[mathjaxinline]B=[/mathjaxinline]</p>
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<span class="trailing_text" id="trailing_text_SHM-005-1b_4_1">[mathjaxinline]\text { cm}[/mathjaxinline]</span>
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<div id="display_SHM-005-1b_4_1" class="equation">`{::}`</div>
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<p><b class="bfseries">(Part b)</b> We can also describe the position of the block as a function of time with an equation of the form </p>
<table id="a0000000009" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000010">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle x(t) = C\cos {(\omega _0 t + \phi )} \, ,[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p>
where [mathjaxinline]C[/mathjaxinline] is the amplitude and [mathjaxinline]\phi[/mathjaxinline] is the phase constant. </p>
<p>
Determine the values of the amplitude and phase constant for this simple harmonic oscillator. Choose the value of [mathjaxinline]\phi[/mathjaxinline] in [mathjaxinline][-\pi , \pi ][/mathjaxinline]. Enter your numerical answers such that they agree with the units presented below. </p>
<p>
<p style="display:inline">[mathjaxinline]C=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_SHM-005-1b_5_1" class="text-input-dynamath capa_inputtype inline textline">
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<span class="trailing_text" id="trailing_text_SHM-005-1b_5_1">[mathjaxinline]\text { cm}[/mathjaxinline]</span>
<span class="status unanswered" id="status_SHM-005-1b_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<div id="display_SHM-005-1b_5_1" class="equation">`{::}`</div>
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<p>
<p style="display:inline">[mathjaxinline]\phi =[/mathjaxinline]</p>
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<span class="trailing_text" id="trailing_text_SHM-005-1b_6_1">[mathjaxinline]\text { rad}[/mathjaxinline]</span>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<div id="display_SHM-005-1b_6_1" class="equation">`{::}`</div>
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<h2 class="hd hd-2 unit-title">L40Q4: Amplitude Phase and Period in Simple Harmonic Motion</h2>
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Graphical Determination of Amplitude, Phase, Period, Frequency
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The position of a simple harmonic oscillator with respect to time, [mathjaxinline]x(t)[/mathjaxinline], is shown in the graph above. Use the graph to determine each of the following parameters of motion: the period [mathjaxinline]T[/mathjaxinline], the angular frequency [mathjaxinline]\omega _0[/mathjaxinline], the amplitude [mathjaxinline]A[/mathjaxinline], and the phase constant [mathjaxinline]\phi[/mathjaxinline] for the solution [mathjaxinline]x(t) = A\cos ({\omega _0 t + \phi })[/mathjaxinline]. Choose the value of [mathjaxinline]\phi[/mathjaxinline] to be in the range [mathjaxinline]-\pi \le \phi \le \pi[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]T=[/mathjaxinline]</p>
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<span class="trailing_text" id="trailing_text_SHM-005-2_2_1">[mathjaxinline]\text { s}[/mathjaxinline]</span>
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<p style="display:inline">[mathjaxinline]\omega _0 =[/mathjaxinline] </p>
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<span class="trailing_text" id="trailing_text_SHM-005-2_3_1">[mathjaxinline]\text { rad/s}[/mathjaxinline]</span>
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<p style="display:inline">[mathjaxinline]A=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\phi =[/mathjaxinline]</p>
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<span class="trailing_text" id="trailing_text_SHM-005-2_5_1">[mathjaxinline]\text { rad}[/mathjaxinline]</span>
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<h2 class="hd hd-2 unit-title">L40Q5: Angular Frequency and Period</h2>
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Angular Frequency and Period
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A certain object of mass [mathjaxinline]m[/mathjaxinline] experiences a net force that varies with position [mathjaxinline]x[/mathjaxinline] according to [mathjaxinline]F_{\mathrm{net}}(x) = - \dfrac {a b}{H} x[/mathjaxinline], where [mathjaxinline]a[/mathjaxinline], [mathjaxinline]b[/mathjaxinline] and [mathjaxinline]H[/mathjaxinline] are all positive constants. Because the net force is a linear restoring force, this object will undergo simple harmonic motion if displaced from equilibrium. Express your answers to the questions in terms of [mathjaxinline]a[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], [mathjaxinline]H[/mathjaxinline], and [mathjaxinline]m[/mathjaxinline]. </p>
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What is the angular frequency of this object's simple harmonic motion? </p>
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<p style="display:inline">[mathjaxinline]\omega _0 =[/mathjaxinline] </p>
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What is the period of the resulting simple harmonic motion? </p>
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<p style="display:inline">[mathjaxinline]T =[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">L40v3: Graphical Representations of Velocity and Acceleration</h2>
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<h2 class="hd hd-2 unit-title">L40Q6: The Acceleration Plot</h2>
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The Acceleration Plot
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A block of mass [mathjaxinline]m[/mathjaxinline] is attached to a horizontal and ideal spring of spring constant [mathjaxinline]k[/mathjaxinline]. A plot of its x-component of the acceleration as a function of time is shown in the figure above. </p>
<p><b class="bfseries">(Part a)</b> What is [mathjaxinline]\omega _0[/mathjaxinline], the angular frequency of the oscillations. <p style="display:inline">[mathjaxinline]\omega _0=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls39_ls39_09_2_1" class=" capa_inputtype inline textline">
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<span class="trailing_text" id="trailing_text_ls_ls39_ls39_09_2_1">[mathjaxinline]\mathrm{(1/s)}[/mathjaxinline]</span>
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<p><b class="bfseries">(Part b)</b> What is [mathjaxinline]A[/mathjaxinline], the amplitude of the oscillations? (The maximum displacement of the block with respect to the equilibrium position) </p>
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<p style="display:inline">[mathjaxinline]A=[/mathjaxinline]</p>
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<span class="trailing_text" id="trailing_text_ls_ls39_ls39_09_3_1">[mathjaxinline]\mathrm{(cm )}[/mathjaxinline]</span>
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<p><b class="bfseries">(Part c)</b> Write the mathematical expression of [mathjaxinline]x(t)[/mathjaxinline], the position of the block as a function of time. Express your answer in terms of the numerical parameters you found in parts (a) and (b), [mathjaxinline]t[/mathjaxinline], and the corresponding trigonometric function sin( ) or cos( ). </p>
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<p style="display:inline">[mathjaxinline]x(t)=[/mathjaxinline]</p>
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<span class="trailing_text" id="trailing_text_ls_ls39_ls39_09_4_1">[mathjaxinline]\mathrm{(cm)}[/mathjaxinline]</span>
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<div id="display_ls_ls39_ls39_09_4_1" class="equation">`{::}`</div>
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