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<h2 class="hd hd-2 unit-title">Introduction to Coupled Oscillators</h2>
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<p>
Thus far, we have undergone an in-depth analysis of simple harmonic oscillators, including damped and driven systems. Now we move from single-oscillator systems, to multi-oscillator systems. </p><p>
We refer to a system of two or more oscillators that are connected to each other as a "coupled" system. The process of analyzing the motion of a coupled system of oscillators is one with which we are familiar—it involves deriving equations of motion, solving for a general solution, and using initial conditions to solve for the time evolution of the system. </p><p>
However, coupled systems involve several equations of motion, which can be cumbersome to work with. Therefore, we introduce "easier" methods to solve these equations of motion, namely using a matrix to describe the coupling. </p><p>
One aspect of coupled systems that is different from single-oscillator systems is that there is more than one mode of natural oscillation. We will learn about the characteristics of these so-called "normal modes" of oscillation for coupled systems of oscillators. </p>
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<h2 class="hd hd-2 unit-title">L6v1: Coupled Oscillator Examples</h2>
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<h3 class="hd hd-2">L6v1: Coupled Oscillator Examples</h3>
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<h2 class="hd hd-2 unit-title">L6v2: Normal Modes</h2>
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<h2 class="hd hd-2 unit-title">L6v3: Identification of More Normal Modes</h2>
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<h3 class="hd hd-2">L6v3: Identification of More Normal Modes</h3>
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<h2 class="hd hd-2 unit-title">L6Q1: Identifying Normal Modes I</h2>
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Identifying Normal Modes - I
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<p><b class="bfseries">(Part a)</b> Consider a system of 3 identical masses, attached in series with 4 identical springs. Select ALL diagrams that represent a normal mode oscillation of the system (not all normal modes are necessarily depicted). The lengths of the red arrows represents the relative amplitude of the motion of each block and the direction of the arrows indicates which masses are moving in the same or opposite directions at any given time. </p>
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<text>a)</text>
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Identifying Normal Modes - II
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<p><b class="bfseries">(Part b)</b> Consider a system of two coupled masses, attached to springs, constrained to move vertically like pistons. The masses are attached to each other by a massless string, which provides approximately constant tension in the direction of the string. Note that the motion constraint means that only the vertical component of the string tension matters. Select ALL diagrams that represent normal mode oscillations of the system (not all normal modes are necessarily depicted). </p>
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<h2 class="hd hd-2 unit-title">L6Q2: Free Parameters</h2>
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Counting Equations and Parameters - I
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Consider the following systems. Identify the number of equations of motion that must be solved, and the number of free parameters that are required to completely solve the motion of all of the masses in each system. </p>
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Note: The "free parameters" are the set of initial conditions that need to be specified in order to exactly describe the motion. </p>
<p><b class="bfseries">(Part a)</b> A system of 3 identical masses, attached in series to 4 identical springs, where the masses can only move horizontally. </p>
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<p style="display:inline">Number of equations of motion:</p>
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<code>2/3</code>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
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<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
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<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
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<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Counting Equations and Parameters - II
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<p><b class="bfseries">(Part b)</b> A system of two coupled masses, attached to springs, constrained to move vertically like pistons. The masses are attached to each other by a massless string, which provides approximately constant tension in the direction of the string. </p>
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<p style="display:inline">Number of equations of motion:</p>
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<p style="display:inline">Number of free parameters:</p>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L6Q3: Determining the Equation of Motion</h2>
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Understanding Force Diagrams
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Consider the following system of two coupled masses [mathjaxinline]m_{1}[/mathjaxinline] and [mathjaxinline]m_{2}[/mathjaxinline], attached to springs with spring constants [mathjaxinline]k_{1}[/mathjaxinline] and [mathjaxinline]k_{2}[/mathjaxinline], constrained to move vertically like pistons. The masses are attached to each other by a massless string, which provides approximately constant tension with magnitude [mathjaxinline]T=\kappa L[/mathjaxinline] in the direction of the string. Assume that at equilibrium the two masses are at the same height. Then, both masses are displaced downward from their equilibrium positions with [mathjaxinline]m_2[/mathjaxinline] being displaced by a larger distance, as shown below. </p>
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<p><b class="bfseries">(Part a)</b> Consider the following pair of force diagrams, which show ALL of the forces on each hanging mass. Each mass experiences four forces: (1) the spring force [mathjaxinline]F_{s}[/mathjaxinline], (2) the tension force along the direction of the string [mathjaxinline]T[/mathjaxinline], (3) the normal force due to the side of the piston [mathjaxinline]F_{N}[/mathjaxinline], which counteracts the horizontal force due to the string, and (4) the force due to gravity [mathjaxinline]F_{g}[/mathjaxinline]. </p>
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<p><b class="bfseries">(Part i)</b> Which forces do you need to consider to determine the rest equilibrium position? Select ALL that apply. </p>
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<text>a) [mathjaxinline]F_{s}[/mathjaxinline]</text>
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<text>b) [mathjaxinline]T[/mathjaxinline]</text>
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<text>c) [mathjaxinline]F_{N}[/mathjaxinline]</text>
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<text>d) [mathjaxinline]F_{g}[/mathjaxinline]</text>
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<p><b class="bfseries">(Part ii)</b> Which forces do you need to consider to determine the equation of motion of the system, when the masses are displaced a small distance from equilibrium? </p>
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<text>a) [mathjaxinline]F_{s}[/mathjaxinline]</text>
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<text>b) [mathjaxinline]T[/mathjaxinline]</text>
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<text>c) [mathjaxinline]F_{N}[/mathjaxinline]</text>
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<text>d) [mathjaxinline]F_{g}[/mathjaxinline]</text>
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Force Components
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<p><b class="bfseries">(Part b)</b> Which of the following is the correct expression for the magnitude of the component of the string tension in the [mathjaxinline]\pm y[/mathjaxinline] direction, [mathjaxinline]|T_{y}|[/mathjaxinline], in terms of the relative displacement of the masses when [mathjaxinline]m_1[/mathjaxinline] and [mathjaxinline]m_2[/mathjaxinline] are displaced by [mathjaxinline]y_1[/mathjaxinline] and [mathjaxinline]y_2[/mathjaxinline], respectively? Define down to be positive so that both [mathjaxinline]y_1[/mathjaxinline] and [mathjaxinline]y_2[/mathjaxinline] are positive quantities and recall that [mathjaxinline]m_2[/mathjaxinline] has been displaced farther as shown in the figure. Use the fact that the relative displacement of the masses is very small compared to their separation [mathjaxinline]L[/mathjaxinline]. </p>
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<text> a) [mathjaxinline]|T_{y}| = T \frac{(y_{2} + y_{1})}{L}[/mathjaxinline]</text>
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<text> b) [mathjaxinline]|T_{y}| = T \frac{(y_{2} - y_{1})}{L}[/mathjaxinline]</text>
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<text> c) [mathjaxinline]|T_{y}| = T \frac{(y_{1} - y_{2})}{L}[/mathjaxinline]</text>
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Equations of Motion
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<p><b class="bfseries">(Part c)</b> What are the equations of motion for each mass in the system? Express your answers in terms of the displacements from the equilibrium position <code>y_1</code> for [mathjaxinline]y_{1}[/mathjaxinline], <code>y_2</code> for [mathjaxinline]y_{2}[/mathjaxinline], <code>m_1</code> for [mathjaxinline]m_{1}[/mathjaxinline], <code>m_2</code> for [mathjaxinline]m_{2}[/mathjaxinline], <code>k_1</code> for [mathjaxinline]k_{1}[/mathjaxinline], <code>k_2</code> for [mathjaxinline]k_{2}[/mathjaxinline], <code>kappa</code> for [mathjaxinline]\kappa[/mathjaxinline], and <code>g</code>. Define the positive direction for displacements and forces to point downward. </p>
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<p style="display:inline">[mathjaxinline]m_{1}\ddot{y}_1=[/mathjaxinline]</p>
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<div id="display_w3_lect_07_01c_2_1" class="equation">`{::}`</div>
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<p style="display:inline">[mathjaxinline]m_{2}\ddot{y}_2=[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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</tr>
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
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<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
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<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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