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<h2 class="hd hd-2 unit-title">Intro to Pulleys and Constraints </h2>
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<p> This lesson covers advanced techniques to deal with complex pulley systems, especially those including two or more constraints. As with the rest of this week's content, the lesson focuses more on building problem solving skills with solved examples and longer exercises.
</p><p> As you go through the lesson, you will find this list of reading from the textbook complementary to the videos and exercises, and on many occasions these readings will contain relevant material to the presentation:</p><ul><li> Worked Example on Pulleys and Ropes Constraints Conditions: <a href="/courses/course-v1:MITx+8.01.1x+3T2018/pdfbook/0/chapter/8/39"><i class="it">chapter 8, example 8.9</i></a></li></ul>
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<h2 class="hd hd-2 unit-title">L12v1: Pulley Problems - Part I, Set up the Equations</h2>
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<p><b>L12WE1: Worked Example: Pulley Problem. </b></p><p> In the figure, pulley A is fixed to the ceiling. An ideal rope is attached at one end
to block 1 of mass \(m_1\), it passes around a second pulley, labelled B, and its other end is fixed to the
ceiling. A second ideal rope attaches pulley B to a second block of mass \(m_2\). As a result, pulley B and block 2 move together.
</p><p> The goal of the problem is to calculate the accelerations of blocks 1 and 2.</p><p><img src="/assets/courseware/v1/b8c847c2f85369ee7582b40a0eea06e6/asset-v1:MITx+8.01.1x+3T2018+type@asset+block/images_ropes_and_pulley_intro.svg" align="middle" height="300" width="300"/></p><p>The solution of this problem is divided into four parts:
<ul><li> Part I : Set up the system of equations.</li><li> Part II: Constraint condition - find the relationship between the accelerations.</li><li> Part III: Constraint condition using a virtual displacement argument.</li><li> Part IV: Solving the system of equations.</li></ul>
</p><p> The video below will help you to set up the equations needed to answer the question. </p>
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<h3 class="hd hd-2">L12v1: Pulley Problems</h3>
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<h2 class="hd hd-2 unit-title">L12v2: Pulley Problem - Part II, Constraint Condition</h2>
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<h3 class="hd hd-2">L12v2: Constraint Condition</h3>
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<h2 class="hd hd-2 unit-title">L12v3: Pulley Problem - Part III, Constraints and Virtual Displacement Arguments</h2>
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<h2 class="hd hd-2 unit-title">L12v4: Pulley Problem - Part IV, Solving the System of Equations</h2>
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<h2 class="hd hd-2 unit-title">L12WE2: Three Pulleys</h2>
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<p><b>L12WE2: Three Pulleys</b></p><p> The following is another worked example of a complex pulley system. Feel free to try the problem yourself or hit "Check" once to be able to see the answer.</p>
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L12WE2: Three Pulleys (Worked Example)
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<p>
The configuration shown in the figure consists of 2 blocks and three ideal, massless pulleys. The blocks and pulleys are connected with ideal ropes (massless and constant length). Find the magnitude of the acceleration of block 2 assuming that [mathjaxinline]m_1&lt;m_2[/mathjaxinline]. Express your answer using some or all of the following variables: m_1 for [mathjaxinline]m_1[/mathjaxinline], m_2 for [mathjaxinline]m_2[/mathjaxinline], and g for standard gravitational acceleration. </p>
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<p style="display:inline">[mathjaxinline]a_2 =[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">L12WE3/L12v5 Worked Example 2 Blocks and 2 Pulleys</h2>
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Blocks with Moveable Pulleys
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<p>Two blocks with pulleys attached lie on a frictionless surface and are connected by a massless rope strung over the pulleys as shown at right. A force of magnitude \(F\) is applied to the second block.</p>
<p>Draw a free body diagram for Block 1 and Block 2.</p>
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<h3 class="hd hd-2">L12v5: Worked Example - 2 Blocks and 2 Pulleys</h3>
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<h2 class="hd hd-2 unit-title">L12Q1: Two Blocks, Three Pulleys and an Incline</h2>
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L12Q1-1: Two blocks, three pulleys and an incline - Part 1
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<p>
The left rope in the figure is attached at one end to block 1 of mass [mathjaxinline]m_1[/mathjaxinline] that can slide along a frictionless incline surface of angle [mathjaxinline]\theta[/mathjaxinline] with respect to the horizontal. The rope passes over a fixed pulley, pulley A, and then around a movable pulley, pulley B. The other end of the rope is attached to the ceiling at point P. The right rope is attached at one end to the center of pulley B, passes around a second movable pulley, pulley C, and its other end is attached to the ceiling at point Q. </p>
<p>
The goal of this problem is to calculate the acceleration of block 2 of mass [mathjaxinline]m_2[/mathjaxinline], attached to pulley C by a third ideal rope. </p>
<p><b class="bfseries">(Part a)</b> In this part you will set up the relationship between the components of the accelerations of box 1 and box 2. </p>
<p>
Consider the two coordinate systems shown in the figure below. Remember that we can choose different coordinate systems for each object. Here we will choose the [mathjaxinline]+x[/mathjaxinline]-axis pointing down the incline with its zero at the center of pulley A and use this coordinate to measure [mathjaxinline]x_1[/mathjaxinline], the position of block 1. The [mathjaxinline]+y[/mathjaxinline] axis, pointing vertically down and with its zero at the ceiling, measures [mathjaxinline]y_ A[/mathjaxinline], [mathjaxinline]y_ B[/mathjaxinline], [mathjaxinline]y_ C[/mathjaxinline], and [mathjaxinline]y_2[/mathjaxinline], the positions of the centers of pulleys A, B , and C as well as block 2, respectively. </p>
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<p>
The accelerations of the objects are given by the following (Note that the [mathjaxinline]\hat{i}[/mathjaxinline] and [mathjaxinline]\hat{j}[/mathjaxinline] refer to the two different coordinate systems shown above): </p>
<p>
[mathjaxinline]\begin{array}{l} \mbox{acceleration of Pulley B:} &amp; \vec{a}_ B= &amp; a_{By} \, \hat{j}= &amp; \frac{d^2y_ B}{dt^2}\hat{j} \\ \mbox{acceleration of Pulley C: } &amp; \vec{a}_ C= &amp; a_{Cy} \, \hat{j}= &amp; \frac{d^2y_ C}{dt^2}\hat{j} \\ \mbox{acceleration of Block 2: } &amp; \vec{a}_2= &amp; a_{2y} \, \hat{j}= &amp; \frac{d^2y_2}{dt^2}\hat{j} \\ \mbox{acceleration of Block 1: } &amp; \vec{a}_1= &amp; a_{1x} \, \hat{i}= &amp; \frac{d^2x_1}{dt^2}\hat{i} \end{array}[/mathjaxinline] </p>
<p>
<b class="bfseries">Left Rope:</b>
</p>
<p>
Using the constraint that the length of the left rope is constant, find an expression for the ratio [mathjaxinline]\displaystyle \frac{a_{1x}}{a_{By}}[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \frac{a_{1x}}{a_{By}}=[/mathjaxinline] </p>
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<b class="bfseries">Right Rope:</b>
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Using the constraint that the length of the right rope is constant, find an expression for the ratio [mathjaxinline]\displaystyle \frac{a_{By}}{a_{2y}}[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \frac{a_{By}}{a_{2y}} =[/mathjaxinline] </p>
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<b class="bfseries">Relating the motion of blocks 1 and 2:</b>
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Calculate the ratio [mathjaxinline]\displaystyle \frac{a_{1x}}{a_{2y}}[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \frac{a_{1x}}{a_{2y}} =[/mathjaxinline] </p>
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<h3 class="hd hd-3 problem-header" id="ls_ls04_ls04_07_part2-problem-title" aria-describedby="block-v1:MITx+8.01.1x+3T2018+type@problem+block@ls_ls04_ls04_07_part2-problem-progress" tabindex="-1">
L12Q1-2: Two blocks, three pulleys and an incline - Part 2
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In this part of the problem you will set up the equations from Newton's 2nd law. We are assuming that the ropes are massless and that there is no friction between the ropes and the pulleys. As a result, the magnitude of the force of tension along each rope is the same. [mathjaxinline]T_ L[/mathjaxinline] is the magnitude of the force of tension at any point along the left rope, [mathjaxinline]T_ R[/mathjaxinline] is the magnitude of the force of tension at any point along the right rope, and [mathjaxinline]T_ v[/mathjaxinline] is the magnitude of the force of tension at any point along the rope connecting pulley C and block 2. </p>
<p>
Draw the free body force diagram of block 1, pulley B, pulley C and block 2. Then apply Newton's second law to each of the objects. </p>
<p>
[mathjaxinline]\begin{array}{c|c ccl} System &amp; &amp; \sum {\vec{F}} &amp; =&amp; m\vec{a} \\ \hline \mbox{Block 1} &amp; \hat{i}: &amp; \sum {F}_{x}^{\mbox{on 1}} &amp; = &amp; m_1 a_{1x} \\ \mbox{Pulley B} &amp; \hat{j}: &amp; \sum {F}_{y}^{\mbox{on B}} &amp; = &amp; 0 \\ \mbox{Pulley C} &amp; \hat{j}: &amp; \sum {F}_{y}^{\mbox{on C}} &amp; = &amp; 0 \\ \mbox{Block 2} &amp; \hat{j}: &amp; \sum {F}_{y}^{\mbox{on 2}} &amp; = &amp; m_2 a_{2y} \end{array}[/mathjaxinline] </p>
<p>
Complete the [mathjaxinline]\sum F[/mathjaxinline] column of the table above. Express your answer in terms of [mathjaxinline]g[/mathjaxinline], m_1 for [mathjaxinline]m_2[/mathjaxinline], m_2 for [mathjaxinline]m_2[/mathjaxinline], theta for [mathjaxinline]\theta[/mathjaxinline], T_L for [mathjaxinline]T_ L[/mathjaxinline], T_R for [mathjaxinline]T_ R[/mathjaxinline], and T_v for [mathjaxinline]T_ v[/mathjaxinline] as needed. Be sure to write your equations using the choice of coordinate systems shown on the lower drawing. </p>
<p>
<p style="display:inline">[mathjaxinline]\sum {F}_ x^{\mbox{on 1}} =[/mathjaxinline]</p>
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<span class="trailing_text" id="trailing_text_ls_ls04_ls04_07_part2_2_1">[mathjaxinline]\mathrm{= m_1a_{1x}}[/mathjaxinline]</span>
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<p style="display:inline">[mathjaxinline]\sum {F}_{y}^{\mbox{ on B}} =[/mathjaxinline]</p>
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<span class="trailing_text" id="trailing_text_ls_ls04_ls04_07_part2_3_1">[mathjaxinline]\mathrm{= 0}[/mathjaxinline]</span>
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<p style="display:inline">[mathjaxinline]\sum {F}_{y}^{\mbox{on C}} =[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\sum {F}_{y}^{\mbox{on 2}} =[/mathjaxinline]</p>
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<span class="trailing_text" id="trailing_text_ls_ls04_ls04_07_part2_5_1">[mathjaxinline]\mathrm{= m_2a_{2y}}[/mathjaxinline]</span>
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L12Q1-3: Two blocks, three pulleys and an incline - Part 3
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<p><b class="bfseries">(Part c)</b> What is [mathjaxinline]a_{2y}[/mathjaxinline], the y-component of the acceleration of block 2? Express your answer in term of m_1 for [mathjaxinline]m_1[/mathjaxinline], m_2 for [mathjaxinline]m_2[/mathjaxinline], theta for [mathjaxinline]\theta[/mathjaxinline] and [mathjaxinline]g[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]a_{2y} =[/mathjaxinline] </p>
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