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<h2 class="hd hd-2 unit-title">Intro to Differential Analysis</h2>
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<p> In this sequence, we will explore what happens when ropes cannot be modeled to be massless. We'll start by looking at simple examples where the rope is broken into a small number of parts, and consider the parts to behave as point masses where you can draw a free body diagram and apply Newton's second law. We then move on to more complex examples, where the rope needs to be considered as a continuous system. In these examples, the rope is broken into a very large number of small mass "differential" elements, where Newton's second law is applied to obtain a "differential equation", which is then integrated over the whole rope. We usually call this technique: "differential analysis", which we will use in several places in the course where we deal with continuous systems.</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 examples on massive ropes <a href="/courses/course-v1:MITx+8.01.1x+3T2018/pdfbook/0/chapter/8/20"><i class="it">chapter 8, examples 8.3 and 8.4</i></a></li><li> Continuous Systems and Newton’s Second Law as a Differential Equations <a href="/courses/course-v1:MITx+8.01.1x+3T2018/pdfbook/0/chapter/8/25"><i class="it">chapter 8, section 8.5.2</i></a></li><li> Worked Example: Capstan <a href="/courses/course-v1:MITx+8.01.1x+3T2018/pdfbook/0/chapter/8/46"><i class="it">chapter 8, example 8.11</i></a></li></ul>
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<h2 class="hd hd-2 unit-title">L13WE1/L13v1: Massive Rope Hanging Betweetn Trees</h2>
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Tension in Massive Rope
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<p>A rope of mass \(m\) hangs between two trees, making an angle \(\theta\) with the vertical at each end. Since the rope is massive, its tension varies throughout its length.</p>
<p>Your goal in this problem will be to find the magnitude of the tension force in the rope at the ends and midpoint of the rope. Begin by drawing a free-body diagram just for the <strong>left half</strong> of the rope (excluding an infinitesimal end segment, as indicated at right).</p>
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<em>Please draw force vectors with their tails roughly where forces are applied.</em>
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<h3 class="hd hd-2">L13v1: Rope Hanging Betweetn Trees</h3>
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<h2 class="hd hd-2 unit-title">L13Q1: Tension in a Suspended Rope - Part I</h2>
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L13Q1: Tension in a Suspended Rope - Part 1
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<p>
A uniform rope of mass [mathjaxinline]M[/mathjaxinline] and length [mathjaxinline]L[/mathjaxinline] is suspended from a ceiling. The magnitude of the acceleration due to gravity is [mathjaxinline]g[/mathjaxinline]. Choose a coordinate system with the origin at the ceiling and the positive [mathjaxinline]x[/mathjaxinline]-direction pointing downward. </p>
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<img src="/assets/courseware/v1/9a000289867a2a1df370db4a7b5b5430/asset-v1:MITx+8.01.1x+3T2018+type@asset+block/images_hangingrope.svg" width="220"/>
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<p><b class="bfseries">(Part a)</b> What is the tension at the bottom end of the rope? </p>
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<text> less than [mathjaxinline]Mg[/mathjaxinline]</text>
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<text> greater than [mathjaxinline]Mg[/mathjaxinline]</text>
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<p><b class="bfseries">(Part b)</b> What is the tension at the top end of the rope? </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ls_ls04_ls04_04_part1_3_1">
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<text> [mathjaxinline]Mg[/mathjaxinline]</text>
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<text> less than [mathjaxinline]Mg[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L13v2: Differential Analysis of a Massive Rope</h2>
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<h2 class="hd hd-2 unit-title">L13Q2: Tension in a Suspended Rope - Part II</h2>
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L13Q2-1: Tension in a Suspended Rope - Part 2
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<p><b class="bfseries">(Part c)</b> Now consider the system to be a small segment of the rope of length [mathjaxinline]\Delta x=x_2-x_1[/mathjaxinline] whose upper end is at a distance [mathjaxinline]x_1[/mathjaxinline] below the ceiling. This small segment has a mass [mathjaxinline]\Delta m[/mathjaxinline] and it is indicated with darker gray in the figure. </p>
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<img src="/assets/courseware/v1/dd4bc4a59c4b673c676fe84decc9d325/asset-v1:MITx+8.01.1x+3T2018+type@asset+block/images_hangingrope_partc.svg" width="220"/>
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Assuming that the mass is uniformly distributed along the length of the rope, find an expression for [mathjaxinline]\Delta m[/mathjaxinline] in terms of [mathjaxinline]M[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], and Deltax for [mathjaxinline]\Delta x[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\Delta m =[/mathjaxinline] </p>
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L13Q2-2: Tension in a Suspended Rope - Part 3
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<p><b class="bfseries">(Part d)</b> Now, consider three different pieces of the rope, an upper piece above the small segment between [mathjaxinline]x = 0[/mathjaxinline] and [mathjaxinline]x= x_1[/mathjaxinline], the small segment of mass [mathjaxinline]\Delta m[/mathjaxinline] between [mathjaxinline]x=x_1[/mathjaxinline] and [mathjaxinline]x = x_2[/mathjaxinline], and a lower piece below the small segment between [mathjaxinline]x=x_2[/mathjaxinline] and [mathjaxinline]x=L[/mathjaxinline]. </p>
<p>
Draw a free-body force diagram for the small segment of rope of mass [mathjaxinline]\Delta m[/mathjaxinline]. Use as notation [mathjaxinline]T(x_1)[/mathjaxinline] for the magnitude of the force of tension the upper piece exerts on the small segment at [mathjaxinline]x=x_1[/mathjaxinline], and [mathjaxinline]T(x_2)[/mathjaxinline] for the magnitude of the force of tension the lower piece exerts on the small segment at [mathjaxinline]x=x_2[/mathjaxinline]. </p>
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Apply Newton's 2nd law to the small segment of rope of mass [mathjaxinline]\Delta m[/mathjaxinline] to calculate [mathjaxinline]T(x_2)-T(x_1)[/mathjaxinline]. Express your answer in terms of [mathjaxinline]M[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], and Deltax for [mathjaxinline]\Delta x[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]T(x_2) -T(x_1)=[/mathjaxinline] </p>
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If [mathjaxinline]\Delta T[/mathjaxinline] is defined as [mathjaxinline]\Delta T = T(x_2) - T(x_1)[/mathjaxinline], what is the ratio [mathjaxinline]\displaystyle \frac{\Delta T}{\Delta x}[/mathjaxinline]? Express your answer in terms of [mathjaxinline]M[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], and [mathjaxinline]g[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \frac{\Delta T}{\Delta x} =[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">L13Q3: Differential Elements</h2>
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L13Q3-1: Mass Density - Part I
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The objects shown in the figure have a total mass [mathjaxinline]M[/mathjaxinline]. Assuming that the mass is uniformly distributed throughout the objects express the total mass in terms of the linear mass density (mass per unit length [mathjaxinline]\lambda[/mathjaxinline]), areal mass density (mass per unit area [mathjaxinline]\sigma[/mathjaxinline]), or volume mass density (mass per unit volume [mathjaxinline]\rho[/mathjaxinline]) and the geometric factors [mathjaxinline]L, h, R, R_1[/mathjaxinline] and [mathjaxinline]R_2[/mathjaxinline] as needed. Enter lambda for [mathjaxinline]\lambda[/mathjaxinline], sigma for [mathjaxinline]\sigma[/mathjaxinline], rho for [mathjaxinline]\rho[/mathjaxinline], pi for [mathjaxinline]\pi[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> Solid cylinder. <p style="display:inline">[mathjaxinline]M =[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_module_mod3_lp_1_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<div id="display_module_mod3_lp_1_2_1" class="equation">`{::}`</div>
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<p><b class="bfseries">(Part b)</b> Thin rod. <p style="display:inline">[mathjaxinline]M =[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_module_mod3_lp_1_3_1" class="text-input-dynamath capa_inputtype inline textline">
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<div id="display_module_mod3_lp_1_3_1" class="equation">`{::}`</div>
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<p><b class="bfseries">(Part c)</b> Cylindrical Shell <p style="display:inline">[mathjaxinline]M =[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_module_mod3_lp_1_4_1" class="text-input-dynamath capa_inputtype inline textline">
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<div id="display_module_mod3_lp_1_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_module_mod3_lp_1_4_1_dynamath" name="input_module_mod3_lp_1_4_1_dynamath"/>
</div>
</div></div> </p>
<p><b class="bfseries">(Part d)</b> Solid Shpere <p style="display:inline">[mathjaxinline]M =[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_module_mod3_lp_1_5_1" class="text-input-dynamath capa_inputtype inline textline">
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<div id="display_module_mod3_lp_1_5_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_module_mod3_lp_1_5_1_dynamath" name="input_module_mod3_lp_1_5_1_dynamath"/>
</div>
</div></div> </p>
<p><b class="bfseries">(Part e)</b> Thin Disk <p style="display:inline">[mathjaxinline]M =[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_module_mod3_lp_1_6_1" class="text-input-dynamath capa_inputtype inline textline">
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<div id="display_module_mod3_lp_1_6_1" class="equation">`{::}`</div>
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</div></div> </p>
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L13Q3-2: Mass Density - Part II
</h3>
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<p><b class="bfseries">(Part a.)</b> A mass [mathjaxinline]M[/mathjaxinline] is uniformly distributed along a thin rod of length [mathjaxinline]L[/mathjaxinline]. The mass of the small element of rod of length [mathjaxinline]\Delta x[/mathjaxinline] is given by: </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_module_mod3_lp_2_2_1">
<fieldset aria-describedby="status_module_mod3_lp_2_2_1">
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<input type="radio" name="input_module_mod3_lp_2_2_1" id="input_module_mod3_lp_2_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="module_mod3_lp_2_2_1-choice_1-label" for="input_module_mod3_lp_2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_module_mod3_lp_2_2_1">
<text> [mathjaxinline]\Delta m = M/L[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_module_mod3_lp_2_2_1" id="input_module_mod3_lp_2_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="module_mod3_lp_2_2_1-choice_2-label" for="input_module_mod3_lp_2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_module_mod3_lp_2_2_1">
<text> [mathjaxinline]\Delta m =M\Delta x[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_module_mod3_lp_2_2_1" id="input_module_mod3_lp_2_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="module_mod3_lp_2_2_1-choice_3-label" for="input_module_mod3_lp_2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_module_mod3_lp_2_2_1">
<text> [mathjaxinline]\Delta m=(M/L)\Delta x[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_module_mod3_lp_2_2_1" id="input_module_mod3_lp_2_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="module_mod3_lp_2_2_1-choice_4-label" for="input_module_mod3_lp_2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_module_mod3_lp_2_2_1">
<text> [mathjaxinline]\Delta m = ML\Delta x[/mathjaxinline]</text>
</label>
</div>
<span id="answer_module_mod3_lp_2_2_1"/>
</fieldset>
<div class="indicator-container">
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p><b class="bfseries">(Part b.)</b> A mass [mathjaxinline]M[/mathjaxinline] is uniformly distributed throughout a rectangle of length [mathjaxinline]a[/mathjaxinline] and height [mathjaxinline]b[/mathjaxinline]. The mass of a small element of the rectangle of length [mathjaxinline]\Delta x[/mathjaxinline] and height [mathjaxinline]\Delta y[/mathjaxinline] is given by: </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_module_mod3_lp_2_3_1">
<fieldset aria-describedby="status_module_mod3_lp_2_3_1">
<div class="field">
<input type="radio" name="input_module_mod3_lp_2_3_1" id="input_module_mod3_lp_2_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="module_mod3_lp_2_3_1-choice_1-label" for="input_module_mod3_lp_2_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_module_mod3_lp_2_3_1">
<text> [mathjaxinline]\Delta m = M\Delta x\Delta y[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_module_mod3_lp_2_3_1" id="input_module_mod3_lp_2_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="module_mod3_lp_2_3_1-choice_2-label" for="input_module_mod3_lp_2_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_module_mod3_lp_2_3_1">
<text> [mathjaxinline]\Delta m =M(ab)\Delta x\Delta y[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_module_mod3_lp_2_3_1" id="input_module_mod3_lp_2_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="module_mod3_lp_2_3_1-choice_3-label" for="input_module_mod3_lp_2_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_module_mod3_lp_2_3_1">
<text> [mathjaxinline]\Delta m=(M/(ab))\Delta x\Delta y[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_module_mod3_lp_2_3_1" id="input_module_mod3_lp_2_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="module_mod3_lp_2_3_1-choice_4-label" for="input_module_mod3_lp_2_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_module_mod3_lp_2_3_1">
<text> [mathjaxinline]\Delta m = M/(ab)[/mathjaxinline]</text>
</label>
</div>
<span id="answer_module_mod3_lp_2_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_module_mod3_lp_2_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p><b class="bfseries">(Part c.)</b> A mass [mathjaxinline]M[/mathjaxinline] is uniformly distributed throughout a disk of radius [mathjaxinline]R[/mathjaxinline]. Consider the mass element to be the ring of inner radius [mathjaxinline]r[/mathjaxinline] and thickness [mathjaxinline]\Delta r[/mathjaxinline]. The mass of the ring is given by: </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_module_mod3_lp_2_4_1">
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<div class="field">
<input type="radio" name="input_module_mod3_lp_2_4_1" id="input_module_mod3_lp_2_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="module_mod3_lp_2_4_1-choice_1-label" for="input_module_mod3_lp_2_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_module_mod3_lp_2_4_1">
<text> [mathjaxinline]\Delta m = (M/(2\pi R))\Delta r[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_module_mod3_lp_2_4_1" id="input_module_mod3_lp_2_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="module_mod3_lp_2_4_1-choice_2-label" for="input_module_mod3_lp_2_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_module_mod3_lp_2_4_1">
<text> [mathjaxinline]\Delta m=(M/R)\Delta r[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_module_mod3_lp_2_4_1" id="input_module_mod3_lp_2_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="module_mod3_lp_2_4_1-choice_3-label" for="input_module_mod3_lp_2_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_module_mod3_lp_2_4_1">
<text> [mathjaxinline]\Delta m=(2M/R)\Delta r[/mathjaxinline]</text>
</label>
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<h2 class="hd hd-2 unit-title">Demo: Wrapping Friction</h2>
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In this demo, a heavy weight is balanced by wrapping the loose end of the rope a few times around a steel pipe.
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<h3 class="hd hd-2">Demo: Wrapping Friction</h3>
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<h2 class="hd hd-2 unit-title">L13Q4: Wrapping Friction</h2>
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L13Q4: Wrapping Friction.
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<p>
Consider a simpler situation than the one shown in the demonstration. A rope of uniform mass density is passed above the top of a fixed pipe of circular cross section only once as shown in the left figure above. </p>
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The front view of the pipe is shown in the figure on the right. The red dots labeled with <b class="bfseries">B</b> and <b class="bfseries">C</b> are the points where the rope loses contact with the pipe. Points <b class="bfseries">A</b> and <b class="bfseries">D</b> are at the end of the rope. The segment of rope hanging between points <b class="bfseries">A</b> and <b class="bfseries">B</b> has a length [mathjaxinline]2d[/mathjaxinline], and the segment of rope hanging between points <b class="bfseries">C</b> and <b class="bfseries">D</b> has a length of [mathjaxinline]d[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> What is [mathjaxinline]T_ A[/mathjaxinline], the magnitude of the force of tension at point <b class="bfseries">A</b>? </p>
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<p style="display:inline">[mathjaxinline]T_ A =[/mathjaxinline]</p>
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<p><b class="bfseries">(Part b)</b> What is [mathjaxinline]T_ D[/mathjaxinline], the magnitude of the force of tension at point <b class="bfseries">D</b>? </p>
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<p style="display:inline">[mathjaxinline]T_ D =[/mathjaxinline]</p>
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<p><b class="bfseries">(Part c)</b> If [mathjaxinline]T_ B[/mathjaxinline] and [mathjaxinline]T_ C[/mathjaxinline] are the magnitude of the force of tension at the points <b class="bfseries">B</b> and <b class="bfseries">C</b>, respectively, calculate the ratio [mathjaxinline]T_ B/T_ C[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]T_ B/T_ C =[/mathjaxinline]</p>
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