Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap2:2.6_physics_application_balancing_forces [2021/10/31 17:51] – [2.6.1 Self Test] jvbook:chap2:2.6_physics_application_balancing_forces [2022/04/01 21:00] (current) – jv
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 +[[forcestorques|2. Balancing Forces and Torques]]
 +  * [[  2.1 Motivation and Outline| 2.1 Motivation and outline: forces are vectors ]]
 +  * [[  2.2 Sets| 2.2 Sets ]]
 +  * [[  2.3 Groups| 2.3 Groups ]]
 +  * [[  2.4 Fields| 2.4 Fields ]]
 +  * [[  2.5 Vector spaces| 2.5 Vector spaces ]]
 +  * ** 2.6.  Physics application: balancing forces **
 +  * [[  2.7 The inner product | 2.7 The inner product]]
 +  * [[  2.8 Cartesian coordinates | 2.8 Cartesian coordinates]]
 +  * [[  2.9 Cross products --- torques| 2.9 Cross products — torques ]]
 +  * [[ 2.10 Worked example Calder's mobiles| 2.10 Worked example: Calder's mobiles ]]
 +  * [[ 2.11 Problems| 2.11 Problems ]]
 +  * [[ 2.12 Further reading| 2.12 Further reading ]]
 +
 +----
 +
 ===== 2.6 Physics application: balancing forces ===== ===== 2.6 Physics application: balancing forces =====
  
 +<WRAP group>
 <WRAP 120pt left #fig_TugOfWar> <WRAP 120pt left #fig_TugOfWar>
 {{tug-of-war__NicolayBogdanov.jpg|}}\\ {{tug-of-war__NicolayBogdanov.jpg|}}\\
-Tug of War, Nikolay Bogdanov-Belsky, 1939\\ [[https://www.wikiart.org/en/nikolay-bogdanov-belsky/tug-of-war|wikiart]] / public domain}}\\ +Tug of War, Nikolay Bogdanov-Belsky, 1939\\ [[https://www.wikiart.org/en/nikolay-bogdanov-belsky/tug-of-war|wikiart]] / public domain
- +
-{{vector_sums.png}} +
-Figure 2.10: The left diagrams show two and three forces acting on a ring. To the right it is demonstrated that they add to zero.+
 </WRAP> </WRAP>
  
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 For the ring the sums of the forces are illustrated in the right panels of [[#fig_TugOfWar |Figure 2.10]]. For the ring the sums of the forces are illustrated in the right panels of [[#fig_TugOfWar |Figure 2.10]].
 The ring does not move when they add to zero. The ring does not move when they add to zero.
 +</WRAP>
 +
 +<WRAP group>
 +<WRAP 120pt left #fig_vector-sums>
 +{{vector_sums.png}}
 +Figure 2.10: The left diagrams show two and three forces acting on a ring. To the right it is demonstrated that they add to zero.
 +</WRAP>
  
 <WRAP box round #axiom_ForceBalance> **Axiom 2.1:** <wrap em>Force balance</wrap>\\ <WRAP box round #axiom_ForceBalance> **Axiom 2.1:** <wrap em>Force balance</wrap>\\
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 We come back to this point in [[book:chap2:2.9_cross_products_---_torques|Section 2.9]]. We come back to this point in [[book:chap2:2.9_cross_products_---_torques|Section 2.9]].
 </wrap> </wrap>
 +</WRAP>
 +
 +<WRAP group>
 +<WRAP 120pt left #figure_slackline >
 +{{ :book:chap2:force_slackline.png?direct&170 |}}
 +Figure 2.11: For a person balancing on a slackline, the gravitational force $\mathbf F_d$ (d for down)
 +is balanced by forces $\mathbf F_l$ and $\mathbf F_r$ along the line that pull towards the left and right, respectively. See [[#Example_Slackline |Example 2.19]] for further discussion.\\
 +</WRAP>
  
 <WRAP box round #Example_Slackline >**Example 2.19** <wrap em>Balancing on a slackline</wrap> \\ <WRAP box round #Example_Slackline >**Example 2.19** <wrap em>Balancing on a slackline</wrap> \\
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 These forces become huge when the slackline runs almost horizontally. These forces become huge when the slackline runs almost horizontally.
 Every now a then a careless slackliner roots out a tree or fells a pillar. Every now a then a careless slackliner roots out a tree or fells a pillar.
 +</WRAP>
 </WRAP> </WRAP>
  
-<WRAP 120pt left #figure_slackline #figure_freeBodyStaticFriction> +<WRAP goup> 
-{{./Sketch/force_slackline.pdf}} +<WRAP 120pt left #figure_freeBodyStaticFriction >
-Figure 2.11: For a person balancing on a slackline, the gravitational force $\mathbf F_d$ (d for down) +
-is balanced by forces $\mathbf F_l$ and $\mathbf F_r$ along the line that pull towards the left and right, respectively. See [[#Example_Slackline |Example 2.19]] for further discussion.\\ +
 {{free_body_static_friction.png}}\\ {{free_body_static_friction.png}}\\
 {{PhyPhoxStaticFriction.JPG}}\\ {{PhyPhoxStaticFriction.JPG}}\\
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 i.e., $\mu \simeq 0.5$.  i.e., $\mu \simeq 0.5$. 
 Using [[https://phyphox.org/|PhyPhox]] and a cell phone one can easily measure $\theta_c$ and $\mu$ for other combinations of materials. Using [[https://phyphox.org/|PhyPhox]] and a cell phone one can easily measure $\theta_c$ and $\mu$ for other combinations of materials.
-</WRAP>\\+</WRAP>
  
 <WRAP box round> **Example 2.20:** <wrap em>Measuring the static friction coefficient</wrap>\\ <WRAP box round> **Example 2.20:** <wrap em>Measuring the static friction coefficient</wrap>\\
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 By splitting the gravitational force, $m\mathbf g$ acting on a block on a plane into its components parallel and normal to the surface (gray arrows in [[#figure_freeBodyStaticFriction |Figure 2.12]]), By splitting the gravitational force, $m\mathbf g$ acting on a block on a plane into its components parallel and normal to the surface (gray arrows in [[#figure_freeBodyStaticFriction |Figure 2.12]]),
 one finds that in the presence of a force balance $m \mathbf g + \mathbf f + \mathbf F_N = \mathbf 0$ one finds that in the presence of a force balance $m \mathbf g + \mathbf f + \mathbf F_N = \mathbf 0$
-one has\\+one has
 \begin{align*} \begin{align*}
     \left .     \left .
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 When $\theta$ exceeds $\theta_c$ the block starts to slide. When $\theta$ exceeds $\theta_c$ the block starts to slide.
 Hence, one can infer $\gamma$ from measurements of $\theta_c$. Hence, one can infer $\gamma$ from measurements of $\theta_c$.
 +</WRAP>
 </WRAP> </WRAP>
  
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 <WRAP #quest_forceSelftest-rest > Problem 2.14: ** Particles at rest **\\ <WRAP #quest_forceSelftest-rest > Problem 2.14: ** Particles at rest **\\
-There are three forces acting on the center of mass of a body. In which cases does it stay at rest?\\+There are three forces acting on the center of mass of a body. In which cases does it stay at rest? 
 {{ :book:chap2:forceselftest-rest.png?direct&600 |}} {{ :book:chap2:forceselftest-rest.png?direct&600 |}}
 +</WRAP>
  
-</WRAP>\\+----
  
 <WRAP #quest_forceSelftest-vectorSum >Problem 2.15: ** Graphical sum of vectors **\\ <WRAP #quest_forceSelftest-vectorSum >Problem 2.15: ** Graphical sum of vectors **\\
-Determine the sum of the vectors. In which cases is the resulting vector vertical to the horizontal direction?\\ +Determine the sum of the vectors. In which cases is the resulting vector vertical to the horizontal direction? 
-{{forceSelftest-vectorSum1.pdf}}\\ + 
-{{forceSelftest-vectorSum2.pdf}}\\ +{{ :book:chap2:forceselftest-vectorsum1.png?direct&600 |}} 
-</WRAP>\\+{{ :book:chap2:forceselftest-vectorsum2.png?direct&600 |}} 
 +</WRAP> 
 + 
 +----
  
 <WRAP #quest_forceSelftest-Muskelprotz >Problem 2.16: ** Towing a stone **\\ <WRAP #quest_forceSelftest-Muskelprotz >Problem 2.16: ** Towing a stone **\\
 Three Scottish muscleman ((In highland games one still uses Imperial Units. A hundredweight (cwt) amounts to eight stones (stone) that each have a mass of $14$ pounds(lb). A pound-force (lbg) amounts to the gravitational force acting on a pound. One can solve this problem without converting units.)) try to tow a stone with mass $M=20\text{cwt}$ from a field. Each of them gets his own rope, and he can act a maximal force of Three Scottish muscleman ((In highland games one still uses Imperial Units. A hundredweight (cwt) amounts to eight stones (stone) that each have a mass of $14$ pounds(lb). A pound-force (lbg) amounts to the gravitational force acting on a pound. One can solve this problem without converting units.)) try to tow a stone with mass $M=20\text{cwt}$ from a field. Each of them gets his own rope, and he can act a maximal force of
-$300\text{lbg}$ as long as the ropes run in directions that differ by at least $30^\circ$\\+$300\text{lbg}$ as long as the ropes run in directions that differ by at least $30^\circ$
  
   -  Sketch the forces acting on the stone and their sum. By which ratio is the force exerted by three men larger than that of a single man?   -  Sketch the forces acting on the stone and their sum. By which ratio is the force exerted by three men larger than that of a single man?
   -  The stone counteracts the pulling of the men by a static friction force $\mu M g$, where $g$ is the gravitational acceleration. What is the maximum value that the friction coefficient $\mu$ may take when the men can move the stone?   -  The stone counteracts the pulling of the men by a static friction force $\mu M g$, where $g$ is the gravitational acceleration. What is the maximum value that the friction coefficient $\mu$ may take when the men can move the stone?
-</WRAP>\\+</WRAP>
  
 ~~DISCUSSION|Questions, Remarks, and Suggestions~~ ~~DISCUSSION|Questions, Remarks, and Suggestions~~
book/chap2/2.6_physics_application_balancing_forces.1635699083.txt.gz · Last modified: 2021/10/31 17:51 by jv