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--- book:chap2:2.11_problems [2021/11/08 17:26] – external edit 127.0.0.1
+++ book:chap2:2.11_problems [2022/04/01 21:30] (current) – jv
@@ Line 1: / Line 1: @@
+[[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| 2.6.  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 **
+  * [[ 2.12 Further reading| 2.12 Further reading ]]
+----
 ===== 2.11 Problems =====
 ==== 2.11.1 Rehearsing Concepts ====
-<wrap #quest_forces-08>Problem 2.27:</wrap>
+<wrap #quest_forces-08 >Problem 2.27:</wrap>
 **Tackling tackles and pulling pulleys**
@@ Line 15: / Line 31: @@
 ++ Hint: | $\quad$ The power is defined here as the change of $Mg\, z(t)$ and $(M+m)\,g\,z(t)$,
 per unit time, respectively. Verify by dimensional analysis that this is a meaningful definition.++
 ==== 2.11.2 Practicing Concepts ====
@@ Line 74: / Line 92: @@
 \end{align*}
 Altogether these are $8$ equations to determine the two components of $\mathbf q_0$, $l_1$, $\dots$ $l_3$, and the angles $\alpha$, $\beta$ and $\gamma$. Determine $\mathbf q_0$.
 ------
@@ Line 79: / Line 98: @@
 <wrap #quest_forces-11>Problem 2.29: </wrap>** Torques acting on a ladder **
-The sketch in the margin shows the setup of a ladder leaning to the roof of a hut.
+[[#fig_leaning-ladder |Figure 2.27]] shows the setup of a ladder leaning to a wall.
 The indicated angle from the downwards vertical to the ladder is denoted as $\theta$.
 There is a gravitational force of magnitude $Mg$ acting of a ladder of mass $M$.
-At the point where it leans to the roof there is a normal force of magnitude $F_r$
+At the point where it leans to the wall there is a normal force $\mathbf N$
-acting from the roof to the ladder.
+acting from the wall to the ladder.
-At the ladder feet there is a normal force to the ground of magnitude $F_g$,
+At the ladder feet there is a normal force to the ground  $\vec f$,
-and a tangential friction force of magnitude $\gamma F_f$.
+and a tangential friction force of magnitude $\gamma_1 f$.
-This is again the sketch to the ladder leaning to the roof of a hut.
-The angle from the downwards vertical to the ladder is denoted $\theta$.
-There is a gravitational force of magnitude $Mg$ acting of a ladder.
-At the point where it leans to the roof there is a normal force of magnitude $F_r$.
-At the ladder feet there is a normal force to the ground of magnitude $F_g$,
-and a tangential friction force of magnitude $F_f$.
-<WRAP 120pt left>
+<WRAP 120pt left #fig_leaning-ladder >
-{{10_Leaning_ladder_setup.png}}
+{{:book:chap2:10_leaning_ladder_setup.png}}
-[[https://commons.wikimedia.org/wiki/File:Leaning_ladder_setup.svg|original: Bradley, vector: Sarang / wikimedia]], public domain\\
+<wrap lo>
-Figure 2.27: Setup for [[#quest_forces-11 |Problem 2.29]]: leaning a ladder to a roof.
+[[https://commons.wikimedia.org/wiki/File:Leaning_ladder_setup.svg|based on original: Bradley, vector: Sarang / wikimedia]], public domain</wrap>\\
+Figure 2.27: Setup for [[#quest_forces-11 |Problem 2.29]]: leaning a ladder to a wall.
 </WRAP>
-  -  In principle there also is a friction force $\gamma_r \, F_r$ acting at the contact from the ladder to the roof. Why is it admissible to neglect this force?\\ (Remark: There are at least two good arguments).\\
+  -  In principle there also is a friction force $\gamma_2 \, N$ acting at the contact from the ladder to the wall. Why is it admissible to neglect this force?\\ Remark: There are at least two good arguments.
-  -  Determine the vertical and horizontal force balance for the ladder.Is there a unique solution?\\
+  -  Determine the vertical and horizontal force balance for the ladder. Is there a unique solution?
-  -  The feet of the ladder start sliding when $F_f$ exceeds the maximum static friction force $\gamma F_g$. What does this condition entail for the angle $\theta$?\\ Assume that $\gamma \simeq 0.3$ What does this imply for the critical angle $\theta_c$.\\
+  -  The feet of the ladder start sliding when $\gamma_1 f$ exceeds the maximum static friction force $\gamma_s f$.  Which constraints do the force balance and this condition entail for the angles $\theta$ where the ladder leans at the wall?
-  -  Where does the mass of the ladder enter the discussion? Do you see why?\\
+  -  Where does the mass of the ladder enter the discussion? Do you see why?
-  -  Determine the torque acting on the ladder. Does it matter whether you consider the torque with respect to the contact point to the roof, the center of mass, or the foot of the ladder?\\
+  -  Determine the torque acting on the ladder. Does it matter whether you consider the torque with respect to the contact point to the wall, the center of mass, or the foot of the ladder?
-  -  The ladder slides when the modulus of the friction force $F_f$ exceeds a maximum value $\mu_S F_g$ where $\mu$ is the static friction coefficient for of the ladder feet on the ground. For metal feet on a wooden ground it takes a value of $\mu_S \simeq 2$. What does that tell about the angels where the ladder starts to slide?\\
+  -  Determine the threshold of sliding based on the balance of torques. \\ For metal feet on a wooden ground it takes a value of $\gamma_s \simeq 2$. For a slippery smooth ground it can be as small as $\gamma_s \simeq 0.3$. What does that tell about the range angles where the ladder starts to slide?
-  -  Why does a ladder commonly starts sliding when when a man has climbed to the top? Is there anything one can do against it? Is that even true, or just an urban legend?
+  - :!: Why does a ladder commonly start sliding when when a man has climbed to the top? Is there anything one can do against it? Is that even true, or just an urban legend?
 -----
@@ Line 135: / Line 149: @@
-The more traffic you encounter when it becomes dark the more important it becomes to make your bikes visible. Retro-reflectors fixed in the sparks enhance the visibility to the sides. They trace a path of a curtate trochoid that is characterized by the ratio $\rho$ of the reflectors distance $d$ to the wheel axis and the wheel radius $r$. A small stone in the profile traces a cycloid ($\rho=1$). Animations of the trajectories can be found at [[https://en.wikipedia.org/wiki/Trochoid]] and \\
+The more traffic you encounter when it becomes dark the more important it becomes to make your bikes visible. Retro-reflectors fixed in the sparks enhance the visibility to the sides. They trace a path of a curtate trochoid that is characterized by the ratio $\rho$ of the reflectors distance $d$ to the wheel axis and the wheel radius $r$. A small stone in the profile traces a cycloid ($\rho=1$). Animations of the trajectories can be found at [[https://en.wikipedia.org/wiki/Trochoid]], \\
-[[http://katgym.by.lo-net2.de/c.wolfseher/web/zykloiden/zykloiden.html]].
+[[http://katgym.by.lo-net2.de/c.wolfseher/web/zykloiden/zykloiden.html]],
+and in the  [[sage:plot:animate-cycloids|Sage playground]]
+where it is shown how to generate the following plots:
-<WRAP 120pt left>
+{{ :sage:plot:p04_animate-cycloids_d1.0.gif?500 |}}
-{{05_trochoids.png}}
+{{ :sage:plot:p04_animate-cycloids_flag.gif?500 |}}
-based on [[https://commons.wikimedia.org/wiki/File:Zykloiden.svg|Kmhkmh Zykloiden]],
-[[https://creativecommons.org/licenses/by/4.0|CC BY 4.0]]
-</WRAP>
 A trochoid is most easily described in two steps: