Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

User Tools

Site Tools

Trace:
book:chap2:2.11_problems

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revisionPrevious revision
Next revision		Previous revision
book:chap2:2.11_problems [2021/11/08 17:06] – [2.11.1 Rehearsing Concepts] jvbook: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 Problems =====
  
 ==== 2.11.1 Rehearsing Concepts ==== ==== 2.11.1 Rehearsing Concepts ====
 +
 +<wrap #quest_forces-08 >Problem 2.27:</wrap> 
 +**Tackling tackles and pulling pulleys**
  
 <WRAP 150pt left> <WRAP 150pt left>
 {{:book:chap2:08_pulleys.png?direct:180|}} {{:book:chap2:08_pulleys.png?direct:180|}}
 </WRAP> </WRAP>
-<wrap #quest_forces-08>Problem 2.27:</wrap>  
-**Tackling tackles and pulling pulleys** 
  
   -  Which forces are required to hold the balance in the left and the right sketch?   -  Which forces are required to hold the balance in the left and the right sketch?
   -  Let the sketched person and the weight have masses of $m=75\text{kg}$ and $M=300\text{kg}$, respectively. Which power is required then to haul the line at a speed of $1 m/s$.   -  Let the sketched person and the weight have masses of $m=75\text{kg}$ and $M=300\text{kg}$, respectively. Which power is required then to haul the line at a speed of $1 m/s$.
-Hint: 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.+++ 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 ==== ==== 2.11.2 Practicing Concepts ====
  
-<wrap #quest_forces-3balancedForces>Problem 2.28: </wrap>** Angles between three balanced forces **\\+<wrap #quest_forces-3balancedForces>Problem 2.28: </wrap>** Angles between three balanced forces **
  
 We consider three masses $m_1$, $m_2$, and $m_3$. With three ropes they are attached to a ring at position $\mathbf q_0$. The ropes with the attached masses hang over the edge of a table at the fixed positions We consider three masses $m_1$, $m_2$, and $m_3$. With three ropes they are attached to a ring at position $\mathbf q_0$. The ropes with the attached masses hang over the edge of a table at the fixed positions
Line 36: Line 56:
 Multiplying this equation with $\hat{\boldsymbol e}_1$, $\dots$ $\hat{\boldsymbol e}_3$ provides three equations Multiplying this equation with $\hat{\boldsymbol e}_1$, $\dots$ $\hat{\boldsymbol e}_3$ provides three equations
 that are linear in $\cos \theta_{ij}$. The first one is $ 0 = M_1 + M_2 \, \cos\theta_{12} + M_3 \, \cos\theta_{13}$. Find the other two equation, and solve the equations as follows. that are linear in $\cos \theta_{ij}$. The first one is $ 0 = M_1 + M_2 \, \cos\theta_{12} + M_3 \, \cos\theta_{13}$. Find the other two equation, and solve the equations as follows.
-\\+
 From the equation that is given above you find $\cos\theta_{12}$ in terms of $\cos\theta_{13}$. From the equation that is given above you find $\cos\theta_{12}$ in terms of $\cos\theta_{13}$.
-\\+
 Inserting this into the other equation involving $\cos\theta_{12}$ (and rearranging terms) Inserting this into the other equation involving $\cos\theta_{12}$ (and rearranging terms)
 provides $\cos\theta_{23}$ in terms of $\cos\theta_{13}$. provides $\cos\theta_{23}$ in terms of $\cos\theta_{13}$.
-\\+
 Inserting this into the third equation provides Inserting this into the third equation provides
 \begin{align*} \begin{align*}
       \cos\theta_{13} = \frac{M_2^2 - M_1^2 - M_3^2}{2 \, M_1 \, M_3}       \cos\theta_{13} = \frac{M_2^2 - M_1^2 - M_3^2}{2 \, M_1 \, M_3}
-\end{align*}\\+\end{align*}
  
 **b)** Which angle $\theta_{23}$ do you find when $M_1 = M_2 = M_3$? **b)** Which angle $\theta_{23}$ do you find when $M_1 = M_2 = M_3$?
 The three forces have the same absolute value in this case. The three forces have the same absolute value in this case.
-Which symmetry argument does then also provide the value of the angle?\\+Which symmetry argument does then also provide the value of the angle?
  
 **c)** Determine also the other two angles $\theta_{13}$ and $\theta_{12}$. **c)** Determine also the other two angles $\theta_{13}$ and $\theta_{12}$.
 They can also be found from a symmetry argument without calculation. They can also be found from a symmetry argument without calculation.
-\\ + 
-Hint: The angles do not care which mass you denote as $1$, $2$, and $3$.\\+++ Hint:| The angles do not care which mass you denote as $1$, $2$, and $3$.++
  
 **d)** Note that we found the angles $\theta_{ij}$ without referring to the positions $\mathbf q_1$, $\dots$ $\mathbf q_3$! Make a sketch what this implies for the position of the ring, and how $\mathbf q_0$ changes qualitatively upon changing a mass. **d)** Note that we found the angles $\theta_{ij}$ without referring to the positions $\mathbf q_1$, $\dots$ $\mathbf q_3$! Make a sketch what this implies for the position of the ring, and how $\mathbf q_0$ changes qualitatively upon changing a mass.
Line 71: Line 91:
       \beta = \frac{3\pi}{2} - \gamma - \theta_{13}       \beta = \frac{3\pi}{2} - \gamma - \theta_{13}
 \end{align*} \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 $\mathbf q_0$.\\+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 $\mathbf q_0$. 
  
 ------ ------
  
-<wrap #quest_forces-11>Problem 2.29: </wrap>** Torques acting on a ladder **\\+<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 $\theta$. 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 $M$. 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> </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?
  
 ----- -----
  
-<wrap #quest_forces-14>Problem 2.30: </wrap>** Walking a yoyo **\\+<wrap #quest_forces-14>Problem 2.30: </wrap>** Walking a yoyo **
  
 <WRAP 120pt right> <WRAP 120pt right>
Line 131: Line 147:
  
 <wrap #quest_2Dmotion-05>Problem 2.31: </wrap>** Retro-reflector paths on bike wheels ** <wrap #quest_2Dmotion-05>Problem 2.31: </wrap>** Retro-reflector paths on bike wheels **
-\\ 
  
-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 \\ 
-[[http://katgym.by.lo-net2.de/c.wolfseher/web/zykloiden/zykloiden.html]]. 
  
-<WRAP 120pt left> +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]], \\ 
-{{05_trochoids.png}} +[[http://katgym.by.lo-net2.de/c.wolfseher/web/zykloiden/zykloiden.html]], 
-based on [[https://commons.wikimedia.org/wiki/File:Zykloiden.svg|Kmhkmh Zykloiden]], +and in the  [[sage:plot:animate-cycloids|Sage playground]] 
-[[https://creativecommons.org/licenses/by/4.0|CC BY 4.0]] +where it is shown how to generate the following plots: 
-</WRAP>+ 
 +{{ :sage:plot:p04_animate-cycloids_d1.0.gif?500 |}} 
 +{{ :sage:plot:p04_animate-cycloids_flag.gif?500 |}}
  
 A trochoid is most easily described in two steps: A trochoid is most easily described in two steps:
Line 155: Line 170:
       \mathbf D(\theta) = \begin{pmatrix} - d \, \sin( \varphi + \theta ) \\ d \, \cos( \varphi + \theta ) \end{pmatrix} \, .       \mathbf D(\theta) = \begin{pmatrix} - d \, \sin( \varphi + \theta ) \\ d \, \cos( \varphi + \theta ) \end{pmatrix} \, .
 \end{align*} \end{align*}
-What is the meaning of $\varphi$ in this equation?\\+What is the meaning of $\varphi$ in this equation?
  
 **b)** The length of the track of a trochoid can be determined by integrating the modulus of its velocity over time, **b)** The length of the track of a trochoid can be determined by integrating the modulus of its velocity over time,
Line 171: Line 186:
       L = 2 \, r \: \int_{0}^{\theta} \mathrm{d} \theta \: \left|  \cos\frac{\varphi+\theta}{2} \right|       L = 2 \, r \: \int_{0}^{\theta} \mathrm{d} \theta \: \left|  \cos\frac{\varphi+\theta}{2} \right|
 \end{align*} \end{align*}
-How long is one period of the track traced out by a stone picked up by the wheel profile?\\+How long is one period of the track traced out by a stone picked up by the wheel profile?
  
  
 ==== 2.11.3 Mathematical Foundation ==== ==== 2.11.3 Mathematical Foundation ====
  
-<wrap #quest_group-01>Problem 2.32:</wrap> ** The natural numbers modulo $n$ are a group **\\+<wrap #quest_group-01>Problem 2.32:</wrap> ** The natural numbers modulo $n$ are a group **
  
 We consider here groups $G_n$ where the combined action of group elements can be represented as a sum of two numbers modulo $n \in \mathbb N$. We consider here groups $G_n$ where the combined action of group elements can be represented as a sum of two numbers modulo $n \in \mathbb N$.
Line 187: Line 202:
 ----- -----
  
-<wrap #quest_group-02>Problem 2.33:</wrap> ** Groups with four elements **\\+<wrap #quest_group-02>Problem 2.33:</wrap> ** Groups with four elements **
  
 In [[#quest_group-01 |Problem 2.32]] we encountered the group $G_n$. In [[#quest_group-01 |Problem 2.32]] we encountered the group $G_n$.
Line 200: Line 215:
  
 ----- -----
-<wrap #quest_vector-conicSection>Problem 2.34:</wrap> ** Conic Sections **\\+<wrap #quest_vector-conicSection>Problem 2.34:</wrap> ** Conic Sections **
  
 <WRAP 120pt right #fig_vector-KeplerOrbits> <WRAP 120pt right #fig_vector-KeplerOrbits>
Line 222: Line 237:
   -  Determine the vector $\mathbf a$ that points from the vertex of the double cone to the point where the plane intersects the axis of the double cone.   -  Determine the vector $\mathbf a$ that points from the vertex of the double cone to the point where the plane intersects the axis of the double cone.
   -  Describe the points in the intersection as sum of $\mathbf a$ and a vector $\mathbf b$ that lies in the plane.   -  Describe the points in the intersection as sum of $\mathbf a$ and a vector $\mathbf b$ that lies in the plane.
-  -  :!: Determine the length of the vector $\mathbf b$ as function of the angle $\theta$ that characterizes the direction of $\mathbf b$ in $\mathsf P$. How can this expression be used to plot the functions shown in [[#fig_vector-KeplerOrbits |Figure 2.28]]?\\+  -  :!: Determine the length of the vector $\mathbf b$ as function of the angle $\theta$ that characterizes the direction of $\mathbf b$ in $\mathsf P$. How can this expression be used to plot the functions shown in [[#fig_vector-KeplerOrbits |Figure 2.28]]?
  
 ----- -----
  
-<wrap #quest_forces-3 >Problem 2.35:</wrap> ** Linear dependence of three vectors in 2D **\\+<wrap #quest_forces-3 >Problem 2.35:</wrap> ** Linear dependence of three vectors in 2D **
  
 In the lecture I pointed out that every vector $\mathbf v = (v_1, v_2)$ of a two-dimensional vector space In the lecture I pointed out that every vector $\mathbf v = (v_1, v_2)$ of a two-dimensional vector space
Line 235: Line 250:
 In this exercise we revisit this statement for $\mathbb{R}^2$ with the standard forms of vector addition and multiplication by scalars. In this exercise we revisit this statement for $\mathbb{R}^2$ with the standard forms of vector addition and multiplication by scalars.
  
-**a)** Provide a triple of vectors $\mathbf a$, $\mathbf b$ and $\mathbf v$ such that $\mathbf v$ can //not// be represented as a scalar combination of $\mathbf a$ and $\mathbf b$.\\+**a)** Provide a triple of vectors $\mathbf a$, $\mathbf b$ and $\mathbf v$ such that $\mathbf v$ can //not// be represented as a scalar combination of $\mathbf a$ and $\mathbf b$.
  
 **b)** To be specific we henceforth fix **b)** To be specific we henceforth fix
Line 248: Line 263:
 \[ \[
 \mathbf v = \alpha \: \mathbf a + \beta \: \mathbf b \mathbf v = \alpha \: \mathbf a + \beta \: \mathbf b
-\]\\+\]
  
 **c)** Consider now also a third vector **c)** Consider now also a third vector
Line 258: Line 273:
 What is the general constraints on  $(\alpha, \beta, \gamma)$  such that What is the general constraints on  $(\alpha, \beta, \gamma)$  such that
 $\mathbf v =  \alpha \: \mathbf a + \beta \: \mathbf b + \gamma \mathbf c$.\\ $\mathbf v =  \alpha \: \mathbf a + \beta \: \mathbf b + \gamma \mathbf c$.\\
-What does this imply on the number of solutions?\\+What does this imply on the number of solutions?
  
 **d)** Discuss now the linear dependence of the vectors $\mathbf a$, $\mathbf b$ and $\mathbf c$ **d)** Discuss now the linear dependence of the vectors $\mathbf a$, $\mathbf b$ and $\mathbf c$
Line 265: Line 280:
 \mathbf 0 =  \alpha \: \mathbf a + \beta \: \mathbf b + \gamma \mathbf c \mathbf 0 =  \alpha \: \mathbf a + \beta \: \mathbf b + \gamma \mathbf c
 \] \]
-How are the constraints for the null vector related to those obtained in part c)?\\+How are the constraints for the null vector related to those obtained in part c)?
  
 ----- -----
  
-<wrap #quest_numberField>Problem 2.36:</wrap> ** Algebraic number fields **\\+<wrap #quest_numberField>Problem 2.36:</wrap> ** Algebraic number fields **
  
 Consider the set $\mathbb{K} = \mathbb{Q} + I \mathbb{Q}$ with $I^2 \in \mathbb{Q}$. Consider the set $\mathbb{K} = \mathbb{Q} + I \mathbb{Q}$ with $I^2 \in \mathbb{Q}$.
Line 289: Line 304:
   -  Consider $I = \sqrt{10}$. Show that $\mathbb{K}$ is a field that is different from $\mathbb{Q}$.\\   -  Consider $I = \sqrt{10}$. Show that $\mathbb{K}$ is a field that is different from $\mathbb{Q}$.\\
   -  Consider $I = \sqrt{8}$. In this case  $\mathbb{K}$  is //not// a field! Why?\\   -  Consider $I = \sqrt{8}$. In this case  $\mathbb{K}$  is //not// a field! Why?\\
-  -  :!: Find the general rule: For which natural numbers $n$ does $I = \sqrt{n}$ provide a non-trivial field?\\ Remark: Non-trivial means here different from $\mathbb{Q}$.\\+  -  :!: Find the general rule: For which natural numbers $n$ does $I = \sqrt{n}$ provide a non-trivial field?\\ Remark: Non-trivial means here different from $\mathbb{Q}$.
  
 ----- -----
  
-<wrap #quest_forces-20>Problem 2.37:</wrap> ** Bases for polynomials **\\+<wrap #quest_forces-20>Problem 2.37:</wrap> ** Bases for polynomials **
  
 We consider the set of polynomials $\mathbb{P}_N$ of degree $N$ with real coefficients $p_n$, $n \in \{0, \dots, N \}$, We consider the set of polynomials $\mathbb{P}_N$ of degree $N$ with real coefficients $p_n$, $n \in \{0, \dots, N \}$,
Line 311: Line 326:
       \quad \text{ and } \quad       \quad \text{ and } \quad
       c   \cdot \mathbf{p} = \left( \sum_{k=0}^N (c\, p_k) \, x^k \right) \, .       c   \cdot \mathbf{p} = \left( \sum_{k=0}^N (c\, p_k) \, x^k \right) \, .
-\end{align*}\\+\end{align*}
  
 **b)** Demonstrate that **b)** Demonstrate that
Line 317: Line 332:
 \mathbf{p} \cdot \mathbf{q} = \left(  \int_0^1 \mathrm{d} x \left( \sum_{k=0}^N p_k \, x^k  \right) \, \left( \sum_{j=0}^N q_j \, x^j  \right) \right) \, , \\ \mathbf{p} \cdot \mathbf{q} = \left(  \int_0^1 \mathrm{d} x \left( \sum_{k=0}^N p_k \, x^k  \right) \, \left( \sum_{j=0}^N q_j \, x^j  \right) \right) \, , \\
 \end{align*} \end{align*}
-establishes a scalar product on this vector space.\\+establishes a scalar product on this vector space.
  
 **c)** Demonstrate that the three polynomials **c)** Demonstrate that the three polynomials
 $\mathbf b_0 = (1)$, $\mathbf b_1 = (x)$ and $\mathbf b_2 = (x^2)$ $\mathbf b_0 = (1)$, $\mathbf b_1 = (x)$ and $\mathbf b_2 = (x^2)$
-form a basis of the vector space $\mathbb{P}_2$: For each polynomial $\mathbf p$ in $\mathbb{P}_2$ there are real numbers $x_k$, $k\in\{0,1,2\}$, such that $\mathbf p = x_0 \, \mathbf b_0 + x_1 \, \mathbf b_1 + x_2 \, \mathbf b_2$. However, in general we have  $x_i \neq \mathbf p \cdot \mathbf b_i$. Why is that?\\ +form a basis of the vector space $\mathbb{P}_2$: For each polynomial $\mathbf p$ in $\mathbb{P}_2$ there are real numbers $x_k$, $k\in\{0,1,2\}$, such that $\mathbf p = x_0 \, \mathbf b_0 + x_1 \, \mathbf b_1 + x_2 \, \mathbf b_2$. However, in general we have  $x_i \neq \mathbf p \cdot \mathbf b_i$. Why is that? 
-Hint: Is this an orthonormal basis?\\+ 
 +++ Hint: | $\quad$ Is this an orthonormal basis? ++
  
-**d)** Demonstrate that the three vectors $\hat{\boldsymbol e}_0 = (1)$, $\hat{\boldsymbol e}_1 = \sqrt{3} \, (2\, x-1) $ and $\hat{\boldsymbol e}_2 = \sqrt{5} \, ( 6\, x^2 - 6\, x + 1)$ are orthonormal.\\+**d)** Demonstrate that the three vectors $\hat{\boldsymbol e}_0 = (1)$, $\hat{\boldsymbol e}_1 = \sqrt{3} \, (2\, x-1) $ and $\hat{\boldsymbol e}_2 = \sqrt{5} \, ( 6\, x^2 - 6\, x + 1)$ are orthonormal.
  
 **e)** Demonstrate that every vector $\mathbf p \in \mathbb{P}_2$ can be written as a scalar combination of $( \hat{\boldsymbol e}_0,  \hat{\boldsymbol e}_1, \hat{\boldsymbol e}_2  )$, **e)** Demonstrate that every vector $\mathbf p \in \mathbb{P}_2$ can be written as a scalar combination of $( \hat{\boldsymbol e}_0,  \hat{\boldsymbol e}_1, \hat{\boldsymbol e}_2  )$,
Line 330: Line 346:
       \mathbf p = ( \mathbf p \cdot \hat{\boldsymbol e}_0 ) \, \hat{\boldsymbol e}_0 + ( \mathbf p \cdot \hat{\boldsymbol e}_1 ) \, \hat{\boldsymbol e}_1 + ( \mathbf p \cdot \hat{\boldsymbol e}_2 ) \, \hat{\boldsymbol e}_2 \, .       \mathbf p = ( \mathbf p \cdot \hat{\boldsymbol e}_0 ) \, \hat{\boldsymbol e}_0 + ( \mathbf p \cdot \hat{\boldsymbol e}_1 ) \, \hat{\boldsymbol e}_1 + ( \mathbf p \cdot \hat{\boldsymbol e}_2 ) \, \hat{\boldsymbol e}_2 \, .
 \end{align*} \end{align*}
-Hence,  $( \hat{\boldsymbol e}_0,  \hat{\boldsymbol e}_1, \hat{\boldsymbol e}_2  )$ form an orthonormal basis of $\mathbb{P}_2$.\\+Hence,  $( \hat{\boldsymbol e}_0,  \hat{\boldsymbol e}_1, \hat{\boldsymbol e}_2  )$ form an orthonormal basis of $\mathbb{P}_2$.
  
-**f)** Find a constant $c$ and a vector $\hat{\boldsymbol n}_1$, such that  $\hat{\boldsymbol n}_0 = (c \, x)$ and $\hat{\boldsymbol n}_1$ form an orthonormal basis of $\mathbb{P}_1$.\\+**f)** Find a constant $c$ and a vector $\hat{\boldsymbol n}_1$, such that  $\hat{\boldsymbol n}_0 = (c \, x)$ and $\hat{\boldsymbol n}_1$ form an orthonormal basis of $\mathbb{P}_1$.
  
 ----- -----
  
-<wrap #quest_forces-21>Problem 2.38:</wrap> ** Systems of linear equations **\\+<wrap #quest_forces-21>Problem 2.38:</wrap> ** Systems of linear equations **
  
 A system of $N$ linear equations of $M$ variables $x_1$, $\dots$ $x_M$ comprises A system of $N$ linear equations of $M$ variables $x_1$, $\dots$ $x_M$ comprises
Line 360: Line 376:
       c   \cdot \mathbf{p} &= \bigl[ c\, p_0 =  c\,p_1 \, x_1 + c\,p_2 \, x_2 + \dots + c\,p_M \, x_M \bigr] \, .       c   \cdot \mathbf{p} &= \bigl[ c\, p_0 =  c\,p_1 \, x_1 + c\,p_2 \, x_2 + \dots + c\,p_M \, x_M \bigr] \, .
 \end{align*} \end{align*}
-How do these operations relate to the operations performed in Gauss elimination to solve the system of linear equations?\\+How do these operations relate to the operations performed in Gauss elimination to solve the system of linear equations?
  
 **b)** The system of linear equations can also be stated in the following form **b)** The system of linear equations can also be stated in the following form
Line 381: Line 397:
 ==== 2.11.4 Transfer and Bonus Problems, Riddles ==== ==== 2.11.4 Transfer and Bonus Problems, Riddles ====
  
-<wrap #quest_2Dmotion-01>Problem 2.39:</wrap> ** Crossing a river **\\+<wrap #quest_2Dmotion-01>Problem 2.39:</wrap> ** Crossing a river **
  
 A ferry is towed at the bank of a river of width $B=100\;$m A ferry is towed at the bank of a river of width $B=100\;$m
Line 389: Line 405:
  
   -  When will it arrive at the other bank when it always heads straight to the other side? (In other words, at any time its velocity is perpendicular to the river bank.) How far will it drift downstream on its journey?   -  When will it arrive at the other bank when it always heads straight to the other side? (In other words, at any time its velocity is perpendicular to the river bank.) How far will it drift downstream on its journey?
-  -  In which direction (i.e. angle of velocity relative to the downstream velocity of the river) must the ferryman head to reach exactly at the opposite side of the river? Determine first the general solution. What happens when you try to evaluate it for the given velocities?\\+  -  In which direction (i.e. angle of velocity relative to the downstream velocity of the river) must the ferryman head to reach exactly at the opposite side of the river? Determine first the general solution. What happens when you try to evaluate it for the given velocities?
  
 ----- -----
  
-<wrap #quest_forces-18>Problem 2.40:</wrap> ** Piling bricks **\\+<wrap #quest_forces-18>Problem 2.40:</wrap> ** Piling bricks **
  
 At Easter and Christmas Germans consume enormous amounts of chocolate. At Easter and Christmas Germans consume enormous amounts of chocolate.
Line 405: Line 421:
  
   -  We consider $N$ bars of length $l$ piled on a table. What is the maximum amount that the topmost bar can reach beyond the edge of the table.   -  We consider $N$ bars of length $l$ piled on a table. What is the maximum amount that the topmost bar can reach beyond the edge of the table.
-  -  The sketch above shows the special case $N=4$. However, what about the limit $N \to \infty$?\\+  -  The sketch above shows the special case $N=4$. However, what about the limit $N \to \infty$?
  
 ----- -----
  
-<wrap #quest_forces-15> Problem 2.41: </wrap> ** Where does the bike go? **\\+<wrap #quest_forces-15> Problem 2.41: </wrap> ** Where does the bike go? **
  
 <WRAP 120pt left> <WRAP 120pt left>
Line 424: Line 440:
 ----- -----
  
-<wrap #quest_2Dmotion-06>Problem 2.42: </wrap>** Hypotrochoids, roulettes, and the Spirograph **\\+<wrap #quest_2Dmotion-06>Problem 2.42: </wrap>** Hypotrochoids, roulettes, and the Spirograph **
  
 A roulette is the curve traced by a point (called the generator or pole) A roulette is the curve traced by a point (called the generator or pole)
Line 450: Line 466:
  
 ~~DISCUSSION|Questions, Remarks, and Suggestions~~ ~~DISCUSSION|Questions, Remarks, and Suggestions~~
 +
book/chap2/2.11_problems.1636387564.txt.gz · Last modified: 2021/11/08 17:06 by jv