Chapter_3_Assignment.sagews

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# Problems to work on your own:

The following are list of problems for you to work offline.  These will **not be collected** but quiz questions may be based on these questions.  Additionally, these are very basic questions that will help you learn how to work the problems.  Partial solutions for **all** of these problems are included in the text on pages 620-669.  Problems that start with a *P* are preliminary problems (fairly basic skills) and problems starting with *F* are fundamental problems, that are a little more challenging but cover the fundamental skills you need to solve the problems in the chapter.

P3-1, P3-2

F3-1, F3-5, F3-6, F3-8, F3-10

Worked Example

I will work through example 3.4 on page 95 in your text to show you how you can make use of SageMath. The next several cells will walk you through the steps of how to set the problem up.

Determine the required length of cord AC so that the 8-kg lamp can be suspended in the position shown. The undeformed length of the spring AB is 0.4 m, and the spring stiffness is 300 N/m.

1) Facts

The sketch of the problem is shown above. If the image was not embedded in the cell above I would include a hand-drawn sketch to show how things look and include all relevant information on the sketch.

Known quantities The mass of the lamp $m_l=8\ kg$ The relaxed length of the spring is $\ell'_{AB} = 0.4 m$ The spring constant is $k_{AB} = 300\ N/m$ The cord AC makes an angle $\theta = 30^{\circ}$ with respect to the negative x-axis The total horizontal distance between B and C is $x_{BC}=2.0 m$

Question Find the length of cord AC

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2) Lacking

We are lacking the following quantities:

• tension of the spring $T_{AB},

• the weight of the lamp $W_L$,

• the tension in the cord $T_{AC},

• the stretched length of the spring $s_{AB}$,

• the stretched length of the spring $\ell_{AB}$,

• and the length of the cord AC $\ell_{AC}$ that we are solving for.

---

3) Approximations and Assumptions

We will make the following simplifying assumptions:

• The object is near the surface of the Earth so the gravitational force is $F_{grav} \approx m g$

• The mass of the cord and the spring is much smaller than the mass of the lamp so we can ignore the weigh of the cord and lamp

4) Representations

The free body diagrams below are the visual representations needed to solve this problem. In fact, the point of all previous steps is to create a free body diagram, which is essentially the equation in graphic form.

The free body diagram is displayed below

How to solve the problem?

• Choose ring as the system - system is in equilibrium so sum of all forces must be zero

• Add up forces in the x-direction and set equal to zero

  • $+\rightarrow\ \ \sum F_x = 0$

• Add up forces in the y-direction and set equal to zero

  • • $+\uparrow\ \ \sum F_y = 0$

• Find the length of the spring $\ell_{AB}$ needed to supply force $T_{AB}$

• Use the fact that $2.0 m - \ell_{AB} =$ bottom leg of triangle formed by AC to find the length of cord AC $\ell_{AC}$

• Use the solve() command in SageMath to find $\ell_{AC}$

5) Solve Problem

Below is the math used to find the resultant force

#This cell shows how to calculate the forces and angles for this example
var('T_AC, T_AB, L_AB, L_AC')

m_L=8.0
g=9.8
theta_1=30/180*pi
L_sp=0.4
k_AB = 300

#Weight of lamp
W_L = m_L*g


#Set up equations with unknowns
## Sum of forces in x-direction are zero
eq1=T_AB -T_AC*cos(theta_1)==0
## Sum of forces in y-direction are zero
eq2=T_AC*sin(theta_1) - W_L ==0
## Spring force is = k*s
eq3=T_AB==k_AB*(L_AB - L_sp)
## Equation for L_AC
eq4=2.0==L_AC*cos(theta_1)+L_AB


#Solve for unknowns
solve([eq1,eq2,eq3,eq4],[T_AB,T_AC,L_AB,L_AC])


# I evaluated the cell before typing the stuff below, then copied and pasted the numerical results for T_AB, T_AC, L_AB, and L_AC

T_AB=n(392/5*sqrt(3),digits=4)
T_AC=n((784/5),digits=4)

L_AB=n(98/375*sqrt(3) + 2/5,digits=3)
L_AC=n(16/15*sqrt(3) - 196/375,digits=3)

pretty_print("Summary of Results")
pretty_print("The magnitude of the force in cord AC is $F_{AC}= $", T_AC," N")
pretty_print("The magnitude of the force in cord AB is $F_{AB}= $", T_AB," N")
pretty_print("The stretched length of the spring  is $\ell_{AB} = $", L_AB, " m")
pretty_print("The length cord AC is $\ell_{AC} = $", L_AC, " m")

(T_AC, T_AB, L_AB, L_AC) [[T_AB == 392/5*sqrt(3), T_AC == (784/5), L_AB == 98/375*sqrt(3) + 2/5, L_AC == 16/15*sqrt(3) - 196/375]]

Summary of Results

The magnitude of the force in cord AC is $F_{AC}=$ $\displaystyle 156.8$ N

The magnitude of the force in cord AB is $F_{AB}=$ $\displaystyle 135.8$ N

The stretched length of the spring is $\ell_{AB} =$ $\displaystyle 0.853$ m

The length cord AC is $\ell_{AC} =$ $\displaystyle 1.32$ m

Summary

See the cell above for the summary of the results

6) Evaluate the answer

Is this answer reasonable?

There are many ways to check this result. One way is to ask, if the spring was unstretched, how long would AC be? We know AC will have to be smaller than this distance because the spring must be stretched.

With $\ell'_{AB}=0.40\ m$, then $2.0 m = \ell_{AC2}\cos(\theta_1)+\ell'_{AB}$

The calculations in the cell below show that $\ell_{AC2}=1.85\ m$ in this case. Since our length of 1.32 m is less than 1.85 m, our results seem reasonable to me.

# Evaluations for example problem
var('L_AC2')
eq5=2.0==L_AC2*cos(theta_1)+L_sp
solve(eq5,L_AC2)

n(16/15*sqrt(3),digits=4)

L_AC2 [L_AC2 == 16/15*sqrt(3)] 1.848

Problem 1

• Complete problem 3-6 on page 98 of your text

• All calculations must be done in SageMath.

1. Write up the first several steps of the problem solving format, which include 1) Facts, 2) Lacking Information, and 3) Approximations and Assumptions (none needed now).

2. Your Facts section must include a sketch of the problem with all relevant information indicated on the diagram.

3. For part (4) of the problem solving format, Representations, draw a free body diagram. Redraw figure (b) below. I've given you a template for this below if you want to use Tikz. Skip figure (c) for now. If you want to draw the free body diagram and insert a picture, create a markdown cell (start the cell with %md), and use <img src="filename" width="600">. Replace filename with the name of your photo and adjust the width so the free body diagram is easily readable.

4. Describe how you are going to solve the problem. Don't just copy the text from the text. Try paraphrasing in your own words. This is good practice. Use a markdown cell for this. I've included a rough template below.

5. Write up your calculations using SageMath. Include all equations and use comments to explain your steps.

6. Summarize and Evaluate your result by answering "Is this answer reasonable?" Use a markdown cell. I've included a rough template for this.

%md

# 1) Facts

# 2) Lacking

# 3) Assumptions and Approximations

- Assume the system is in static equilibrium
- Assume no other forces (such as gravity) are affecting the system

# 4) Representations


### How to solve the problem?

1) Facts

2) Lacking

3) Assumptions and Approximations

• Assume the system is in static equilibrium

• Assume no other forces (such as gravity) are affecting the system

4) Representations

How to solve the problem?

5) Solve

Insert code for calculating the answer below

#Insert your calculations here

%md

## Summary
- Place your summary of your answers here

## Evaluate Answer:
- Write a credible argument as to why your results are reasonable.

### Is this answer reasonable?

Summary

• Place your summary of your answers here

Evaluate Answer:

• Write a credible argument as to why your results are reasonable.

Is this answer reasonable?

Problem 2

• Complete problem 3-13 on page 100 of your text

• All calculations must be done in SageMath.

1. Write up the first several steps of the problem solving format, which include 1) Facts, 2) Lacking Information, and 3) Approximations and Assumptions (none needed now).

2. Your Facts section must include a sketch of the problem with all relevant information indicated on the diagram.

3. For part (4) of the problem solving format, Representations, draw a free body diagram. Redraw figure (b) below. I've given you a template for this below if you want to use Tikz. Skip figure (c) for now. If you want to draw the free body diagram and insert a picture, create a markdown cell (start the cell with %md), and use <img src="filename" width="600">. Replace filename with the name of your photo and adjust the width so the free body diagram is easily readable.

4. Describe how you are going to solve the problem.

5. Write up your calculations using SageMath. Include all equations and use comments to explain your steps.

6. Summarize and Evaluate your result by answering "Is this answer reasonable?" Use a markdown cell. I've included a rough template for this.

%md

# 1) Facts

# 2) Lacking

# 3) Assumptions and Approximations

- Assume the system is in static equilibrium
- Assume no other forces (such as gravity) are affecting the system

# 4) Representations


### How to solve the problem?

1) Facts

2) Lacking

3) Assumptions and Approximations

• Assume the system is in static equilibrium

• Assume no other forces (such as gravity) are affecting the system

4) Representations

How to solve the problem?

5) Solve

Insert code for calculating the answer below

#Insert code to solve the problem here

Problem 3

• Complete problem 3-14 on page 100 of your text

• All calculations must be done in SageMath.

1. Write up the first several steps of the problem solving format, which include 1) Facts, 2) Lacking Information, and 3) Approximations and Assumptions (none needed now).

2. Your Facts section must include a sketch of the problem with all relevant information indicated on the diagram.

3. For part (4) of the problem solving format, Representations, draw a free body diagram. Redraw figure (b) below. I've given you a template for this below if you want to use Tikz. Skip figure (c) for now. If you want to draw the free body diagram and insert a picture, create a markdown cell (start the cell with %md), and use <img src="filename" width="600">. Replace filename with the name of your photo and adjust the width so the free body diagram is easily readable.

4. Describe how you are going to solve the problem.

5. Write up your calculations using SageMath. Include all equations and use comments to explain your steps.

6. Summarize and Evaluate your result by answering "Is this answer reasonable?" Use a markdown cell. You should use your previous work as a template for this problem

Problem 4

• Complete problem 3-43 on page 112 of your text

• All calculations must be done in SageMath.

1. Write up the first several steps of the problem solving format, which include 1) Facts, 2) Lacking Information, and 3) Approximations and Assumptions (none needed now).

2. Your Facts section must include a sketch of the problem with all relevant information indicated on the diagram.

3. For part (4) of the problem solving format, Representations, draw a free body diagram. Redraw figure (b) below. I've given you a template for this below if you want to use Tikz. Skip figure (c) for now. If you want to draw the free body diagram and insert a picture, create a markdown cell (start the cell with %md), and use <img src="filename" width="600">. Replace filename with the name of your photo and adjust the width so the free body diagram is easily readable.

4. Describe how you are going to solve the problem.

5. Write up your calculations using SageMath. Include all equations and use comments to explain your steps.

6. Summarize and Evaluate your result by answering "Is this answer reasonable?" Use a markdown cell. You should use your previous work as a template for this problem

%md

## Summary
- Place your summary of your answers here

## Evaluate Answer:
- Write a credible argument as to why your results are reasonable.

### Is this answer reasonable?

Summary

• Place your summary of your answers here

Evaluate Answer:

• Write a credible argument as to why your results are reasonable.

Is this answer reasonable?

%md

# <img src="./figures/homework.png" width="64"> Problem 5 - Summary of Assignment

- Describe what you learned in this homework and what things you already knew from previous classes.  List any insights or deeper understanding you gained while completing this assignment.

### Things I used that I already knew:
- Explain (briefly) what skills you already knew that you used here.  
*e.g. I used sines and cosines which I learned about in a trigonometry course*



### Things I learned completing this assignment:
- What new skills did you learn while completing this assignment?
*e.g. I learned how to calculate Cartesian components of a vector*



### Insights and understanding gained:
- What insights or understanding did you gain by completing this assigment?
*e.g. I now have a better understanding of how Cartesian vector notation is related to the scalar vector notation and what those two types of notation mean*

Problem 5 - Summary of Assignment

• Describe what you learned in this homework and what things you already knew from previous classes. List any insights or deeper understanding you gained while completing this assignment.

Things I used that I already knew:

• Explain (briefly) what skills you already knew that you used here. e.g. I used sines and cosines which I learned about in a trigonometry course

Things I learned completing this assignment:

• What new skills did you learn while completing this assignment? e.g. I learned how to calculate Cartesian components of a vector

Insights and understanding gained:

• What insights or understanding did you gain by completing this assigment? e.g. I now have a better understanding of how Cartesian vector notation is related to the scalar vector notation and what those two types of notation mean