# Kerja Gaya Gesek

Marshanda Tiana Sarjito
10219066
mrshndtianasarjito@gmail.com
https://github.com/MarshandaTiana

Kerja yang dilakukan oleh gaya gesek merupakan bentuk kerja yang tidak diharapkan karena energi yang dikeluarkan, biasanya dalam bentuk panas atau bunyi yang dilepas ke lingkungan, tidak dapat dimanfaatkan lagi oleh sistem sehingga energi sistem berkurang.

## Gerak Benda di Atas Lantai Mendatar Kasar

Sistem yang ditinjau adalah suatu benda yang bergerak di atas lantai mendatar kasar. Benda diberi kecepatan awal tertentu dan bergerak melambat sampai berhenti karena adanya gaya gesek kinetis antara benda dan lantai kasar.

### Parameter

Beberapa parameter yang digunakan adalah seperti pada tabel berikut ini.

Tabel 1. Simbol beserta satuan dan artinya

Simbol | Satuan | Arti
:-: | :-: | :-
$t$ | s | waktu
$v_0$ | m/s | kecepatan awal
$x_0$ | m | posisi awal
$v$ | m/s | kecepatan saat $t$
$x$ | m | posisi saat $t$
$a$ | m/s$^2$ | percepatan
$\mu_k$ | - | koefisien gesek kinetis
$f_k$ | N | gaya gesek kinetis
$m$ | kg | massa benda
$F$ | N | total gaya yang bekerja
$N$ | N | gaya normal
$w$ | N | gaya gravitasi

Simbol-simbol pada Tabel 1 akan diberi nilai kemudian saat diimplementasikan dalam program.

## Persamaan

Persamaan-persamaan yang akan digunakan adalah seperti dicantumkan pada bagian ini.

### Kinematika

Hubungan antara antara kecepatan $v$, kecepatan awal $v_0$
, percepatan $a$, dan waktu $t$ diberikan oleh

\begin{equation}\label{eqn1}\tag{1}
v = v_0 + at
\end{equation}

Posisi benda $x$ bergantung pada posisi awal $x_0$, kecepatan awal
$v_0$, percepatan $a$, dan waktu $t$ melalui hubungan

\begin{equation}\label{eqn2}\tag{2}
x = x_0 + v_0t + \frac{1}{2}at^2
\end{equation}

Selain kedua persamaan sebelumnya, terdapat pula persamaan berikut

\begin{equation}\label{eqn3}\tag{3}
v^2 = v_0^2 + 2a(x-x_0)
\end{equation}

yang menghubungkan kecepatan $v$ dengan kecepatan awal $v_0$, percepatan $a$, dan jarak yang ditempuh $x-x_0$.

### Dinamika

Hukum Newton I menyatakan bahwa benda yang semula diam akan tetap diam dan yang semula bergerak dengan kecepatan tetap akan tetap bergerak dengan kecepatan tetap bila tidak ada gaya yang bekerja pada benda atau jumlah gaya-gaya yang bekerja sama dengan nol

\begin{equation}\label{eqn4}\tag{4}
\sum F = 0
\end{equation}

Bila ada gaya yang bekerja pada benda bermassa $m$ atau jumlah gaya-gaya tidak nol

\begin{equation}\label{eqn5}\tag{5}
\sum F = ma
\end{equation}

maka keadaan gerak benda akan berubah melalui percepatan $a$, dengan $m$ > 0 dan $a \neq 0$

### Usaha

Usaha oleh suatu gaya $F$ dengan posisi awal $x_0$ dan posisi akhir $x$ dapat diperoleh melalui

\begin{equation}\label{eqn6}\tag{6}
W = \int_{x_0}^{x} F dx
\end{equation}

atau dengan

\begin{equation}\label{eqn7}\tag{7}
W = \Delta K
\end{equation}

dengan $K$ adalah energi kinetik. Persamaan \eqref{eqn7} akan memberikan gaya oleh semua gaya. Dengan demikian bila $F$ adalah satu-satunya gaya yang bekerja pada benda, maka persamaan ini akan menjadi Persamaan \eqref{eqn6}

## Sistem

Ilustrasi sistem perlu diberikan agar dapat terbayangan dan memudahkan penyelesaian masalah. Selain itu juga perlu disajikan diagram gaya-gaya yang bekerja pada benda.

### Ilustrasi

Sistem yang benda bermassa $m$ bergerak di atas lantai kasar dapat digambarkan seperti berikut ini.

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<br/>

Gambar <a name='fig1'>1</a>. Sistem benda bermassa $m$ begerak di atas lantai
mendatar kasar dengan koefisien gesek kinetis $\mu_k$.
```

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<br/>

Gambar <a name='fig1'>1</a>. Sistem benda bermassa $m$ begerak di atas lantai
mendatar kasar dengan koefisien gesek kinetis $\mu_k$.

Keadaan akhir benda, yaitu saat kecepatan $v=0$ diberikan pada bagian kanan Gambar 1 dengan warna abu-abu.

### Diagram Gaya

Diagram gaya-gaya yang berja pada benda perlu dibuat berdasarkan informasi dari Gambar 1 dan Tabel 1, yang diberikan berikut ini.

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<br>

Gambar <a name='fig2'>2</a>. Diagram gaya-gaya yang bekerja pada benda
bermassa $m$.
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<br>

Gambar <a name='fig2'>2</a>. Diagram gaya-gaya yang bekerja pada benda
bermassa $m$.

Terlihat bahwa pada arah $y$ terdapat gaya normal $N$ dan gaya gravitasi $w$, sedangkan pada arah $x$ hanya terdapat gaya gesek kinetis $f_k$ yang melawan arah gerak benda. Arah gerak benda diberikan oleh arah kecepatan $v$.

## Metode Numerik

Integrasi suatu fungsi $f(x)$ berbentuk

\begin{equation}\label{eqn8}\tag{8}
A = \int_{a}^{b} f(x) dx
\end{equation}

dapat didekati dengan

\begin{equation}\label{eqn9}\tag{9}
A \approx \sum_{i=0}^{N} f [\frac{1}{2}(x_i+x_{i+1} )]\Delta x
\end{equation}

yang dikenal sebagai metode persegi titik tengah, dimana

\begin{equation}\label{eqn10}\tag{10}
\Delta x = \frac{b-a}{N}
\end{equation}

dengan $N$ adalah jumlah partisi. Variabel $x_i$ pada Persamaan \eqref{eqn9} diberikan oleh

\begin{equation}\label{eqn11}\tag{11}
x_i = a + i\Delta x
\end{equation}

dengan $i = 0,...,N$.

### Penyelesaian

Penerapan 1, 2, 3, 4, dan 5 pada gambar 2 akan menghasilkan

\begin{equation}\label{eqn12}\tag{12}
f_k = \mu_kmg
\end{equation}

dan usahanya adalah

\begin{equation}\label{eqn13}\tag{13}
W = \int_{x_0}^{x} f_k dx
  = \int_{x_0}^{x} \mu_kmg dx
  = mg\int_{x_0}^{x} \mu_k dx
\end{equation}

dengan koefisien gesek statisnya dapat merupakan fungsi dari posisi $\mu_k = \mu_k(x)$

```python
import numpy as np
import matplotlib.pyplot as plt
plt.ion()

# set integral lower and upper bounds
a = 0
b = 1

# generate x
x = [1, 2, 3, 4, 5]

# generate y from numerical integration
y = [1, 2, 3, 5, 6]

## plot results
fig, ax = plt.subplots()
ax.scatter(x, y)
ax.set_xlabel("$x - x^0$")
ax.set_ylabel("y")

from IPython import display
from IPython.core.display import HTML
HTML('''
<div>
Gambar <a name='fig3'>3</a>. Kurva antara usaha $W$ dan jarak tempuh $x - x_0$.
</div>
''')
```

## Diskusi

Berdasarkan Gambar 3 dapat dijelaskan bahwa dengan $\mu_k=\mu_k(x)$ maka kurva $W(x)$ tidak lagi linier karena dipengaruhi oleh sejauh mana perhitungan kerja dilakukan.

## Kesimpulan

Perhitungan kerja dengan $\mu_k=\mu_k(x)$ telah dapat dilakukan

## Referensi

J. A. C. Martins, J. T. Oden, F. M. F. Simões, "A study of static and kinetic friction", International Journal of Engineerting Science, vol 28, no 1, p 29-92, 1990, url https://doi.org/10.1016/0020-7225(90)90014-A.

Carl Rod Nave, "Friction", HyperPhysics, 2017, url http://hyperphysics.phy-astr.gsu.edu/hbase/frict.html#fri [20220419].

Wikipedia contributors, "Friction", Wikipedia, The Free Encyclopedia, 12 April 2022, 00:33 UTC, url https://en.wikipedia.org/w/index.php?oldid=1082223658 [20220419].

Tia Ghose, Ailsa Harvey, "What is friction?", Live Science, 8 Feb 2022, url https://www.livescience.com/37161-what-is-friction.html [20220419].

```python

```


# Python Programming

**What is Python?** source: [python.org](https://www.python.org/doc/essays/blurb/)

Python is an interpreted, object-oriented, high-level programming language with dynamic semantics. Its high-level built in data structures, combined with dynamic typing and dynamic binding, make it very attractive for Rapid Application Development, as well as for use as a scripting or glue language to connect existing components together. Python's simple, easy to learn syntax emphasizes readability and therefore reduces the cost of program maintenance. Python supports modules and packages, which encourages program modularity and code reuse. The Python interpreter and the extensive standard library are available in source or binary form without charge for all major platforms, and can be freely distributed.

Often, programmers fall in love with Python because of the increased productivity it provides. Since there is no compilation step, the edit-test-debug cycle is incredibly fast. Debugging Python programs is easy: a bug or bad input will never cause a segmentation fault. Instead, when the interpreter discovers an error, it raises an exception. When the program doesn't catch the exception, the interpreter prints a stack trace. A source level debugger allows inspection of local and global variables, evaluation of arbitrary expressions, setting breakpoints, stepping through the code a line at a time, and so on. The debugger is written in Python itself, testifying to Python's introspective power. On the other hand, often the quickest way to debug a program is to add a few print statements to the source: the fast edit-test-debug cycle makes this simple approach very effective.

**What is Python capable of?***

Long answer short: everything. Most often Python is used for web developments and prototyping, using frameworks such as [Flask](http://flask.palletsprojects.com/en/1.1.x/) and [Django](https://www.djangoproject.com/). Because of its simplicity, it is also popularly used in data science, both in industry and in academia.

In this lab, we mainly cover the following python programming topics:

data types

built-in data structure

I/O in Python

loops in Python

function and class

## Data Types

```python
# integer
a = 1
print('I am an integer:', a)
```

```python
# float -> decimal numbers
a = 1.0
print('I am a float:', a)
```

```python
# strings -> an array of characteristics
a = "GOTO Chicago Fall 2019"
print('I am a string:', a)
```

```python
# boolean -> binary
a = True
print('I am a binary:', a)
```

**Data Type Conversion**

```python
# integer to float
a = 1
print('I was an integer:', a, ', but now I am a float:', float(a))
```

```python
# float to integer
a = 1.0
print('I was a float:', a, ', but now I am an integer:', int(a))
```

```python
# string to integer/float
a = '1.0'
print('I was a string: "'+ a +'", but now I am a number:', eval(a))
```

```python
# integer/float to string
a = 1.0
print('I was a number:', a, ', but now I am a string:', str(a))
```

Additional material for reading if interested: [source](https://www.geeksforgeeks.org/type-conversion-python/)

## Data Structures

```python
# list -> mutable
a = [1, 2, 3, 'a', 'b', 'c']
print('I am a list:', a)
```

```python
# tuple -> immutable
a = (1, 2, 3, 'a', 'b', 'c')
print('I am a tuple:', a)
```

```python
# dictionary
a = {'1': 'a', 2: 'b', 'c': 3}
print('I am a dictionary:', a)
```

**List**

List is mutable, meaning after a list is defined, you can alter the values stored in it. **Tuple** is not mutable, meaning once it's defined, you can't change the values. Let's focus on the list for now.

```python
a = [1, 2, 3]
```

```python
# select an item
a[0]
```

```python
# select a range of items
a[0:2]
```

```python
# add an item
# method 1: append to the end of the list
a.append(4)
a
```

```python
# method 2: insert into a specific position
a.insert(2, 10)
a
```

```python
# remove an item
# method 1: pop the last item in the list
a.pop()
```

```python
a
```

```python
# method 2: remove a specific position
a.pop(2)
```

```python
a
```

```python
# method 3: remove a specific value
a.remove(3)
```

```python
a
```

**Dictionary**

```python
# initiate an empty dictionary
a = dict()
a
```

```python
# add an item
# dictionary[key_name]= value_name
a['my_key'] = 'my_value'
a['my_key_2'] = 'my_value_2'
a
```

```python
# select a value based on key
a['my_key']
```

```python
# remove an item
# method 1: pop the item added last
a.popitem()
```

```python
a
```

```python
# method 2: pop the key
a.pop('my_key')
```

```python
a
```

**Substring**

```python
a = 'I am a string.'
```

```python
# select a sub string out of the main string
a[:5]
```

## Input/Output (I/O) in Python

Imagine you're reading a book, what are the steps for you to finish the process of reading a book? It should all include three steps,

Open the book

Read the book

Close the book

Similar in Python, to read a text file from your local, you need to open a the file, read the content, and close the file. Here is how to do it.

**Step-by-Step Version**

```python
# open the file
path = './data/dummy.txt'
file = open(path, 'r')
```

`r` denotes read only mode.

```python
# read the content
content = file.read()
```

```python
print(content)
```

```python
# close the file
file.close()
```

Now you can work with the content in `content` variable

```python
content
```

```python
# split() is a built-in method with Python's string datatype that splits a string using the sub-string
# specified; if nothing is specified, it splits by a single space by default
content.split('\n')
```

**Question:** What is the output data structure of the split method?

**"All-in-One" or The Clean Version**

```python
# "with" method takes care of closing the file so you don't have to worry about that
with open('./data/dummy.txt', 'r') as file:
    content2 = file.read()
```

```python
content2
```

Though you may have heard about tools like `Pandas` and `Numpy` that take care of I/O for you, knowing how to correctly read files using native Python is also important because data come in different formats.

## Loops in Python

Given that you are already a developer, this section is just to show you how to write for and while loops in Python.

**For Loop**

```python
my_list = [1, 2, 3, '4', '5', '7', True]
```

```python
for item in my_list:                        # loop through the list
    if isinstance(item, str):               # if the item is a string
        print(item, 'is a string.')
    else:
        print(item, 'is not a string.')
```

**While Loop**

```python
i = 0
while i < len(my_list):
    item = my_list[i]
    if isinstance(item, str):               # if the item is a string
        print(item, 'is a string.')
    else:
        print(item, 'is not a string.')
    i += 1
```

**Break and Continue**

`break`: it breaks out of a for loop and continue the code

`continue`: it skips the rest of the code in the current loop and moves onto the next iteration

```python
for item in my_list:
    if not isinstance(item, str):
        continue
        print('this is not a string.. this line is not even executed at all')
    else:
        print(item)
```

**Q&A:** What happens if we swap out `continue` with `break`?

## Function and Class

A `function` is a set of python instructions/statements that take in inputs, do things to them, and output the results out. Note: This is the standard function definition, and not all functions need to take in inputs and/or returns output.

A `class` is a code template that creates objects and it contains one or more functions. If you have experience in Java, it is just like the Java class.

```python
# function example
def my_function(a_input):                        # define a function
    output = str(a_input) + ' is the input'     # do something to the input
    return output                               # return output
```

```python
my_function(5)
```

```python
# class example
class my_class(object):
    """i am a doc string :) """

    def __init__(self, input_str=None):
        """I am the first function to be called when a new object is initiated"""
        self.input_str = str(input_str)
        self.output_str = None

    def do_something(self):
        """I am a separate method that does something in this class
        Note that I don't have to return anything here
        """
        self.output_str = self.input_str + ' is the input'

    def give_me_output(self):
        return self.output_str
```

```python
new_instance = my_class(5)     # at this step, __init__ is called
```

```python
new_instance.do_something()    # here, do_something() is called
```

```python
new_instance.give_me_output()  # here, give_me_output() is called
```

Finally, if you care about code styling a lot, which you should, feel free to check out PEP 8 Python coding styling standards [here](https://www.python.org/dev/peps/pep-0008/).

**Exercise**

If time permits, try build a calculator that does the following in Python.

Addition

Subtraction

Multiplication

Division

```python
class MyCalculator(object):
    """insert your doc string here"""

    def __init__(self):
        pass

    def add(self):
        pass

    def subtract(self):
        pass

    def multiply(self):
        pass

    def divide(self):
        pass
```

# Statistics

Whew, I'm sure the Python section bored you out a little. Let's learn something more fun.

Let's first use Python's `random` module to generate 1,000,000 random numbers.

```python
import random
random.seed(1234)
# by setting a seed, you can ensure every time you run this block of code, the output is the same

dataset = [random.random() for _ in range(1_000_000)]
# underscore means that variable is not important to store

import statistics # let's import this for later
```

```python
len(dataset)
```

```python
dataset[:5]
```

**Arithmetic Mean a.k.a. Average**

```python
def average(arr):
    return sum(arr) / len(arr)
```

```python
mean = average(dataset)
```

```python
mean
```

```python
# validate
statistics.mean(dataset)
```

**Median**

```python
def median(arr):
    arr = sorted(arr)
    median_index = int(len(arr) / 2)
    if len(arr) % 2 == 0:      # if the list has even number of elements
        median = average([arr[median_index], arr[median_index-1]])
    else:
        median = arr[median_index]
    return median
```

```python
median(dataset)
```

```python
# validate
statistics.median(dataset)
```

**Variance**

```python
def variance(arr):
    mean = average(arr)      # note that I am using the average() function here
    variance = sum(
                map(lambda i: (i-mean)**2, arr)
                  ) / (len(arr) - 1)
    return variance
```

`map` operation "maps" a function to every element of a collection passed in the second section.

`lambda` is a Python way to explicitly define small-scale function anonymously; see more [here](https://realpython.com/python-lambda/)

```python
variance(dataset)
```

```python
# validate
statistics.variance(dataset)
```

**Standard Deviation**

```python
def stdv(arr):
    var = variance(arr)
    standard_deviation = var ** 0.5
    return standard_deviation
```

```python
stdv(dataset)
```

```python
# validate
statistics.stdev(dataset)
```

Ok... I got the math now, but what do all of these mean???????

Nomral distribution is a bell-curve shaped distribution, where `mean` equals to `median`. In addition, it is estimated that

68% of the data fall between +/-1 standard deviation>

95% of the data fall between +/-2 standard deviation>

99% of the data fall between +/-3 standard deviation>

To calculate how far the value is away from the sample mean, `z-score` is calculated which we do not cover today. Read more about `z-score` [here](https://www.investopedia.com/terms/z/zscore.asp).

## Let's Make It A Bit More Complicated

We've been focusing on one variable. Let's now look into the descriptive statistics of two variables.

Assuming we have two variables, `X` and `Y`, we can calculate the same descriptive statistics above for both of them such that

         Average

            \begin{equation}
                \bar{X} = \frac{\displaystyle\sum_{i=1}^n x_i}{n}
            \end{equation}

            \begin{equation}
                \bar{Y} = \frac{\displaystyle\sum_{i=1}^n y_i}{n}
            \end{equation}

         Variance

            \begin{equation}
                Var(X) = \sigma_x^2 = \frac{\displaystyle\sum_{i=1}^{n}(x_i - \bar{X})^2} {n-1}
            \end{equation}

            \begin{equation}
                Var(Y) = \sigma_y^2 = \frac{\displaystyle\sum_{i=1}^{n}(y_i - \bar{Y})^2} {n-1}
            \end{equation}

         Standard Deviation

            \begin{equation}
                Stdev(X) = \sqrt{Var(X)}
            \end{equation}

            \begin{equation}
                Stdev(Y) = \sqrt{Var(Y)}
            \end{equation}

**Covariance**

Covariance is a measure of how much two random variables vary together.

\begin{equation}
    cov_{x,y}=\frac{\displaystyle\sum_{i=1}^{N}(x_{i}-\bar{x})(y_{i}-\bar{y})}{N-1}
\end{equation}

If the covariance is positive, it tells us that the two variables are positively related, and vice versa. However, covariance does not have a upper nor lower limit, thus we cannot compare the magnitude of inter-variable relationship.

**Correlation**

Correlation coefficients are used to measure how strong the relationship is between two variables. There are several types of correlation coefficients and Pearson's Correlation is commonly used in describing linear relations.

\begin{equation}
    cor_{x,y} = \frac{cov_{x, y}}{\sigma_x\sigma_y}
\end{equation}

```python
# Let's generate two random variables with the same size
import numpy as np
# numpy (Numerical Python) is a very popular data science tool (https://numpy.org/)

x = np.random.random(1_000_000)
y = np.random.random(1_000_000)
```

```python
len(x), len(y)
```

```python
x[:5]
```

```python
# Let's calculate covariance
np.cov(x, y)
```

The output above is known as `variance-covariance matrix`, where the diagonal represents the variance of each variable, and the other positions represents the covariance of different variable pairs, such that

        X
        Y

        X
        Var(X)
        Cov(X, Y)

        Y
        Cov(X, Y)
        Var(Y)

**Note:** Cov(X, X) is equal to Var(X)

```python
# Let's calculate correlation
np.corrcoef(x, y)
```

The output above is known as `correlation matrix`, where the diagonal represents the correlation of the variable with itself, and the other positions represents the correlation of different variable pairs, such that

        X
        Y

        X
        Corr(X, X)
        Corr(X, Y)

        Y
        Corr(X, Y)
        Corr(Y, Y)

**Note:** Corr(X, X) is 1 -> A variable is perfectly postiively correlated with itself.

# Recap

In this session, we've covered

Basic Python programming

Statistics for single variable

Descriptive statistics for two variables

You should be able to

Complete the Calculator exercise

Explain what descriptive statistics are appropriate for sing variable

Interpret the implications of descriptive statistics

Explain the difference between covariance and correlation

Interpret Pearson's Correlation Coefficient

```python

```


Dinámica

Capítulo 2: Cinemática y Cinética de partículas

Movimiento curvilíneo

2021/02

MEDELLÍN - COLOMBIA

 Text provided under a Creative Commons Attribution license, CC-BY. All code is made available under the FSF-approved MIT license.(c) Carlos Alberto Alvarez Henao

***

***Docente:*** Carlos Alberto Álvarez Henao, I.C. D.Sc.

***e-mail:*** carlosalvarezh@gmail.com

***skype:*** carlos.alberto.alvarez.henao

***Linkedin:*** https://www.linkedin.com/in/carlosalvarez5/

***github:*** https://github.com/carlosalvarezh/Dinamica

***Herramienta:*** [Jupyter](http://jupyter.org/)

***Kernel:*** Python 3.9

***

Tabla de Contenidos

1  Movimiento curvilíneo General

1.1  Definición

1.2  Posición

1.3  Desplazamiento

1.4  Velocidad

1.5  Aceleración

1.6  Comentarios al movimiento curvilíneo general

2  Movimiento curvilíneo: Componente rectangulares

2.1  Posición

2.2  Velocidad

2.3  Aceleración

2.4  Ejemplos movimiento curvilínea componentes rectangulares

2.4.1  Globo atmosférico

 Fuente: Universidade do Porto

## Movimiento curvilíneo General

### Definición

El [movimiento curvilíneo](https://en.wikipedia.org/wiki/Curvilinear_motion) es aquél que se describe cuando una partícula se desplaza a lo largo de una trayectoria curva ([parabólico](https://en.wikipedia.org/wiki/Projectile_motion), [oscilatorio](https://en.wikipedia.org/wiki/Oscillation) o [circular](https://en.wikipedia.org/wiki/Circular_motion)). Se emplearán los conceptos de [análisis vectorial](https://en.wikipedia.org/wiki/Vector_calculus) para describir la posición, velocidad y aceleración de la partícula en las tres dimensiones espaciales. En este capítulo se considerarán también tres tipos de [sistemas de coordenadas](https://en.wikipedia.org/wiki/Coordinate_system) usadas frecuentemente en el análisis de este tipo de movimiento.

### Posición

 Fuente: Hibbeler R. Engineering Mechanics: Dynamics

Sea una partícula situada en un punto de una curva espacial definida por la función de trayectoria $s(t)$. El vector de posición $\vec{\boldsymbol{r}} = \vec{\boldsymbol{r}}(t)$ designará la posición de la partícula, medida con respecto a un punto fijo $O$. Tanto la magnitud como la dirección de este vector cambiarán a medida que la partícula se mueve a lo largo de la curva.

### Desplazamiento

 Fuente: Hibbeler R. Engineering Mechanics: Dynamics

Si durante un breve intervalo $\Delta t$ la partícula se mueve una distancia $\Delta s$ a lo largo de la curva a una nueva posición, definida por $\vec{\boldsymbol{r'}} = \vec{\boldsymbol{r}} +\Delta \vec{\boldsymbol{r}}$, el desplazamiento $\Delta \vec{\boldsymbol{r}}$ representará el cambio de posición de la partícula y se determina mediante la resta vectorial $\Delta \vec{\boldsymbol{r}}= \vec{\boldsymbol{r'}} - \vec{\boldsymbol{r}}$.

### Velocidad

 Fuente: Hibbeler R. Engineering Mechanics: Dynamics

Durante el tiempo $\Delta t$, la velocidad promedio de la partícula es

\begin{equation*}
\vec{\boldsymbol{v}}_{prom}=\frac{\Delta \vec{\boldsymbol{r}}}{\Delta t}
\label{eq:Ec2_1} \tag{2.1}
\end{equation*}

La *velocidad instantánea* se determina cuando $\Delta t \rightarrow 0$, entonces, la dirección de $\Delta \vec{\boldsymbol{r}}$ tiende a la *tangente* a la curva y,

\begin{equation*}
\vec{\boldsymbol{v}}=\lim \limits_{\Delta t \to 0} \frac{\Delta \vec{\boldsymbol{r}}}{\Delta t}=\frac{d{\boldsymbol{r}}}{dt}
\label{eq:Ec2_2} \tag{2.2}
\end{equation*}

Como $\Delta \vec{\boldsymbol{r}}$ será tangente a la curva, la dirección de $\vec{\boldsymbol{v}}$ también es tangente a la curva. La magnitud de $\vec{\boldsymbol{v}}$, conocida como la rapidez, se obtiene al tener en cuenta que la longitud del segmento de línea recta $\Delta \vec{\boldsymbol{r}}$ tiende la longitud de arco $\Delta s$ a medida que $\Delta t \rightarrow 0$, tenemos

\begin{equation*}
\vec{\boldsymbol{v}}=\lim \limits_{\Delta t \to 0} \frac{\Delta \vec{\boldsymbol{r}}}{\Delta t}=\lim \limits_{\Delta t \to 0} \frac{\Delta s}{\Delta t}=\frac{ds}{dt}
\label{eq:Ec2_3} \tag{2.3}
\end{equation*}

### Aceleración

 Fuente: Hibbeler R. Engineering Mechanics: Dynamics

Si la velocidad de la partícula es $\vec{\boldsymbol{v}}$ en el instante $t$ y $\vec{\boldsymbol{v'}}=\vec{\boldsymbol{v}}+\Delta \vec{\boldsymbol{v}}$ en el instante $t + \Delta t$, entonces la aceleración promedio de la partícula durante el intervalo $\Delta t$ es

\begin{equation*}
\vec{\boldsymbol{a}}_{prom}=\frac{\Delta \vec{\boldsymbol{v}}}{\Delta t}
\label{eq:Ec2_4} \tag{2.4}
\end{equation*}

Para estudiar la tasa de cambio en el tiempo, los dos vectores de velocidad en la figura anterior se trazan en la siguiente figura

 Fuente: Hibbeler R. Engineering Mechanics: Dynamics

de modo que sus colas queden en el punto fijo $O'$ y sus cabezas de punta de flecha toquen puntos situados en la curva. Esta curva se llama [hodógrafa](https://en.wikipedia.org/wiki/Hodograph) y describe el lugar geométrico de puntos para la cabeza de punta de flecha del vector de velocidad, así como la trayectoria $s$ describe el lugar geométrico de puntos para la cabeza de punta de flecha del vector de posición.

La *aceleración instantánea* se determina cuando $\Delta t \rightarrow 0$, entonces, en el límite $\Delta \vec{\boldsymbol{v}}$ la tangente tenderá a la *hodógrafa* y por lo tanto

\begin{equation*}
\vec{\boldsymbol{a}}=\lim \limits_{\Delta t \to 0} \frac{\Delta \vec{\boldsymbol{v}}}{\Delta t}=\frac{d{\boldsymbol{v}}}{dt}
\label{eq:Ec2_5} \tag{2.5}
\end{equation*}

este resultado también puede ser escrito en función del vector posición $\vec{\boldsymbol{r}}$ de la siguiente manera

\begin{equation*}
\vec{\boldsymbol{a}}=\frac{d^2{\boldsymbol{r}}}{dt^2}
\label{eq:Ec2_6} \tag{2.6}
\end{equation*}

### Comentarios al movimiento curvilíneo general

- Por definición de la derivada, $\vec{\boldsymbol{a}}$ actúa tangente a la hodógrafa,

 Fuente: Hibbeler R. Engineering Mechanics: Dynamics

y, en general no es tangente a la trayectoria del movimiento.

 Fuente: Hibbeler R. Engineering Mechanics: Dynamics

- Tanto $\Delta \vec{\boldsymbol{v}}$ como $\vec{\boldsymbol{a}}$, deben responder el cambio tanto de magnitud como de dirección de la velocidad $\vec{\boldsymbol{v}}$ a medida que la partícula se mueve de un punto al siguiente a lo largo de la trayectoria.

- Para que la partícula siga cualquier trayectoria curva, el cambio direccional siempre *“cambia”* el vector de velocidad hacia el *“interior”* o *“lado cóncavo”* de la trayectoria, y por consiguiente a no puede permanecer tangente a la trayectoria.

- En conclusión, $\vec{\boldsymbol{v}}$ siempre es tangente a la trayectoria y $\vec{\boldsymbol{a}}$ siempre es tangente a la hodógrafa.

## Movimiento curvilíneo: Componente rectangulares

El movimiento curvilíneo también se puede expresar en función de las coordenadas $x$, $y$, $z$ que cubren su trayectoria.

### Posición

 Fuente: Hibbeler R. Engineering Mechanics: Dynamics

Si la partícula está en el punto $(x, y, z)$ de la trayectoria curva $s$, entonces su posición se definirá por el vector:

\begin{equation*}
\vec{\boldsymbol{r}}=x\vec{\boldsymbol{i}}+y\vec{\boldsymbol{j}}+z\vec{\boldsymbol{k}}
\label{eq:Ec2_7} \tag{2.7}
\end{equation*}

***Comentarios:***

- Cuando la partícula se mueve, las compontes del vector $\vec{\boldsymbol{r}}$ se representan en función del tiempo, es decir, $x(t), y(t), z(t)$. Por lo que $\vec{\boldsymbol{r}}=\vec{\boldsymbol{r}}(t)$

- La magnitud de $\vec{\boldsymbol{r}}$ estará dada por la [norma euclideana](https://en.wikipedia.org/wiki/Euclidean_space#Euclidean_norm):

\begin{equation*}
r=\sqrt{x^2+y^2+z^2}
\label{eq:Ec2_8} \tag{2.8}
\end{equation*}

- La dirección del vector $\vec{\boldsymbol{r}}$ se determina a través del [vector unitario](https://en.wikipedia.org/wiki/Unit_vector):

\begin{equation*}
\vec{\boldsymbol{u}}_r=\frac{\vec{\boldsymbol{r}}}{r}
\label{eq:Ec2_9} \tag{2.9}
\end{equation*}

### Velocidad

 Fuente: Hibbeler R. Engineering Mechanics: Dynamics

Derivando el vector posición respecto al tiempo, se obtendrá el vector velocidad:

\begin{equation*}
\vec{\boldsymbol{v}}=\frac{d\vec{\boldsymbol{r}}}{dt}=\frac{d}{dt}(x\vec{\boldsymbol{i}})+\frac{d}{dt}(y\vec{\boldsymbol{j}})+\frac{d}{dt}(z\vec{\boldsymbol{k}})
\label{eq:Ec2_10} \tag{2.10}
\end{equation*}

***Comentarios:***

- Tener en cuenta que al derivar es necesario derivar tanto la magnitud como la dirección en cada componente, es decir:

\begin{equation*}
\begin{aligned}
\frac{d}{dt}(x\vec{\boldsymbol{i}})=\frac{dx}{dt}\vec{\boldsymbol{i}}+x\frac{d\vec{\boldsymbol{i}}}{dt} \\
\frac{d}{dt}(y\vec{\boldsymbol{j}})=\frac{dy}{dt}\vec{\boldsymbol{j}}+y\frac{d\vec{\boldsymbol{j}}}{dt} \\
\frac{d}{dt}(z\vec{\boldsymbol{k}})=\frac{dz}{dt}\vec{\boldsymbol{k}}+z\frac{d\vec{\boldsymbol{k}}}{dt}
\end{aligned}
\label{eq:Ec2_11} \tag{2.11}
\end{equation*}

- El segundo término del lado derecho en cada una de las ecuaciones anteriores será cero si el marco de referencia es fijo, por lo que la dirección y magnitud de $\vec{\boldsymbol{i}}$ no cambia con el tiempo.

- Es usual representar el vector velocidad también como:

\begin{equation*}
\vec{\boldsymbol{v}}=\frac{d\vec{\boldsymbol{r}}}{dt}=v_x\vec{\boldsymbol{i}}+v_y\vec{\boldsymbol{j}}+v_z\vec{\boldsymbol{k}}
\label{eq:Ec2_12} \tag{2.12}
\end{equation*}

- O usando la notación "punto" que representa las primeras derivadas de $x=x(t)$, $y=y(t)$, $z=z(t)$: $v_x=\dot{x}$, $v_y=\dot{y}$, $v_z=\dot{z}$

- La magnitud de la velocidad es a su vez:

\begin{equation*}
v=\sqrt{v_x^2+v_y^2+v_z^2}
\label{eq:Ec2_13} \tag{2.13}
\end{equation*}

- y el vector unitario, que especifica su dirección y es tangente a la trayectoria, está dado por:

\begin{equation*}
\vec{\boldsymbol{u}}_v=\frac{\vec{\boldsymbol{v}}}{v}
\label{eq:Ec2_14} \tag{2.14}
\end{equation*}

### Aceleración

 Fuente: Hibbeler R. Engineering Mechanics: Dynamics

Sacando la primera derivada del vector velocidad respecto al tiempo (o la segunda derivada del vector posición), se obtiene el vector aceleración:

\begin{equation*}
\vec{\boldsymbol{a}}=\frac{d\vec{\boldsymbol{v}}}{dt}=a_x\vec{\boldsymbol{i}}+a_y\vec{\boldsymbol{j}}+a_z\vec{\boldsymbol{k}}
\label{eq:Ec2_15} \tag{2.15}
\end{equation*}

***Comentarios:***

- Usando la notación "punto":

\begin{equation*}
\begin{aligned}
a_x=\dot{v}_x=\ddot{x} \\
a_y=\dot{v}_y=\ddot{y} \\
a_z=\dot{v}_z=\ddot{z}
\end{aligned}
\label{eq:Ec2_16} \tag{2.16}
\end{equation*}

- La magnitud de la aceleración estará dada por,

\begin{equation*}
a=\sqrt{a_x^2+a_y^2+a_z^2}
\label{eq:Ec2_17} \tag{2.17}
\end{equation*}

- el vector unitario, está dado por:

\begin{equation*}
\vec{\boldsymbol{u}}_a=\frac{\vec{\boldsymbol{a}}}{a}
\label{eq:Ec2_18} \tag{2.18}
\end{equation*}

Como la aceleración representa el cambio tanto de la magnitud como de la dirección de la velocidad, en geneeral, no es tangente a la trayectoria.

### Ejemplos movimiento curvilínea componentes rectangulares

#### Globo atmosférico

 Ejemplo 12.9: Hibbeler R. Engineering Mechanics: Dynamics

En cualquier instante $x=(8t)pies$, donde $t$ está en segundos, define la posición horizontal del globo atmosférico de la figura. Si la ecuación de la trayectoria es $y = x^2/10$, determina la magnitud y dirección de la velocidad y la aceleración cuando $t = 2 s$.

***Solución analítica:***

- ***Velocidad:***

La componente de la velocidad en la dirección $x$ se obtiene como:

$$v_x=\dot{x}=\frac{d}{dt}(8t)=8 pies/s \rightarrow$$

Para determinar la componente de la velocidad en la dirección $y$ es necesario emplear la [regla de la cadena](https://en.wikipedia.org/wiki/Chain_rule) vista en cálculo

$$v_y=\dot{y}=\frac{d}{dt}(x^2/10)=2x\dot{x}/10=2(16)(8)/10=25.6pies/s \uparrow$$

en $t=2s$, la magnitud de la velocidad es:

$$v=\sqrt{(8 pies/s)^2+(25.6pies/s)^2}=26.8 pies/s$$

Para determinar la dirección, se debe tener presente que ésta se calcula como el ángulo respecto al eje $x$ y es tangente a la trayectoria.

 Fuente: Hibbeler R. Engineering Mechanics: Dynamics

$$\theta_v=tan^{-1}\frac{v_y}{v_x}=tan^{-1}\frac{25.6}{8}=72.6^\circ$$

- ***Aceleración:***

Procediendo de la misma forma que para la velocidad, se empleará la Regla de la Cadena para determinar las componentes de la aceleración

$$a_x=\dot{v}_x=\frac{d}{dt}(8)=0$$
$$a_y=\dot{v}_y=\frac{d}{dt}(2x\dot{x}/10)=2(\dot{x})\dot{x}/10+2x\ddot{x}/10$$

\begin{equation*}
\begin{aligned}
a_y &=\dot{v}_y=\frac{d}{dt}(2x\dot{x}/10)=2(\dot{x})\dot{x}/10+2x\ddot{x}/10 \\
    &=2(8)^2/10+2(16)(0)=12.8pies/s^2 \uparrow
\end{aligned}
\end{equation*}

la magnitud estaría dada por

$$a=\sqrt{(0)^2+(12.8)^2}=12.8 pies/s^2$$

y su dirección

$$\theta_a=tan^{-1}\frac{12.8}{0}=90^\circ$$

***Solución computacional***

```python
from sympy import *
t = symbols('t')
init_printing(use_latex='mathjax')
```

Para el cálculo de las componentes de la velocidad tanto en la dirección $x$ como $y$ se procederá de la siguiente forma.

- Se crean las funciones de la posición y la trayectoria

```python
# Ecuación de la posición x en un instante t
x = 8 * t

# Ecuación de la trayectoria
y = x**2 / 10
```

- La velocidad es la derivada del espacio, para cada una de sus componentes, respecto al tiempo

```python
# Cálculo de la velocidad en la dirección x
vx = diff(x,t)
vx
```

$\displaystyle 8$

```python
# Evaluación de la velocidad en x cuando t = 2 (en este caso es una constante)
vx = vx.subs(t,2)
vx
```

$\displaystyle 8$

```python
# calculo de la velocidad en la dirección x, y evaluación en t=2 en una misma línea de código
vx = diff(x,t).subs(t,2)
vx
```

$\displaystyle 8$

```python
# Cálculo de la velocidad en la dirección y
vy = diff(y,t)
vy
```

$\displaystyle \frac{64 t}{5}$

```python
# Evaluación de la velocidad en y cuando t = 2 y representación en punto flotante (Numerical)
vy = N(vy.subs(t,2))
vy
```

$\displaystyle 25.6$

- Se cacula la magnitud euclideana

```python
magVel = sqrt(vx**2 + vy**2).evalf(3)
magVel
```

$\displaystyle 26.8$

```python
magVel = N(sqrt(vx**2 + vy**2),3)
magVel
```

$\displaystyle 26.8$

- Para el cálculo de la dirección hay qué tener en cuenta que los valores resultantes de los cálculos numéricos se expresan en [radianes](https://es.wikipedia.org/wiki/Radi%C3%A1n). Por lo que es necesario convertirlos a grados mediante la fórmula

$$x[grad] = x [rad] \times \frac{180}{\pi}$$

```python
theta = N(atan(vy / vx) * 180 / pi, 3)
theta
```

$\displaystyle 72.6$

Se deja como ejercicio al estudiante el desarrollo del punto que corresponde a la aceleración