```python
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   "source": [
    "# Kerja Gaya Gesek\n",
    "\n",
    "Sparisoma Viridi1, Raden Roro Zahra Auliya S2
\n",
    "Program Studi Sarjana Fisika, Institut Teknologi Bandung
\n",
    "Jalan Gensha 10, Bandung 40132, Indonesia
\n",
    "1dudung@gmail.com, https://github.com/dudung
\n",
    "2rzahraauliya@gmail.com, https://github.com/RRZahra\n",
    "\n",
    "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."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aed2f31a",
   "metadata": {},
   "source": [
    "## Gerak benda di atas lantai mendatar kasar\n",
    "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."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dc4acf61",
   "metadata": {},
   "source": [
    "## Parameter\n",
    "Beberapa parameter yang digunakan adalah seperti pada tabel berikut ini.\n",
    "\n",
    "Tabel 1. Simbol beserta satuan dan artinya.\n",
    "\n",
    "Simbol | Satuan | Arti\n",
    ":- | :- | :-\n",
    "$t$ | s | waktu\n",
    "$v_0$ | m/s | kecepatan awal\n",
    "$x_0$ | m | posisi awal\n",
    "$v$ | m/s | kecepatan saat $t$\n",
    "$x$ | m | waktu saat $t$\n",
    "$a$ | m/s2 | percepatan\n",
    "$\\mu_k$ | - | koefisien gesek kinetis\n",
    "$f_k$ | N | gaya gesek kinetis\n",
    "$m$ | kg | massa benda\n",
    "$F$ | N | total gaya yang bekerja\n",
    "$N$ | N | gaya normal\n",
    "$w$ | N | gaya gravitasi\n",
    "\n",
    "Simbol-simbol pada Tabel [1](#tab1) akan diberi nilai kemudian saat diimplementasikan dalam program."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "12457da3",
   "metadata": {},
   "source": [
    "## Persamaan\n",
    "Persamaan-persamaan yang akan digunakan adalah seperti dicantumkan pada bagian ini."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb54dc5c",
   "metadata": {},
   "source": [
    "### Kinematika\n",
    "Hubungan antara antara kecepatan $v$, kecepatan awal $v_0$, percepatan $a$, dan waktu $t$ diberikan oleh\n",
    "\n",
    "\n",
    "\\begin{equation}\\label{eqn:kinematics-v-a-t}\\tag{1}\n",
    "v = v_0 + at.\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0e52be88",
   "metadata": {},
   "source": [
    "Posisi benda $x$ bergantung pada posisi awal $x_0$, kecepatan awal $v_0$, percepatan $a$, dan waktu $t$ melalui hubungan\n",
    "\n",
    "\n",
    "\\begin{equation}\\label{eqn:kinematics-x-v-a-t}\\tag{2}\n",
    "x = x_0 + v_0 t + \\tfrac12 at^2.\n",
    "\\end{equation}\n"
   ]
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  {
   "cell_type": "markdown",
   "id": "3c8a08fa",
   "metadata": {},
   "source": [
    "Selain kedua persamaan sebelumnya, terdapat pula persamaan berikut\n",
    "\n",
    "\n",
    "\\begin{equation}\\label{eqn:kinematics-v-x-a}\\tag{3}\n",
    "v^2 = v_0^2 + 2a(x - x_0),\n",
    "\\end{equation}\n",
    "\n",
    "yang menghubungkan kecepatan $v$ dengan kecepatan awal $v_0$, percepatan $a$, dan jarak yang ditempuh $x - x_0$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4fbcd3b6",
   "metadata": {},
   "source": [
    "### Dinamika\n",
    "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\n",
    "\n",
    "\n",
    "\\begin{equation}\\label{eqn:newtons-law-1}\\tag{4}\n",
    "\\sum F = 0.\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c237c502",
   "metadata": {},
   "source": [
    "Bila ada gaya yang bekerj pada benda bermassa $m$ atau jumlah gaya-gaya tidak nol\n",
    "\n",
    "\n",
    "\\begin{equation}\\label{eqn:newtons-law-2}\\tag{5}\n",
    "\\sum F = ma,\n",
    "\\end{equation}\n",
    "\n",
    "maka keadaan gerak benda akan berubah melalui percepatan $a$, dengan $m > 0$ dan $a \\ne 0$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "79c05418",
   "metadata": {},
   "source": [
    "### Usaha\n",
    "Usaha oleh suatu gaya $F$ dengan posisi awal $x_0$ dan posisi akhir $x_0$ dapat diperoleh melalui\n",
    "\n",
    "\n",
    "\\begin{equation}\\label{eqn:work-1}\\tag{6}\n",
    "W = \\int_{x_0}^x F dx\n",
    "\\end{equation}\n",
    "\n",
    "atau dengan\n",
    "\n",
    "\n",
    "\\begin{equation}\\label{eqn:work-2}\\tag{7}\n",
    "W = \\Delta K\n",
    "\\end{equation}\n",
    "\n",
    "dengan $K$ adalah energi kinetik. Persamaan ([7](#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 ([6](#eqn6))."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "44901d75",
   "metadata": {},
   "source": [
    "## Sistem\n",
    "Ilustrasi sistem perlu diberikan agar dapat terbayangan dan memudahkan penyelesaian masalah. Selain itu juga perlu disajikan diagram gaya-gaya yang bekerja pada benda."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8060fbca",
   "metadata": {},
   "source": [
    "### Ilustrasi\n",
    "Sistem yang benda bermassa $m$ bergerak di atas lantai kasar dapat digambarkan\n",
    "seperti berikut ini."
   ]
  },
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       "\n",
       "Gambar 1. Sistem benda bermassa $m$ begerak di atas lantai\n",
       "mendatar kasar dengan koefisien gesek kinetis $\\mu_k$.\n"
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\n",
    "\n",
    "Gambar 1. Sistem benda bermassa $m$ begerak di atas lantai\n",
    "mendatar kasar dengan koefisien gesek kinetis $\\mu_k$."
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    "Keadaan akhir benda, yaitu saat kecepatan $v = 0$ diberikan pada bagian kanan Gambar [1](#fig1) dengan warna abu-abu."
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    "### Diagram gaya\n",
    "Diagram gaya-gaya yang berja pada benda perlu dibuat berdasarkan informasi dari Gambar [1](#fig1) dan Tabel [1](#tab1), yang diberikan berikut ini."
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\n",
       "\n",
       "Gambar 2. Diagram gaya-gaya yang bekerja pada benda\n",
       "bermassa $m$.\n"
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    "\n",
    "Gambar 2. Diagram gaya-gaya yang bekerja pada benda\n",
    "bermassa $m$."
   ]
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   "source": [
    "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 $v$."
   ]
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   "cell_type": "markdown",
   "id": "23119498",
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   "source": [
    "## Metode numerik\n",
    "Interasi suatu fungsi $f(x)$ berbentuk\n",
    "\n",
    "\n",
    "\\begin{equation}\\label{eqn:integral-1}\\tag{8}\n",
    "A = \\int_a^b f(x) dx\n",
    "\\end{equation}\n",
    "\n",
    "dapat didekati dengan\n",
    "\n",
    "\n",
    "\\begin{equation}\\label{eqn:integral-2}\\tag{9}\n",
    "A \\approx \\sum_{i = 0}^N f\\left[ \\tfrac12(x_i + x_{i+1}) \\right] \\Delta x\n",
    "\\end{equation}\n",
    "\n",
    "yang dikenal sebagai metode persegi titik tengah, di mana\n",
    "\n",
    "\n",
    "\\begin{equation}\\label{eqn:integral-3}\\tag{10}\n",
    "\\Delta x = \\frac{b - a}{N}\n",
    "\\end{equation}\n",
    "\n",
    "dengan $N$ adalah jumlah partisi. Variabel $x_i$ pada Persamaan ([9](#eqn9)) diberikan oleh\n",
    "\n",
    "\n",
    "\\begin{equation}\\label{eqn:integral-4}\\tag{11}\n",
    "x_i = a + i\\Delta x\n",
    "\\end{equation}\n",
    "\n",
    "dengan $i = 0, \\dots, N$."
   ]
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   "id": "d82ae009",
   "metadata": {},
   "source": [
    "## Penyelesaian\n",
    "Penerapan Persamaan ([1](#eqn1)), ([2](#eqn2)), ([3](#eqn3)), ([4](#eqn4)), dan ([5](#eqn5)) pada Gambar [2](#fig2) akan menghasilkan\n",
    "\n",
    "\n",
    "\\begin{equation}\\label{eqn:friction}\\tag{10}\n",
    "f_k = \\mu_k mg\n",
    "\\end{equation}\n",
    "\n",
    "dan usahanya adalah\n",
    "\n",
    "\n",
    "\\begin{equation}\\label{eqn:friction-work}\\tag{11}\n",
    "\\begin{array}{rcl}\n",
    "W & = & \\displaystyle \\int_{x_0}^x f_k dx \\newline\n",
    "& = & \\displaystyle \\int_{x_0}^x \\mu_k m g dx \\newline\n",
    "& = & \\displaystyle m g \\int_{x_0}^x \\mu_k dx\n",
    "\\end{array}\n",
    "\\end{equation}\n",
    "\n",
    "dengan koefisien gesek statisnya dapat merupakan fungsi dari posisi $\\mu_k = \\mu_k(x)$."
   ]
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   "id": "33c63a26",
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   "outputs": [
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      "text/plain": [
       "
"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "plt.ion()\n",
    "\n",
    "# set integral lower and upper bounds\n",
    "a = 0\n",
    "b = 1\n",
    "\n",
    "# generate x\n",
    "x = [1, 2, 3, 4, 5]\n",
    "\n",
    "# generate y from numerical integration\n",
    "y = [1, 2, 3, 5, 6]\n",
    "\n",
    "## plot results\n",
    "fig, ax = plt.subplots()\n",
    "ax.scatter(x, y)\n",
    "ax.set_xlabel(\"$x - x^0$\")\n",
    "ax.set_ylabel(\"y\")\n",
    "\n",
    "from IPython import display\n",
    "from IPython.core.display import HTML\n",
    "HTML('''\n",
    "
\n",
    "Gambar 3. Kurva antara usaha $W$ dan jarak tempuh $x - x_0$.\n",
    "
\n",
    "''')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6f0ceb31",
   "metadata": {},
   "source": [
    "## Diskusi\n",
    "Berdasarkan Gambar [3](#fig3) dapat dijelaskan bahwa dengan $\\mu_k = \\mu_k(x)$ maka kurva $W(x)$ tidak lagi linier karena dipengaruhi oleh sejauh mana perhitungan kerja dilakukan."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "67fba2fe",
   "metadata": {},
   "source": [
    "## Kesimpulan\n",
    "Perhitungan kerja dengan $\\mu_k = \\mu_k(x)$ telah dapat dilakukan."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "44995bd5",
   "metadata": {},
   "source": [
    "## Referensi\n",
    "1. 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>. \n",
    "1. Carl Rod Nave, \"Friction\", HyperPhysics, 2017, url <http://hyperphysics.phy-astr.gsu.edu/hbase/frict.html#fri> [20220419].\n",
    "2. Wikipedia contributors, \"Friction\", Wikipedia, The Free Encyclopedia, 12 April 2022, 00:33 UTC, url <https://en.wikipedia.org/w/index.php?oldid=1082223658> [20220419].\n",
    "3. Tia Ghose, Ailsa Harvey, \"What is friction?\", Live Science, 8 Feb 2022, url <https://www.livescience.com/37161-what-is-friction.html> [20220419]."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": [],
   "id": "a7da2b33",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}

```

    {'cells': [{'cell_type': 'markdown',
       'id': '1e51ab9a',
       'metadata': {},
       'source': ['# Kerja Gaya Gesek\n',
        '\n',
        'Sparisoma Viridi1, Raden Roro Zahra Auliya S2
\n',
        'Program Studi Sarjana Fisika, Institut Teknologi Bandung
\n',
        'Jalan Gensha 10, Bandung 40132, Indonesia
\n',
        '1dudung@gmail.com, https://github.com/dudung
\n',
        '2rzahraauliya@gmail.com, https://github.com/RRZahra\n',
        '\n',
        '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.']},
      {'cell_type': 'markdown',
       'id': 'aed2f31a',
       'metadata': {},
       'source': ['## Gerak benda di atas lantai mendatar kasar\n',
        '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.']},
      {'cell_type': 'markdown',
       'id': 'dc4acf61',
       'metadata': {},
       'source': ['## Parameter\n',
        'Beberapa parameter yang digunakan adalah seperti pada tabel berikut ini.\n',
        '\n',
        "Tabel 1. Simbol beserta satuan dan artinya.\n",
        '\n',
        'Simbol | Satuan | Arti\n',
        ':- | :- | :-\n',
        '$t$ | s | waktu\n',
        '$v_0$ | m/s | kecepatan awal\n',
        '$x_0$ | m | posisi awal\n',
        '$v$ | m/s | kecepatan saat $t$\n',
        '$x$ | m | waktu saat $t$\n',
        '$a$ | m/s2 | percepatan\n',
        '$\\mu_k$ | - | koefisien gesek kinetis\n',
        '$f_k$ | N | gaya gesek kinetis\n',
        '$m$ | kg | massa benda\n',
        '$F$ | N | total gaya yang bekerja\n',
        '$N$ | N | gaya normal\n',
        '$w$ | N | gaya gravitasi\n',
        '\n',
        'Simbol-simbol pada Tabel [1](#tab1) akan diberi nilai kemudian saat diimplementasikan dalam program.']},
      {'cell_type': 'markdown',
       'id': '12457da3',
       'metadata': {},
       'source': ['## Persamaan\n',
        'Persamaan-persamaan yang akan digunakan adalah seperti dicantumkan pada bagian ini.']},
      {'cell_type': 'markdown',
       'id': 'fb54dc5c',
       'metadata': {},
       'source': ['### Kinematika\n',
        'Hubungan antara antara kecepatan $v$, kecepatan awal $v_0$, percepatan $a$, dan waktu $t$ diberikan oleh\n',
        '\n',
        "\n",
        '\\begin{equation}\\label{eqn:kinematics-v-a-t}\\tag{1}\n',
        'v = v_0 + at.\n',
        '\\end{equation}']},
      {'cell_type': 'markdown',
       'id': '0e52be88',
       'metadata': {},
       'source': ['Posisi benda $x$ bergantung pada posisi awal $x_0$, kecepatan awal $v_0$, percepatan $a$, dan waktu $t$ melalui hubungan\n',
        '\n',
        "\n",
        '\\begin{equation}\\label{eqn:kinematics-x-v-a-t}\\tag{2}\n',
        'x = x_0 + v_0 t + \\tfrac12 at^2.\n',
        '\\end{equation}\n']},
      {'cell_type': 'markdown',
       'id': '3c8a08fa',
       'metadata': {},
       'source': ['Selain kedua persamaan sebelumnya, terdapat pula persamaan berikut\n',
        '\n',
        "\n",
        '\\begin{equation}\\label{eqn:kinematics-v-x-a}\\tag{3}\n',
        'v^2 = v_0^2 + 2a(x - x_0),\n',
        '\\end{equation}\n',
        '\n',
        'yang menghubungkan kecepatan $v$ dengan kecepatan awal $v_0$, percepatan $a$, dan jarak yang ditempuh $x - x_0$.']},
      {'cell_type': 'markdown',
       'id': '4fbcd3b6',
       'metadata': {},
       'source': ['### Dinamika\n',
        '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\n',
        '\n',
        "\n",
        '\\begin{equation}\\label{eqn:newtons-law-1}\\tag{4}\n',
        '\\sum F = 0.\n',
        '\\end{equation}']},
      {'cell_type': 'markdown',
       'id': 'c237c502',
       'metadata': {},
       'source': ['Bila ada gaya yang bekerj pada benda bermassa $m$ atau jumlah gaya-gaya tidak nol\n',
        '\n',
        "\n",
        '\\begin{equation}\\label{eqn:newtons-law-2}\\tag{5}\n',
        '\\sum F = ma,\n',
        '\\end{equation}\n',
        '\n',
        'maka keadaan gerak benda akan berubah melalui percepatan $a$, dengan $m > 0$ dan $a \\ne 0$.']},
      {'cell_type': 'markdown',
       'id': '79c05418',
       'metadata': {},
       'source': ['### Usaha\n',
        'Usaha oleh suatu gaya $F$ dengan posisi awal $x_0$ dan posisi akhir $x_0$ dapat diperoleh melalui\n',
        '\n',
        "\n",
        '\\begin{equation}\\label{eqn:work-1}\\tag{6}\n',
        'W = \\int_{x_0}^x F dx\n',
        '\\end{equation}\n',
        '\n',
        'atau dengan\n',
        '\n',
        "\n",
        '\\begin{equation}\\label{eqn:work-2}\\tag{7}\n',
        'W = \\Delta K\n',
        '\\end{equation}\n',
        '\n',
        'dengan $K$ adalah energi kinetik. Persamaan ([7](#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 ([6](#eqn6)).']},
      {'cell_type': 'markdown',
       'id': '44901d75',
       'metadata': {},
       'source': ['## Sistem\n',
        'Ilustrasi sistem perlu diberikan agar dapat terbayangan dan memudahkan penyelesaian masalah. Selain itu juga perlu disajikan diagram gaya-gaya yang bekerja pada benda.']},
      {'cell_type': 'markdown',
       'id': '8060fbca',
       'metadata': {},
       'source': ['### Ilustrasi\n',
        'Sistem yang benda bermassa $m$ bergerak di atas lantai kasar dapat digambarkan\n',
        'seperti berikut ini.']},
      {'cell_type': 'code',
       'execution_count': 1,
       'id': '479756b2',
       'metadata': {'tags': ['hide_input']},
       'outputs': [{'data': {'text/html': ['\n',
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           '\n',
           "Gambar 1. Sistem benda bermassa $m$ begerak di atas lantai\n",
           'mendatar kasar dengan koefisien gesek kinetis $\\mu_k$.\n'],
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        '\n',
        "Gambar 1. Sistem benda bermassa $m$ begerak di atas lantai\n",
        'mendatar kasar dengan koefisien gesek kinetis $\\mu_k$.']},
      {'cell_type': 'markdown',
       'id': '7018f95c',
       'metadata': {},
       'source': ['Keadaan akhir benda, yaitu saat kecepatan $v = 0$ diberikan pada bagian kanan Gambar [1](#fig1) dengan warna abu-abu.']},
      {'cell_type': 'markdown',
       'id': 'fb277d2b',
       'metadata': {},
       'source': ['### Diagram gaya\n',
        'Diagram gaya-gaya yang berja pada benda perlu dibuat berdasarkan informasi dari Gambar [1](#fig1) dan Tabel [1](#tab1), yang diberikan berikut ini.']},
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       'id': '1d265114',
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        '\n',
        "Gambar 2. Diagram gaya-gaya yang bekerja pada benda\n",
        'bermassa $m$.']},
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       'source': ['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 $v$.']},
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       'source': ['## Metode numerik\n',
        'Interasi suatu fungsi $f(x)$ berbentuk\n',
        '\n',
        "\n",
        '\\begin{equation}\\label{eqn:integral-1}\\tag{8}\n',
        'A = \\int_a^b f(x) dx\n',
        '\\end{equation}\n',
        '\n',
        'dapat didekati dengan\n',
        '\n',
        "\n",
        '\\begin{equation}\\label{eqn:integral-2}\\tag{9}\n',
        'A \\approx \\sum_{i = 0}^N f\\left[ \\tfrac12(x_i + x_{i+1}) \\right] \\Delta x\n',
        '\\end{equation}\n',
        '\n',
        'yang dikenal sebagai metode persegi titik tengah, di mana\n',
        '\n',
        "\n",
        '\\begin{equation}\\label{eqn:integral-3}\\tag{10}\n',
        '\\Delta x = \\frac{b - a}{N}\n',
        '\\end{equation}\n',
        '\n',
        'dengan $N$ adalah jumlah partisi. Variabel $x_i$ pada Persamaan ([9](#eqn9)) diberikan oleh\n',
        '\n',
        "\n",
        '\\begin{equation}\\label{eqn:integral-4}\\tag{11}\n',
        'x_i = a + i\\Delta x\n',
        '\\end{equation}\n',
        '\n',
        'dengan $i = 0, \\dots, N$.']},
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       'metadata': {},
       'source': ['## Penyelesaian\n',
        'Penerapan Persamaan ([1](#eqn1)), ([2](#eqn2)), ([3](#eqn3)), ([4](#eqn4)), dan ([5](#eqn5)) pada Gambar [2](#fig2) akan menghasilkan\n',
        '\n',
        "\n",
        '\\begin{equation}\\label{eqn:friction}\\tag{10}\n',
        'f_k = \\mu_k mg\n',
        '\\end{equation}\n',
        '\n',
        'dan usahanya adalah\n',
        '\n',
        "\n",
        '\\begin{equation}\\label{eqn:friction-work}\\tag{11}\n',
        '\\begin{array}{rcl}\n',
        'W & = & \\displaystyle \\int_{x_0}^x f_k dx \\newline\n',
        '& = & \\displaystyle \\int_{x_0}^x \\mu_k m g dx \\newline\n',
        '& = & \\displaystyle m g \\int_{x_0}^x \\mu_k dx\n',
        '\\end{array}\n',
        '\\end{equation}\n',
        '\n',
        'dengan koefisien gesek statisnya dapat merupakan fungsi dari posisi $\\mu_k = \\mu_k(x)$.']},
      {'cell_type': 'code',
       'execution_count': 17,
       'id': '33c63a26',
       'metadata': {},
       'outputs': [{'data': {'text/html': ['\n',
           '
\n',
           "Gambar 3. Kurva antara usaha $W$ dan jarak tempuh $x - x_0$.\n",
           '
\n'],
          'text/plain': ['<IPython.core.display.HTML object>']},
         'execution_count': 17,
         'metadata': {},
         'output_type': 'execute_result'},
        {'data': {'image/png': 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\n',
          'text/plain': ['
']},
         'metadata': {'needs_background': 'light'},
         'output_type': 'display_data'}],
       'source': ['import numpy as np\n',
        'import matplotlib.pyplot as plt\n',
        'plt.ion()\n',
        '\n',
        '# set integral lower and upper bounds\n',
        'a = 0\n',
        'b = 1\n',
        '\n',
        '# generate x\n',
        'x = [1, 2, 3, 4, 5]\n',
        '\n',
        '# generate y from numerical integration\n',
        'y = [1, 2, 3, 5, 6]\n',
        '\n',
        '## plot results\n',
        'fig, ax = plt.subplots()\n',
        'ax.scatter(x, y)\n',
        'ax.set_xlabel("$x - x^0$")\n',
        'ax.set_ylabel("y")\n',
        '\n',
        'from IPython import display\n',
        'from IPython.core.display import HTML\n',
        "HTML('''\n",
        '
\n',
        "Gambar 3. Kurva antara usaha $W$ dan jarak tempuh $x - x_0$.\n",
        '
\n',
        "''')"]},
      {'cell_type': 'markdown',
       'id': '6f0ceb31',
       'metadata': {},
       'source': ['## Diskusi\n',
        'Berdasarkan Gambar [3](#fig3) dapat dijelaskan bahwa dengan $\\mu_k = \\mu_k(x)$ maka kurva $W(x)$ tidak lagi linier karena dipengaruhi oleh sejauh mana perhitungan kerja dilakukan.']},
      {'cell_type': 'markdown',
       'id': '67fba2fe',
       'metadata': {},
       'source': ['## Kesimpulan\n',
        'Perhitungan kerja dengan $\\mu_k = \\mu_k(x)$ telah dapat dilakukan.']},
      {'cell_type': 'markdown',
       'id': '44995bd5',
       'metadata': {},
       'source': ['## Referensi\n',
        '1. 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>. \n',
        '1. Carl Rod Nave, "Friction", HyperPhysics, 2017, url <http://hyperphysics.phy-astr.gsu.edu/hbase/frict.html#fri> [20220419].\n',
        '2. Wikipedia contributors, "Friction", Wikipedia, The Free Encyclopedia, 12 April 2022, 00:33 UTC, url <https://en.wikipedia.org/w/index.php?oldid=1082223658> [20220419].\n',
        '3. Tia Ghose, Ailsa Harvey, "What is friction?", Live Science, 8 Feb 2022, url <https://www.livescience.com/37161-what-is-friction.html> [20220419].']},
      {'cell_type': 'code',
       'execution_count': [],
       'id': 'a7da2b33',
       'metadata': {},
       'outputs': [],
       'source': []}],
     'metadata': {'kernelspec': {'display_name': 'Python 3 (ipykernel)',
       'language': 'python',
       'name': 'python3'},
      'language_info': {'codemirror_mode': {'name': 'ipython', 'version': 3},
       'file_extension': '.py',
       'mimetype': 'text/x-python',
       'name': 'python',
       'nbconvert_exporter': 'python',
       'pygments_lexer': 'ipython3',
       'version': '3.10.4'}},
     'nbformat': 4,
     'nbformat_minor': 5}

```python

```