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fixed minor typos #1

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43 changes: 28 additions & 15 deletions Lab05_Simple_Harmonic_Oscillator.ipynb
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Expand Up @@ -23,9 +23,17 @@
"## Lab #5 - Simple Harmonic Oscillator \n",
"### What are simple harmonic oscillators?\n",
"\n",
"In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement. If restoring force is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).[\\[1\\]](https://en.wikipedia.org/wiki/Harmonic_oscillator)\n",
"In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium\n",
"position, experiences a restoring force proportional to the displacement. If restoring force is the\n",
"only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes\n",
"simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant\n",
"amplitude and a constant frequency (which does not depend on the\n",
"amplitude).[\\[1\\]](https://en.wikipedia.org/wiki/Harmonic_oscillator)\n",
"\n",
"Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves. In this lab, we will build spring-mass simple harmonic oscillator using common-place materials, and determine the stiffness of the spring based on governing equations of the system.\n"
"Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as\n",
"clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves. In\n",
"this lab, we will build spring-mass simple harmonic oscillator using common materials, and\n",
"determine the stiffness of the spring based on governing equations of the system.\n"
]
},
{
Expand All @@ -34,13 +42,15 @@
"source": [
"### 1-DOF spring-mass system \n",
"\n",
"Figure 1 shows the schematic of spring-mass simple harmonic oscillator. In this system with 1 mass and 1 spring, we have 1 degree of freedom. So, there is 1 differential\n",
"equations that describe the motion of mass. Employing Newton's law, $ F = ma = m \\ddot{x}$ and Hook's law for spring restoring forcce , $F = -kx$, to this sytem, the governing differential is obtained as: \n",
"Figure 1 shows the schematic of spring-mass simple harmonic oscillator. In this system with 1 mass\n",
"and 1 spring, we have 1 degree of freedom. So, there is 1 differential equation that describes the\n",
"motion of mass. Employing Newton's law, $ F = ma = m \\ddot{x}$ and Hook's law for spring restoring\n",
"force , $F = -kx$, to this sytem, the governing differential is obtained as: \n",
"\n",
"$m \\ddot{x} = -kx$ (1)\n",
"\n",
"where $m$ and $k$ denote the mass and spring stiffness respectively. The differential\n",
"equations relate acceleration of mass $\\ddot{x}$ to displacement, $x$. \n",
"where $m$ and $k$ denote the mass and spring stiffness respectively. The differential equations\n",
"relate acceleration of mass $\\ddot{x}$ to displacement, $x$. \n",
"\n",
"\n",
"<center><img src=\"./spring_mass_animation.gif\" alt=\"Drawing\" style=\"width: 100px;\"/></center>\n",
Expand All @@ -54,17 +64,17 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"The solution for ${x}$ will be a combination of simple harmonics, sine and cosine functions, at\n",
"The solution for ${x}$ will be a combination of sine and cosine functions, at\n",
"natural frequency of the system, depending upon initial conditions. Substituting\n",
"\n",
"${x}=A\\sin(\\omega t)$ (2)\n",
"${x}=A\\sin(\\omega t) + B \\cos(\\omega t)$ (2)\n",
"\n",
"Eqn 1 gives\n",
"\n",
"$m \\omega^2 = k$ (3)\n",
"\n",
"where $A$ is amplitude of the sine function and $\\omega$ is the\n",
"natural frequency. Eqn 3 can also be rearranged as \n",
"where $A$ is amplitude of the sine function, $B$ is the amplitude of the cosine function, and\n",
"$\\omega$ is the natural frequency. Eqn 3 can also be rearranged as \n",
"\n",
"$\\omega^2 = k \\frac{1}{m} $ (4)\n"
]
Expand All @@ -75,7 +85,7 @@
"source": [
"## Lab procedure\n",
"\n",
"This Lab requires very simple materials to explore the physics of periodic motion. All you need is a spring and some weights, such as small fishing sinkers. A example is shown in Figure 2 which uses spring removed from spiral notebook. The spring has much smaller mass than added masses, and can be neglected. \n",
"This Lab requires simple materials to explore the physics of periodic motion. You need a spring and some weights, such as small fishing sinkers. A example is shown in Figure 2 which uses spring removed from spiral notebook. The spring has much smaller mass than added masses, and can be neglected. \n",
"\n",
"<center><img src=\"./spring_mass_smartphone.jpg\" alt=\"Drawing\" style=\"width: 300px;\"/></center>\n",
"<center>Figure 2: Simple harmonic spring-mass oscillator using smartphone</center>\n",
Expand Down Expand Up @@ -958,7 +968,7 @@
}
],
"source": [
"# Verification of frequncy of oscillation\n",
"# Verification of frequency of oscillation\n",
"omega_smartphone = 2.39\n",
"t1 = 33.4313;\n",
"t2 = 35.1133\n",
Expand All @@ -977,7 +987,10 @@
"source": [
"### Linear least square fit\n",
"\n",
"Once you have number of data points for $m$ and $\\omega$, you can use a linear regression to determine the slope of the data. The equation (4) predicts that the square of frequncy of oscillation , $\\omega^2$, and inverse of mass, $1/m$, are related by a proportional constant, $k$ which is the stiffness constant of the spring. If we know $k$, the total squared error is as such\n",
"Once you have number of data points for $m$ and $\\omega$, you can use a linear regression to\n",
"determine the slope of the data. Equation (4) predicts that the square of frequency of\n",
"oscillation , $\\omega^2$, and inverse of mass, $1/m$, are related by a proportional constant, $k$,\n",
"the stiffness constant of the spring. If we know $k$, the total squared error is as such\n",
"\n",
"$SSE=\\sum_i^N{(\\omega_i^2-k\\frac{1}{m_i})^2}$ (7)\n",
"\n",
Expand Down Expand Up @@ -1842,7 +1855,7 @@
"\n",
"$k = \\frac{\\omega^2}{m} $ (5)\n",
"\n",
"The propgation of error in $\\sigma_k$ from the varibales $\\omega$ and $m$ with unceratinties $\\sigma_\\omega$ and $\\sigma_m$ repectively can be calculates as:\n",
"The propagation of error in $\\sigma_k$ from the variables $\\omega$ and $m$ with unceratinties $\\sigma_\\omega$ and $\\sigma_m$ repectively can be calculates as:\n",
"\n",
"$\\sigma_k = \\sqrt{\\frac{\\partial{k}}{\\partial \\omega}\\sigma_\\omega^2 + \\frac{\\partial{k}}{\\partial m}\\sigma_m^2}$ (6)\n",
"\n",
Expand Down Expand Up @@ -1882,7 +1895,7 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.8.6"
"version": "3.8.3"
}
},
"nbformat": 4,
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