Orthopedic some concepts

Simple Harmonic Motion (SHM)

• Periodic motion repeats at equal time intervals.
• SHM is a specific type of periodic motion adhering to certain conditions.

Conditions for SHM

• A position of stable equilibrium exists.
• A restoring force acts upon the object.
• Restoring force is proportional to displacement from equilibrium: (F = -kx).

Key Definitions

• Amplitude ((A)): Maximum displacement from equilibrium.
• Period ((T)): Time to complete one oscillation.
• Frequency ((f)): Oscillations per unit time; (f = \frac{1}{T}); measured in Hertz (Hz).
• Angular Frequency ((\omega)): (\omega = 2\pi f = \frac{2\pi}{T}).

Equation of SHM

• Displacement (x) at time (t): (x = A\ cos(\omega t + \phi)), where (A) is amplitude, (\omega) is angular frequency, and (\phi) is the phase constant.

Velocity in SHM

• Velocity (v) as a function of time: (v = \frac{dx}{dt} = -\omega A\ sin(\omega t + \phi)).
• Maximum velocity: (v_{max} = \omega A).

Acceleration in SHM

• Acceleration (a) as a function of time: (a = \frac{dv}{dt} = -\omega^2 A\ cos(\omega t + \phi)).
• Acceleration can also be expressed (a = -\omega^2 x).
• Maximum acceleration: (a_{max} = \omega^2 A).

Example Problem

• Given displacement (x = A\ cos(\omega t + \frac{\pi}{6})).
  • At (t = 0), displacement (x = A\ cos(\frac{\pi}{6}) = 0.866A).
  • Velocity as a function of time: (v = -\omega A\ sin(\omega t + \frac{\pi}{6})).
  • Acceleration as a function of time: (a = -\omega^2 A\ cos(\omega t + \frac{\pi}{6})).

Radiative Heat Transfer

• Heat transfer occurs through emission of EM waves: (E = \hbar \omega = h\nu).
• EM waves are emitted by changes in electronic configuration, molecular vibration, and molecular rotation.
• No medium required; important at high temperature differences.

Thermal Radiation

• Radiation emitted by a body due to its temperature.
• All bodies emit thermal radiation above absolute zero.

Black Body

• Idealized object that absorbs all incident radiation.
• Emits maximum possible radiation at a given temperature.
• Diffuse emitter: emits uniformly in all directions.

Black Body Radiation

• Described by Planck's law: (E_{b\lambda}(\lambda,T) = \frac{2hc^2}{\lambda^5 (e^{\frac{hc}{\lambda k T}} - 1)}).
  • (E_{b\lambda}) is spectral emissive power.
  • (h) is Planck's constant ((6.626 \times 10^{-34}) J.s), (c) is speed of light ((3.0 \times 10^8) m/s), (k) is Boltzmann's constant ((1.38 \times 10^{-23}) J/K).

Stefan-Boltzmann Law

• Total energy emitted per unit area: (E_b = \sigma T^4).
  • (E_b) is total emissive power.
  • (\sigma) is Stefan-Boltzmann constant ((5.67 \times 10^{-8}) W/m(^2)K(^4)).

Wien's Displacement Law

• Wavelength of maximum emission: (\lambda_{max} T = b).
  • (\lambda_{max}) is wavelength of maximum emission.
  • (b) is Wien's displacement constant ((2.898 \times 10^{-3}) m.K).

Surface Properties

• Emissivity ((\epsilon)): Ratio of radiation emitted by surface to black body; (0 \le \epsilon \le 1); (\epsilon = \frac{E}{E_b}).
• Absorptivity ((\alpha)): Fraction of incident radiation absorbed; (0 \le \alpha \le 1); (\alpha = \frac{\text{Absorbed Radiation}}{\text{Incident Radiation}}).
• Reflectivity ((\rho)): Fraction of incident radiation reflected; (0 \le \rho \le 1); (\rho = \frac{\text{Reflected Radiation}}{\text{Incident Radiation}}).
• Transmissivity ((\tau)): Fraction of incident radiation transmitted; (0 \le \tau \le 1); (\tau = \frac{\text{Transmitted Radiation}}{\text{Incident Radiation}}).
• Relation: (\alpha + \rho + \tau = 1). For opaque surfaces ((\tau = 0)), (\alpha + \rho = 1).

Grey Body

• Radiative properties ((\alpha, \rho, \tau, \epsilon)) are independent of wavelength.

Kirchhoff's Law

• Emissivity equals absorptivity at a given temperature and wavelength: (\epsilon = \alpha).

Configuration Factor

• Fraction of radiation from one surface that strikes another.
• Also known as shape factor, view factor, or angle factor.

Matplotlib Introduction

• Comprehensive library for creating static, animated, and interactive visualizations in Python.

Key Features

• Creates various plots: line, scatter, bar, histograms, and more.
• Customizable visualizations: labels, legends, titles, and styles.
• Exports graphics in various formats.
• Integrates with Python packages like NumPy and Pandas.

Importing Matplotlib

• Use pyplot submodule: import matplotlib.pyplot as plt.
• Standard alias plt.
• Example: import matplotlib.pyplot as plt; import numpy as np.

Creating a Simple Plot

• Create figures using pyplot.
• Example:

  plt.plot([1, 2, 3, 4])
  plt.ylabel('some numbers')
  plt.show()
  

• Shows a line plot with x-axis 0-3 and y-axis 1-4.

Adding Titles and Labels

• Utilize title, xlabel, ylabel functions.
• Example:

  plt.plot([1, 2, 3, 4])
  plt.xlabel('x axis')
  plt.ylabel('y axis')
  plt.title('My first plot')
  plt.show()
  

Plotting Multiple Plots

• Call plot function multiple times in the same figure.
• Example: plt.plot([1, 2, 3, 4], [1, 4, 9, 16]); plt.show().

Adding a Legend

• Pass labels to plot and call legend function.

plt.plot([1, 2, 3, 4], [1, 4, 9, 16], label='Line 1')
plt.plot([1, 2, 3, 4], [1, 2, 3, 4], label='Line 2')
plt.legend()
plt.show()

Plotting Styles

• Use an optional third argument in plot for color and line type.
• Example: plt.plot([1, 2, 3, 4], [1, 4, 9, 16], 'ro') ('ro' = red circles).

Controlling Axis Limits

• Use axis function with [xmin, xmax, ymin, ymax].
• Example: plt.axis([0, 6, 0, 20]).

Working with NumPy Arrays

• Matplotlib works with NumPy arrays.
• Example:

  x = np.linspace(0, 2 * np.pi, 100)
  y = np.sin(x)
  plt.plot(x, y)
  plt.show()
  

Multiple Plots in a Figure

• Example:

  x = np.linspace(0, 2 * np.pi, 100)
  y1 = np.sin(x)
  y2 = np.cos(x)
  fig, ax = plt.subplots()
  ax.plot(x, y1, label='sin(x)')
  ax.plot(x, y2, label='cos(x)')
  ax.legend()
  plt.show()
  

  • The subplots function creates a figure and a set of axes.
  • The plot function plots the data on the axes.
  • A legend is added.

Common Plot Types

• Line plots, scatter plots, bar charts, histograms, box plots, violin plots.
• Example (Scatter Plots):

  x = np.random.rand(100)
  y = np.random.rand(100)
  colors = np.random.rand(100)
  sizes = 100 * np.random.rand(100)
  plt.scatter(x, y, c=colors, s=sizes, alpha=0.5)
  plt.show()
  

  • Creates a scatter plot with 100 random points (color/size are also random).
• Example (Bar Charts):

  x = np.arange(5)
  y = [3, 7, 2, 9, 5]
  plt.bar(x, y)
  plt.show()
  

  • Creates a bar chart with 5 bars (height aligned to defined y values).

Customization

• Modifiable color, line type, font size, and other aspects.
• Color Customization: Use color within plot function. Example: plt.plot(x, y, color='red'). (creates red line plot).
• Line Type Customization: Use linestyle within plot function. Example: plt.plot(x, y, linestyle='dashed'). (creates dashed line plot).