Simple Harmonic Motion (SHM)

Simple Harmonic Motion (SHM)

• SHM is a type of periodic motion with specific conditions.

Conditions for SHM

• There needs to be a stable equilibrium point.
• A restoring force acts on the object.
• The restoring force is proportional to the displacement from equilibrium: (F = -kx).

Key Terms

• Amplitude (A): Maximum displacement from equilibrium.
• Period (T): Time for one complete 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 = A\ cos(\omega t + \phi)), where (x) is displacement, (A) is amplitude, (\omega) is angular frequency, (t) is time, and (\phi) is the phase constant.

Velocity in SHM

• Velocity: (v = \frac{dx}{dt} = -\omega A\ sin(\omega t + \phi)).
• Maximum velocity: (v_{max} = \omega A) when (sin(\omega t + \phi) = 1).

Acceleration in SHM

• Acceleration: (a = \frac{dv}{dt} = -\omega^2 A\ cos(\omega t + \phi)) or (a = -\omega^2 x).
• Maximum acceleration: (a_{max} = \omega^2 A) when (cos(\omega t + \phi) = 1).

Example scenario:

• Displacement of a body executing SHM is expressed as (x = A\ cos(\omega t + \frac{\pi}{6})).
• At (t = 0), the displacement is (x = A\ cos(\frac{\pi}{6}) = A\frac{\sqrt{3}}{2} = 0.866A).
• Velocity as a function of time: (v = \frac{dx}{dt} = -\omega A\ sin(\omega t + \frac{\pi}{6})). Acceleration as a function of time: (a = \frac{dv}{dt} = -\omega^2 A\ cos(\omega t + \frac{\pi}{6})).

Radiative Heat Transfer

• Heat transfers due to EM wave emission.
• These waves are emitted due to changes in electronic configuration, molecular vibration/rotation.
• It does not require a medium.
• More relevant when the temperature difference is high.
• Energy equation: ( E = \hbar \omega = h\nu ).

Thermal Radiation

• Emission by a body because of its temperature.
• All bodies emit thermal radiation above absolute zero.

Black Body

• An idealized object that absorbs all incident radiation.
• Emits the maximum possible radiation at a given temp.
• Emits radiation uniformly in all directions (diffuse emitter).

Black Body Radiation

• Spectral distribution is given 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.
  • ( \lambda ) is the wavelength.
  • ( T ) is the absolute temperature.
  • ( h ) is Planck's constant (( 6.626 \times 10^{-34} ) J.s).
  • ( c ) is the 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 emissive power.
  • ( \sigma ) is the Stefan-Boltzmann constant (( 5.67 \times 10^{-8} ) W/m(^2)K(^4)).
  • ( T ) is the absolute temperature.

Wien's Displacement Law

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

Properties

Emissivity ((\epsilon))

• Ratio of radiation emitted by a surface to that of a black body at the same temperature.
• ( 0 \le \epsilon \le 1 )
• Equation: (\epsilon = \frac{E}{E_b}).

Absorptivity ((\alpha))

• Fraction of incident radiation absorbed.
• ( 0 \le \alpha \le 1 )
• Equation: (\alpha = \frac{\text{Absorbed Radiation}}{\text{Incident Radiation}}).

Reflectivity ((\rho))

• Fraction of incident radiation reflected.
• ( 0 \le \rho \le 1 )
• Equation: (\rho = \frac{\text{Reflected Radiation}}{\text{Incident Radiation}}).

Transmissivity ((\tau))

• Fraction of radiation transmitted.
• ( 0 \le \tau \le 1 )
• Equation: (\tau = \frac{\text{Transmitted Radiation}}{\text{Incident Radiation}}).

Relation

• (\alpha + \rho + \tau = 1).
• For opaque surfaces, ( \tau = 0 ) and (\alpha + \rho = 1).

Grey Body

• One whose radiative properties are independent of wavelength.

Kirchhoff's Law

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

Configuration Factor

• Fraction of radiation leaving one surface that strikes another; also called shape factor, view factor, or angle factor.

Matplotlib Quickstart Guide

Introduction

• Matplotlib is a comprehensive library for creating static, animated, and interactive visualizations in Python.

Key Features

• Creates various plots including line, scatter, bar, and histograms
• Customizes visualizations with labels, legends, titles, and styles.
• Exports in various file formats.
• Integrates with NumPy and Pandas.

Importing Matplotlib

• The pyplot submodule is commonly used.
• Example: import matplotlib.pyplot as plt

Simple Plot

• Create using pyplot.
• Example: shows a line plot with y-axis values from 1 to 4.

Adding Titles and Labels

• Use title, xlabel, and ylabel functions.

Plot Multiple Plots

• Call the plot function multiple times.

Adding A legend

• Pass labels to the plot function, then call the legend function to display label.

Plotting Styles

• A third argument to the plot function controls the color and line type.
• Example: 'ro' plots red circles.

Controlling Axis Limits

• The axis function takes a list [xmin, xmax, ymin, ymax].
• The function specifies axis boundaries

Working with NumPy Arrays

• Matplotlib can plot with NumPy arrays Example: generates a sine wave plot.

Common Plot Types

• Matplotlib supports line plots, scatter plots, bar charts, histograms, box plots, violin plots, etc.

Scatter Plots

• Displays relationships between two sets of data points.

Bar Charts

• Displays categorical data with rectangular bars
• The height of each bar corresponds to the value being visualized.

Customization

• Options: Change color, line type, font size, etc.

Colors

• The color argument in the plot function changes colors.
• Example: color='red' creates a red line.

Line Types

• The linestyle argument in the plot function changes types.
• Example: linestyle='dashed' creates a dashed line.

The Time Value of Money

Definition

• Money available now is worth more than the same amount in the future due to its earning potential.

Formula

• (FV = PV (1 + r)^n)
  • FV = Future Value
  • PV = Present Value
  • r = Interest Rate
  • n = Number of Compounding Periods

Time Value of Money Example

• $10,000 invested for one year at 10% interest. The future value is:
  • (FV = $10,000 (1+0.10)^1 = $11,000)

Why it Matters

• Inflation reduces purchasing power over time.
• Opportunity Cost money can be invested for returns immediately.
• Uncertainty the future is uncertain for investment.

Net Present Value (NPV)

Definition

• The difference between the present value of cash inflows and outflows over a period.

Formula

• (NPV = \sum_{t=1}^{T} \frac{C_t}{(1+r)^t} - C_0)
  • $C_t$ = Net cash flow during the period t
  • $C_0$ = Initial investment
  • r = Discount rate
  • t = Number of time periods

NPV Example

• A company invests $1M, expects to generate $250,000/year for five years, discount rate is 10%.
  • (NPV = \frac{$250,000}{(1+0.10)^1} + \frac{$250,000}{(1+0.10)^2} + \frac{$250,000}{(1+0.10)^3} + \frac{$250,000}{(1+0.10)^4} + \frac{$250,000}{(1+0.10)^5} - $1,000,000)
  • (NPV = $1,040,000 - $1,000,000 = $40,000)
  • NPV is $40,000, indicating the investment is expected to generate a positive return.

Why it Matters

• Profitability helps determine if a project is expected to be profitable.
• Comparison NPV compares different investment options.
• Decision Making basis for investment decisions.