Waves and Oscillations Study Notes

Waves and Oscillations

Basic Reference Information

• Resource Textbooks:   - Physics for Engineers by Giasuddin Ahmad (Part-1)   - University Physics by Sears, Zemansky, Young & Freedman

• Compilation Prepared by:   - Dr. Md. Abu Saklayen   - Nipa Roy   - Md. Asaduzzaman

• Affiliation:   - Institute of Natural Sciences, United International University

• Online References:   - Web references provided on slides

Basics for Waves and Oscillations

• Key Topics Covered:   - Differentiation   - Vector Analysis   - Basics of Motion

Derivative Formulas

General Formulas

1. Definition of Derivative:
      $rac{du}{dx} = ext{lim}_{h o 0} rac{f(x+h) - f(x)}{h}$

2. Derivative of a Constant (k):
      $rac{d(k)}{dx} = 0$

3. Derivative of Functions:    - Power Rule:
        $rac{d(x^n)}{dx} = nx^{n-1}$    - Constant Times a Function:
        $rac{d(c imes f(x))}{dx} = c imes f'(x)$    - Sum Rule:
        $rac{d(u + v)}{dx} = rac{du}{dx} + rac{dv}{dx}$
      - Difference Rule:
        $rac{d(u - v)}{dx} = rac{du}{dx} - rac{dv}{dx}$
      - Product Rule:
        $rac{d(uv)}{dx} = u rac{dv}{dx} + v rac{du}{dx}$
      - Quotient Rule:
        $rac{d rac{u}{v}}{dx} = rac{v rac{du}{dx} - u rac{dv}{dx}}{v^2}$    - Trigonometric Derivatives:
        - $rac{d}{dx}[ ext{sin}(x)] = ext{cos}(x)$      - $rac{d}{dx}[ ext{cos}(x)] = - ext{sin}(x)$
      - Exponential Rule:
        - $rac{d(b^x)}{dx} = b^x ext{ln}(b)$    - Chain Rule:
        - $rac{d}{dx}[f(g(x))] = f'(g(x)) rac{dg}{dx}$

Linear and Rotary Motion

Linear Motion Equations

1. Position as a Function of Time:    - $S = x_1 + x$

2. Velocity and Acceleration
      - $v = rac{x_1 - x_0}{t}$
      - $a = rac{ ext{change in velocity}}{t}$

3. Equations of Motion:
      - $V_1 = V_0 + at$
      - $S = V_0 t + rac{1}{2} a t^2$

rotary motion equations

1. Angular Displacement (for rotation):    - $θ = θ_0 + ωt + rac{1}{2} αt^2$
      - $ω = ω_0 + αt$

2. Centripetal Acceleration:
      - $a_c = rac{v^2}{r} = ω^2r$

Simple Harmonic Motion (SHM)

Definitions and Characteristics

• Periodic Motion: Movement that is repeated at constant time intervals (e.g., clock hands, wheels of a car).

• Oscillatory Motion: A specific periodic motion where an object moves to and from a fixed position (e.g., a pendulum).

• Simple Harmonic Motion (SHM): A type of oscillatory motion characterized by a restoring force proportional to the negative of the displacement from the equilibrium position.

Equations and Concepts

• For a system undergoing SHM, the displacement can be represented as:   - $x = A ext{sin(ωt + φ)}$   - Amplitude (A): Maximum displacement from equilibrium.   - Period (T): The time taken for one complete cycle of the motion.   - Frequency (f): Number of cycles per unit time, defined as
      $f = rac{1}{T}$

• Graphical Representation of SHM: Displays changes in position, velocity, and acceleration over time.

Mass-Spring System in SHM

Hooke’s Law

• Hooke's Law states that the restoring force (F) is proportional to the displacement (x) from the equilibrium position:   - $F = -kx$   - Where k is the spring constant, with units of N/m, indicating the stiffness of the spring.

Energy Considerations in SHM

1. Kinetic Energy (KE):
      - $KE = rac{1}{2}mv^2$

2. Potential Energy (PE):
      - $PE = rac{1}{2}kx^2$

3. Total Energy (E):
      - Total mechanical energy in an ideal mass-spring system is constant:    - $E = KE + PE = ext{constant}$

Sample Problems

1. Problem Example: A block of mass (m = 0.42 kg) attached to a spring (stretched by 2.1 cm) experiences an acceleration of 9.0 m/s² when released. Determine the spring constant.    - Solution:
      - Using Hooke’s law, calculate:
        $k = rac{ma}{x} = rac{0.42 imes 9.0}{0.021} = 180 ext{ N/m}$

2. Energy Calculations: Given k = 19.6 N/m and A = 0.100 m, calculate various energy states, total energy, and velocity at specific displacements.    - a) Total Energy:
        - $E = rac{1}{2} k A^2 = rac{1}{2} imes 19.6 imes (0.1)^2 = 9.8 imes 10^{-2} ext{ J}$

3. SHM of a Loudspeaker: Given amplitude and frequency, find maximum speed of the diaphragm, location of maximum speed.
      - Formula to derive maximum speed ( ext{max}):
        $V_{ ext{max}} = Aω$

Pendulum Motion and Energy

1. Simple Pendulum: Describes motion where restoring force is provided by gravity.    - The period of a simple pendulum is given by:      $T = 2 ext{π} rac{ ext{l}}{g}$
      - Where l is the length of the pendulum and g is the acceleration due to gravity.

Damped Oscillations

Definition

• Damped harmonic motion refers to cases where oscillations decay due to frictional forces, resulting in a gradual decrease of amplitude over time until the motion stops.

Equations

• The equation governing damped motion incorporates damping constant (b):   $m rac{d^2x}{dt^2} + b rac{dx}{dt} + kx = 0$

LC Circuit in SHM

1. Overview:    - An LC circuit consists of an inductor (L) and capacitor (C), can exhibit SHM characteristics.

2. Equations in LC Circuits:
      - The governing equation for the oscillation in LC circuits is analogous to that of SHM:    - $rac{d^2Q}{dt^2} + rac{1}{LC}Q = 0$    - Where Q is the charge on the capacitor.

Additional Topics

• Resonant frequencies of circuits and related SHM.

• Characteristics of different types of damping - underdamped, critically damped, and overdamped motions.

Practice Questions

1. Analyze parameters of damped oscillations using provided m=400 gm, k=100 N/m, b=45 gm/s to determine conditions and behavior exhibited in oscillatory motion.

2. Derive complete solutions for oscillating systems under various damping conditions.