Waves Short Note

• Waves and Vibrations

Course Information

• Course: PHY1114

• Instructor: K. A. S. Lakshan

• Department: Physics

• Faculty: Science

• Institution: University of Ruhuna

• Year: 2025

References

1. University Physics by F.W. Sears, M.W. Zemansky, H.D. Young

2. Fundamentals of Physics by D. Halliday, R. Resnick, J. Walker

3. Physics for Scientists and Engineers by Serway/Jewett

4. Vibration and Waves by Benjamin Crowell

Topics Covered

1. Oscillations

2. Wave Motion

3. Ultrasound

4. The Doppler Effect

5. Information & Data Transfer

1. Oscillations

• Definition: A motion that repeats itself after regular intervals is called periodic or harmonic motion.

• Oscillatory Motion: If the motion is back and forth repeatedly about a fixed position (called equilibrium or mean position), the motion is termed oscillatory or vibratory.

Types of Oscillations

1. Mechanical Oscillations: Examples include vibrating strings, swinging pendulums.

2. Electrical (or Electromagnetic) Oscillations: Examples include oscillations in an LC circuit, radio waves.

3. Acoustic Oscillations: Examples include vibrating tuning forks and speaker diaphragms.

4. Elastic Oscillations: Examples include the vibrations of springs or rubber bands.

5. Thermal Oscillations: Examples consist of periodic heating and cooling cycles.

2. Simple Harmonic Motion (SHM)

• Definition: An oscillating body is said to execute simple harmonic motion if the force (F) acting on the mass is directed towards a fixed point and its magnitude is proportional to the distance to the mass from the fixed point (x).

• Equation: $F = -k x$

  • Here, $k$ represents the spring constant.

Conditions for Mechanical Oscillations

• A system must possess:

  1. Elasticity: Provides a restoring force that allows the system to return to equilibrium.

  2. Inertia: Causes the system to overshoot from the equilibrium position.

Parameters of SHM

• Angular Frequency: Given by the relationship $ext{w} = 2 ext{π}f$

  • where $f$ is the frequency in oscillations per second.

Spring-Mass System

• The general solution to the equation of motion is:
  $m rac{d^2 x}{dt^2} + kx = 0$

• Rewriting it, we have:
  $rac{d^2 x}{dt^2} + rac{k}{m}x = 0$

• The natural oscillating frequency is: $ext{w}^2 = rac{k}{m}$

  • Thus, the solution can be expressed as:
    $x(t) = a ext{sin}( ext{w} t) + b ext{cos}( ext{w} t)$

  • where $a$ and $b$ are constants determined by initial conditions.

3. Energy in SHM

• Potential Energy at maximum displacement:
  $U = rac{1}{2} k x^2$

• Kinetic Energy at maximum speed:
  $K = rac{1}{2} mv^2$

• Total energy in the system remains constant:
  $E = rac{1}{2} k x_m^2$

• The energy remains constant due to the exchange between kinetic and potential energy during SHM.

4. Example Problems

Example 1: Spring-Mass System

• A 200-g block attached to a spring oscillates with a period of 0.250 s and has total energy of 2.00 J.
  Part (a): Calculating the force constant of the spring:

  • Formula:
    $k = rac{4 ext{π}^2 m}{T^2}$

  • Substituting values:
    $k = rac{0.200 ext{kg} imes (2 ext{π})^2}{(0.250s)^2} = 126 ext{N/m}$

  Part(b): Calculating the amplitude:

  • Formula:
    $E = rac{1}{2} k A^2$

  • Rearranging:
    $A = ext{sqrt} rac{2E}{k}$

  • Substituting values:
    $A = ext{sqrt} rac{2 imes 2.00J}{126N/m} = 0.178 m$

Example 2: Unknown Mass on Spring

• Given mass $m$ has period $T$, and an unknown mass $m'$ has period $T'$. To find spring constant and unknown mass.
  Part (a):

  • Formula for angular frequency:
    $ext{W} = rac{2 ext{π}}{T}$

  • Formula for spring constant:
    $k = m ext{W}^2$

  Part (b): Using the values from (a) to find $m'$:

  • Rearranging yields:
    $m' = k rac{T'^2}{4 ext{π}^2}$

Example 3: Block-Spring System

• Given a block-spring system oscillates with an amplitude of 3.50 cm and spring constant of 250 N/m.
  (a) Total energy can be calculated using maximum potential or kinetic energy:
  $E = rac{1}{2} k A^2 = rac{1}{2} imes 250 ext{N/m} imes (0.035m)^2 = 0.153J$

  (b) Maximum speed is calculated:
  $v_{ ext{max}} = A ext{w} = ext{A} imes ext{sqrt} rac{k}{m}$

  (c) Maximum acceleration:
  $a_{ ext{max}} = rac{k}{m}A$

  • Results in 17.5 m/s².

5. Block Connected to Two Springs

• Involves demonstrating that a block connected to two springs exhibits SHM with specified periods given certain configurations.

• Requires application of earlier stated principles for SHM and spring-mass systems to derive results concerning motion and energy.