Oscillation

1. Simple harmonic motion:

• a) Displacement, amplitude, period, frequency, angular frequency and phase difference

• ⇒ Displacement (x):
• The distance of the oscillating particle from its equilibrium position at any instant.
•  Unit: meters (m).
• ⇒ Amplitude (A):
• The maximum displacement of the oscillating particle from its equilibrium position.
• Unit: meters (m).
• 
• Figure 1 Amplitude mentioned in a wave
• ⇒ Period (T):
• The time taken for one complete oscillation or cycle.
• Unit: seconds (s).
• ⇒ Frequency (f):
• The number of complete oscillations per second.
• [math]f = \frac{1}{T}[/math]
• Unit: hertz (Hz).

• b) Angular Frequency (ω):

• The rate of change of angular displacement during oscillation.
• [math] ω = \frac{2π}{T} = 2πf [/math]
• Unit: radians per second (rad/s).
• ⇒ Phase Difference:
• The relative displacement of two oscillating particles as a fraction of their cycle, expressed in radians (∅) or as an angle.
• 
• Figure 2 Phase difference between two waves

• c)   I) Simple Harmonic Motion (SHM)

• ⇒ Defining Equation:
• Simple harmonic motion occurs when the restoring force is proportional to the displacement and directed toward the equilibrium position.
• The acceleration of the particle is given by:
• [math]a = -ω^2 x [/math]
• Where:
• a = acceleration (m/s²),
• ω = angular frequency (rad/s),
• x = displacement from equilibrium (m).

• II) Techniques and procedures used to determine the period/frequency of simple harmonic oscillations:
• The period (T) and frequency (f) of simple harmonic oscillations can be determined experimentally using various techniques. These methods apply to systems such as pendulums, springs, or vibrating objects that exhibit simple harmonic motion (SHM). Below are the primary techniques and procedures used:
• – Direct Measurement Using a Stopwatch
• – Using an Oscilloscope (for Electrical Oscillations)
• – Photogate Timer or Motion Sensor
• – Video Analysis
• – Resonance Method (For Driven Oscillations)
• – Mathematical Analysis Based on Theory
• – Simple pendulum
• – Mass attach to a spring
• ⇒ For a Simple Pendulum:
• – Measure the length (L) of the string.
• – Displace the pendulum by a small angle and release.
• – Use a stopwatch to measure the time for multiple oscillations (N).
• – Calculate the period (T) using:
• [math] T = \frac{total time}{N} [/math]
• 
• Figure 4 simple pendulum system
• – For small amplitudes:
• [math] T = 2π \sqrt{\frac{L}{g}}[/math]
• ⇒ For a Mass-Spring System:
• Hang a known mass on a spring and displace it.
• Measure the time for multiple oscillations.
• Calculate the period:
• [math]T = 2π \sqrt{\frac{m}{k}}[/math]
• ​
• Figure 5 Mass spring system
• Where:
• – m = mass,
• – k = spring constant.

• d)    Solutions to SHM Equation:

• The displacement (x) as a function of time (t) can be expressed as:
• [math]x = Acos(ωt) \quad \text{ or } \quad x = Asin(ωt) [/math]
• Where:
• – x = displacement,
• – A = amplitude,
• – ω = angular frequency,
• – t = time (s).

• e)    Velocity in SHM:

• The velocity of the particle at any point during SHM is:
• [math]v = ±ω \sqrt{A^2 – X^2} [/math]
• Where:
• – v = velocity (m/s)
• – A = amplitude,
• – x = displacement.
• The maximum velocity occurs when x=0 (equilibrium position):
• [math] v_{max} = ω [/math]

• f)  Period Independence from Amplitude (Isochronous Oscillator)

• In SHM, the period of oscillation is independent of the amplitude A.
• [math]T = \frac{2π}{ω} [/math]
• This property makes SHM isochronous, meaning oscillations occur with the same period regardless of amplitude.

• g)    Graphical Relationships

• ⇒ Displacement vs. Time:
• Sine or cosine wave.
• Maximum displacement: x=±Ax.
• 
• Figure 3 sine and cosine wave
• ⇒ Velocity vs. Time:
• A cosine or sine wave shifted by π/2 radians relative to displacement.
• Maximum velocity:[math]v = ±ωA[/math].
• ⇒ Acceleration vs. Time:
• A sine or cosine wave shifted by π\piπ radians relative to displacement.
• Maximum acceleration:[math]a = ±ω^2 A[/math]

• h)   Techniques to Determine the Period/Frequency of SHM

• 2. Energy of a simple Harmonic Oscillator:

• A simple harmonic oscillator exhibits a continuous interchange between kinetic energy (KE) and potential energy (PE) as it oscillates. The total mechanical energy ([math]E_{total}[/math]​) remains constant (assuming no damping).

• a)   Interchange Between Kinetic and Potential Energy

• ⇒ Total Energy:
• The total mechanical energy in SHM is constant and given by:
• [math]E_{total} = \frac{1}{2} kA^2[/math]
• Where:
• – k = spring constant (N/m) or stiffness,
• – A = amplitude of oscillation (maximum displacement).
• Alternatively, in terms of angular frequency ω and mass mmm:
• [math]E_{total} = \frac{1}{2} kω^2 A^2 [/math]
• ⇒ Kinetic Energy (KE):
• Kinetic energy is highest when the object is at equilibrium (x=0) and velocity is maximum ([math]v = ±ωA[/math]):
• [math]KE = \frac{1}{2} mv^2 = \frac{1}{2} mω^2 (A^2-x^2) [/math]
• ⇒ Potential Energy (PE):
• Potential energy is highest at maximum displacement (x=±A), where the velocity is zero:
• [math]PE = \frac{1}{2} kx^2 = \frac{1}{2} mω^2 x^2 [/math]
• ⇒ Energy Exchange:
• At x=0 (equilibrium):
• – KE is maximum, PE is zero.
• At x=±A (extremes):
• – PE is maximum, KE is zero.
• At intermediate positions:
• – Both KE and PE are non-zero, and their sum equals ​.

• b)   Energy-Displacement Graphs:

• ⇒ Kinetic Energy (KE) vs. Displacement (x):
• [math]KE = \frac{1}{2} mω^2 (A^2-x^2) [/math]
• Parabolic shape.
• KE is maximum [math]\frac{1}{2} mω^2 A^2 [/math] at x=0 and decreases to zero at x=±A.
• 
• Figure 6 Energy – displacement graph
• ⇒ Potential Energy (PE) vs. Displacement (x):
• [math]PE = \frac{1}{2} mω^2 x^2 [/math]
• Parabolic shape.
• PE is zero at x=0 and maximum ([math] \frac{1}{2} mω^2 A^2[/math]) at x=±A.
• ⇒ Total Energy ([math]E_{total}[/math]) vs. Displacement (x):
• [math]E_{total} = \frac{1}{2} mω^2 A^2 [/math]
• A horizontal line on the graph, as total energy is independent of displacement.

• 3 Damping:

• a) Free and Forced Oscillations

• ⇒ Free Oscillations:
• A system oscillates without any external force after being displaced from equilibrium.
• The system’s natural frequency ([math]f_{natural}[/math]​) is determined by its physical properties.
• Example: A pendulum swinging in air or a mass-spring system in a vacuum.
• ⇒ Forced Oscillations:
• An external periodic force drives the system.
• The system oscillates at the frequency of the driving force ([math]f_{driving}[/math]), not its natural frequency.
• Example: A child pushed on a swing at regular intervals.
• 
• Figure 7 Forced oscillation

• b)    I) Effects of Damping on an Oscillatory System

• ⇒ Damping:
• Damping occurs when energy is lost from an oscillating system, typically due to resistive forces such as friction or air resistance.
• The rate of energy loss affects the system’s amplitude and frequency over time.
• ⇒ Types of Damping:
• Light Damping:
• – Amplitude decreases gradually, but oscillations persist.
• Critical Damping:
• – The system returns to equilibrium in the shortest time without oscillating.
• Heavy (Overdamping):
• – The system returns to equilibrium slowly, without oscillating.

• b-II) Observing Forced and Damped Oscillations

• Techniques to study forced and damped oscillations include:

• Mass-Spring System:
• – Apply an external periodic force using a motorized device.
• – Introduce damping by submerging the spring in a viscous fluid like oil or by using air resistance.
• Tuning Fork or Loudspeaker:
• – Drive a tuning fork or a loudspeaker diaphragm at different frequencies.
• – Measure amplitude changes with a vibration sensor.
• Pendulum in Air and Water:
• – Compare the decay of oscillations for a pendulum in air and submerged in water.

• c)     Resonance and Natural Frequency

• Natural Frequency ([math]f_{natural} [/math] ​):
• – The frequency at which a system naturally oscillates when undisturbed.
• Resonance:
• – Occurs when the driving frequency matches the natural frequency of the system ([math]f_{driving} = f_{natural}[/math]).
• At resonance:
• – Amplitude becomes maximum.
• – Energy transfer from the driving force to the system is most efficient.
• Practical Observations:
• – Resonance can be observed by varying the driving frequency and recording the resulting amplitudes.

• d)    Amplitude-Driving Frequency Graphs for Forced Oscillators

• Characteristics:
• – Below Natural Frequency: Low amplitude, oscillations lag behind driving force.
• – At Natural Frequency: Maximum amplitude, oscillations in phase with the driving force.
• – Above Natural Frequency: Low amplitude, oscillations lead the driving force.
• Effects of Damping:
• – Increased damping reduces the peak amplitude and broadens the resonance curve.
• – In lightly damped systems, the resonance peak is sharper.

• e)    Practical Examples of Forced Oscillations and Resonance

• Examples of Forced Oscillations:
• – A washing machine drum driven by a motor.
• – A bridge oscillating due to wind or marching troops.
• Examples of Resonance:
• – Musical Instruments: Resonance amplifies sound in a guitar or violin body.
• – Tacoma Narrows Bridge: Destroyed by wind-induced resonance.
• – Microwave Ovens: Water molecules resonate at the microwave frequency to heat food.
• – Radio Tuning: Adjusting the circuit to resonate at a specific radio frequency.