Simple Harmonic Motion

• SCED 135 covers Waves and Optics

Syllabus Overview

• Periodic Motion: Describes oscillations, simple harmonic motion, energy, applications, pendulums, damped/forced oscillations and resonance.
• Mechanical Waves: Encompasses types of mechanical waves, periodic waves, math descriptions, speeds, energy, wave interference, superposition, standing waves and normal modes.
• Sound and Hearing: Includes sound waves, speed of sound waves, sound intensity, standing sound waves, resonance, interference, beats, Doppler effect and shock waves.
• Nature and Propagation of Light: Addresses nature, reflection/refraction, total internal reflection, dispersion, polarization, scattering and Huygen’s Principle.
• Geometric Optics and Optical Instruments: Covers reflection/refraction at surfaces, thin lenses, cameras, eye, magnifiers, microscopes and telescopes.
• Interference: Includes interference, coherent sources, two-source interference, intensity, thin films and the Michelson interferometer.
• Diffraction: Features Fresnel/Fraunhofer diffraction, single slit diffraction, intensity, multiple slits, diffraction gratings, X-Ray diffraction, apertures, resolving power, and holography.

Periodic Motion

• Periodic motion or oscillation is a back and forth repetitive motion that is sinusoidal with a stable equilibrium position.
• Many kinds of motion like pendulums, musical vibrations, and pistons in car engines repeat themselves and this behavior is called periodic motion or oscillation.
• Galileo's pendulum experiments (1638), Huygens, Newton & others developed further analyses in the study of SHM.
• Examples include pendulums (clocks, seismometers), vibrations and waves (sea waves, earthquakes, tides, orbits of plants and moons), and vibrations of molecules and structures (aircraft fuselage, musical instruments, bridges).
• A restoring force exerted by a spring is displaced from its equilibrium position, which tends to restore the object to the equilibrium position causing oscillation or periodic motion.

Describing Oscillation

• Amplitude (A) refers to the maximum magnitude of displacement from equilibrium.
• Period (T) signifies the time for one complete cycle, while frequency (f) is the number of cycles per unit time.
• Angular frequency (ω) is equivalent to 2π times the frequency (ω = 2πf).
• Frequency and period are reciprocals (f = 1/T and T = 1/f).

Simple Harmonic Motion

• Simple Harmonic Motion (SHM) is a vibrating system where the restoring force is directly proportional to the negative displacement.
• The restoring force is directly proportional to the displacement from equilibrium in simple harmonic motion.
• A body that undergoes simple harmonic motion is called a harmonic oscillator.

Simple Harmonic Motion Equations

- F = -kx

- F = -mw²x

- k = mw²

- ω = √k/m

- w = √ k/m (simple harmonic motion)

- f = w/2π = 1/2π√k/m (simple harmonic motion)

- T = 1/f = 2π/w = 2π√ m/k (simple harmonic motion)

- x = A cos (wt + φ) (displacement in SHM)

- x (t) = Acos (wt + φ)

- v (t) = -vmaxsin (wt + φ)

- a(t) = -amax Cos (wt + φ)

- Xmax = A

- Umax = Aω

- amax = Aw².

Energy in Simple Harmonic Motion

• E = 1/2mvx²+ 1/2kx2 = 1/2kA² = constant (total mechanical energy in SHM)

Application of Simple Harmonic Motion

• Fnet = k(∆l − x) + (-mg) = -kx

Simple Pendulum

• A simple pendulum is an idealized model that consists of a point mass suspended by a massless, unstretchable string.

Simple Pendulum Equations

- Fe = -mg sin0

- w = √ k/m = √mg/Lm = √g/L (simple pendulum, small amplitude)

- f = ω/2π = 1/2π√g/L (simple pendulum, small amplitude)

- T = 1/f = 2π/w == 2π√L/g (simple pendulum, small amplitude)

Physical Pendulum

• A physical pendulum is any real pendulum that uses an extended body, as contrasted to the idealized model of the simple pendulum with all the mass concentrated at a single point.

Physical Pendulum Equations

- ω = √mgd/I (physical pendulum, small amplitude)

- T = 2π√ I/mgd (physical pendulum, small amplitude)

Damped Oscillations

• Damping refers to the decrease in amplitude caused by dissipative forces, the motion is called damped oscillation.

Types of Damping

• Critical Damping: Returns to equilibrium without oscillation.
• Overdamping: Returns to equilibrium more slowly than critical damping.
• Underdamping: Oscillates with steadily decreasing amplitude.

Forced Oscillations and Resonance

• Forced oscillation or a driven oscillation occurs when a periodically varying driving force with angular frequency is applied to a damped harmonic oscillator.
• Resonance relates to the amplitude and is a function of the driving frequency and reaches a peak at a driving frequency close to the natural frequency of the system.