Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Classical mechanics → Oscillations and waves ↓


Simple Harmonic Motion


Simple Harmonic Motion (SHM) is a type of periodic motion or oscillatory motion where the restoring force is proportional to the displacement and acts in the opposite direction to the displacement. It is the simplest type of oscillatory motion and can be explained through various practical examples and physics applications.

Introduction to simple harmonic motion

Simple harmonic motion occurs in many physical systems and forms the basis for understanding more complex types of wave motion. When we talk about SHM, it usually relates to vibrations in mechanical systems, sound waves, light waves, and even oscillations found in electrical circuits. Understanding SHM is important to understand how oscillations work in different contexts.

Main features of SHM

There are several characteristics that define simple harmonic motion:

  • Restoring force: The force that brings the object back to the equilibrium position is proportional to the displacement. It is expressed as F = -kx, where k is a constant known as the spring constant, and x is the displacement from the equilibrium position.
  • Equilibrium position: It is the central position where the net force acting on the object is zero.
  • Periodicity: The motion is periodic, which means it repeats in a regular cycle. The time taken for one complete cycle of the motion is called the period (T).
  • Amplitude: Maximum displacement from the equilibrium position.
  • Frequency: The number of cycles per unit time is the frequency (f), which is related to the period by f = 1/T.

Mathematical formulation of S.H.M.

Equation of motion

The motion of an object in simple harmonic motion can be described by a second-order differential equation derived from Hooke's law. The general form of the equation is:

m * d²x/dt² + kx = 0

Where:

  • m is the mass of the object.
  • k is the spring constant.
  • x is the displacement.

Solution of a differential equation

The solution to this differential equation is:

x(t) = A * cos(ωt + φ)

Where:

  • A is the dimension.
  • ω (angular frequency) is given by √(k/m).
  • t is the time variable.
  • φ is the phase angle.

Energy in simple harmonic motion

Energy is conserved in SHM and can be viewed as a continuous exchange between potential energy and kinetic energy. At maximum displacement, the energy of the system is entirely potential. In contrast, when the object passes through the equilibrium position, all the energy is kinetic.

The manifestations of these energies in the system are as follows:

  • Potential Energy (U): U = (1/2)kx²
  • Kinetic Energy (K): K = (1/2)m(v²), where v = ωA * sin(ωt + φ)

Total mechanical energy: Total energy is the sum of potential energy and kinetic energy:

E = U + K = (1/2)kA²

Note that this total energy is constant and independent of time.

Graphical representation

Simple harmonic motion can be represented visually through a displacement-time graph. Below is an SVG representation showing a harmonic oscillator at different points in time.

-A +A

Each circular position represents a critical point in the dance of harmonic motion: from maximum positive displacement (+A) through equilibrium (0) to maximum negative displacement (-A).

Applications of simple harmonic motion

SHM is fundamental to many areas of physics and engineering. Here are some applications:

  • Pendulum: A simple pendulum exhibits SHM for small angles. The period of the pendulum depends only on its length and the gravitational acceleration, which is described as:
    • T = 2π√(L/g)
  • Mass-spring system: The classic example is a mass attached to a spring, where the mass oscillates back and forth when the system is disturbed from its equilibrium position.
  • Electrical circuits: LC or LCR circuits represent the electrical analog of SHM where charge, current, voltage oscillate.

Practical example: spring-mass system

Consider a mass (m) attached to a spring with stiffness constant (k). The system exhibits simple harmonic motion when displaced from equilibrium and released.

Assuming there is no damping, the equation of motion is:

m * d²x/dt² = -kx

Solving this gives:

x(t) = A * cos(ωt)

Where:

  • ω = √(km/m)
  • A is the maximum amplitude of displacement

The period (T) and frequency (f) can be found from:

T = 2π√(m/k) f = 1/T

Conclusion

Simple harmonic motion is a foundational concept in physics that provides insight into periodic and oscillatory phenomena. By understanding SHM, one not only understands the mechanics of vibrations and waves, but also applies these principles in a variety of scientific and engineering scenarios, from designing time-keeping devices to understanding molecular dynamics.

Whether studying a swinging pendulum, stringing a guitar, or designing an electrical circuit, the principles of simple harmonic motion serve as a guide to understanding and harnessing the power of oscillation in many forms.


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Simple Harmonic Motion

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