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16.3 Simple Harmonic Motion: A Special Periodic Motion

3 min read•june 18, 2024

is the back-and-forth movement of objects around a central point. It's seen in everyday things like swings and pendulums. This motion follows predictable patterns, making it a key concept in physics for understanding oscillations and waves.

The notes dive into the math behind simple harmonic motion, covering important terms like , , and . They also explain how mass and force affect oscillations, and describe the energy changes that happen during this type of motion.

Simple Harmonic Motion

Characteristics of harmonic motion

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  • occurs when a particle oscillates about an with a directly proportional to the displacement and acting in the opposite direction
    • expressed as F=−kxF = -kx, where kk is the () and xx is the displacement from equilibrium
    • Examples include a mass attached to a spring, a , or a vibrating string ()
  • (AA) represents the maximum displacement of the particle from its equilibrium position
    • Measured in units of length (m, cm)
  • (TT) is the time taken for one complete oscillation or cycle
    • Measured in seconds (s)
    • Examples: a pendulum with a period of 2 s, a vibrating string with a period of 0.01 s
  • Frequency (ff) is the number of complete oscillations per unit time
    • Measured in (Hz)
    • Related to period by the equation f=1Tf = \frac{1}{T}
    • Examples: a tuning fork with a frequency of 440 Hz, a pendulum with a frequency of 0.5 Hz

Period factors in oscillators

  • The period of a is determined by its mass (mm) and the force constant (kk)
    • Period is calculated using the equation T=2πmkT = 2\pi\sqrt{\frac{m}{k}}
  • Mass and period have a direct relationship
    • As mass increases, the period increases, resulting in slower oscillations
    • Examples: a pendulum with a heavier bob will have a longer period, a more massive spring-mass system will oscillate more slowly
  • Force constant and period have an inverse relationship
    • As the force constant increases, the period decreases, leading to faster oscillations
    • Examples: a stiffer spring will result in a shorter period, a pendulum with a shorter length will have a higher force constant and oscillate more quickly

Equations for harmonic motion

  • Position as a function of time: x(t)=Acos⁡(ωt+ϕ)x(t) = A\cos(\omega t + \phi)
    • AA is the amplitude
    • ω\omega is the , given by ω=2πT=2πf\omega = \frac{2\pi}{T} = 2\pi f
    • ϕ\phi is the , determined by initial conditions (initial position and velocity)
    • Example: a mass on a spring with an amplitude of 0.1 m and a period of 2 s will have a position function x(t)=0.1cos⁡(πt)x(t) = 0.1\cos(\pi t)
  • Velocity as a function of time: v(t)=−Aωsin⁡(ωt+ϕ)v(t) = -A\omega\sin(\omega t + \phi)
    • Maximum velocity occurs when the particle passes through the equilibrium position
    • Velocity is zero at the extremes of the motion (maximum displacement)
    • Example: a pendulum will have its maximum velocity at the lowest point of its swing and zero velocity at the highest points
  • Acceleration as a function of time: a(t)=−Aω2cos⁡(ωt+ϕ)a(t) = -A\omega^2\cos(\omega t + \phi)
    • Acceleration is directly proportional to the displacement and always directed towards the equilibrium position
    • Maximum acceleration occurs at the extremes of the motion (maximum displacement)
    • Acceleration is zero when the particle passes through the equilibrium position
    • Example: a vibrating string will experience maximum acceleration at its maximum displacement and zero acceleration at its equilibrium position

Energy in Simple Harmonic Motion

  • is a fundamental principle in SHM
  • The total energy of the system remains constant, alternating between and elastic potential energy
  • At the equilibrium position, the system has maximum kinetic energy and minimum potential energy
  • At the extremes of motion, the system has maximum elastic potential energy and minimum kinetic energy
  • The interplay between these energy forms drives the continuous oscillation in SHM
  • occurs when energy is gradually lost from the system, reducing the amplitude of oscillations over time
  • is a phenomenon where an oscillating system responds with large amplitude to a periodic driving force of a specific frequency

Connection to Wave Motion

  • Simple harmonic motion is closely related to , as many waves can be described as a collection of harmonic oscillators

Key Terms to Review (29)

Amplitude: Amplitude is the maximum displacement of a wave or oscillating object from its equilibrium position. It measures the extent of change in the wave's physical quantity such as height or pressure.
Amplitude: Amplitude refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium. It plays a crucial role in understanding how energy is transferred in oscillatory systems, impacting the characteristics of waves and sounds.
Angular frequency: Angular frequency is a measure of how quickly an object oscillates or rotates, typically expressed in radians per second. It relates to the periodic motion of systems, connecting the time taken to complete a full cycle (the period) with how many cycles occur per unit of time (the frequency). This concept plays a crucial role in understanding various physical phenomena involving oscillations and circular motion.
Beat frequency: Beat frequency is the frequency at which two waves of slightly different frequencies interfere with each other, resulting in a modulation pattern perceived as a periodic variation in amplitude. It is calculated as the absolute difference between the frequencies of the two interfering waves.
Critical damping: Critical damping occurs when a damping force is applied to an oscillating system, bringing it to rest in the shortest possible time without oscillation. It represents the threshold between overdamping and underdamping.
Damping: Damping refers to the process of reducing or controlling the amplitude of an oscillating or vibrating system over time. It involves the dissipation of energy, which causes the system to gradually come to rest or a steady-state condition.
Energy conservation: Energy conservation is the principle that energy cannot be created or destroyed, only transformed from one form to another. This fundamental concept underscores the importance of understanding how energy changes forms in various processes, allowing for the analysis of mechanical systems, the impacts of energy use on the environment, and the oscillatory motion seen in physical systems.
Equilibrium Position: The equilibrium position refers to the stable or balanced state of a system where the net force acting on the system is zero, and the system remains at rest or in a state of constant motion. This concept is particularly important in the study of oscillations, simple harmonic motion, energy of oscillators, and damped motion.
Force Constant: The force constant, also known as the spring constant, is a measure of the stiffness of a spring or any elastic object. It represents the amount of force required to cause a unit displacement of the object from its equilibrium position. The force constant is a crucial parameter in the study of simple harmonic motion, as it determines the frequency and period of oscillation.
Frequency: Frequency is a fundamental concept in physics that describes the number of occurrences of a repeating event per unit of time. It is a crucial parameter in various areas of study, including radiation, oscillations, waves, sound, and electromagnetic phenomena.
Harmonic Oscillator: A harmonic oscillator is a system that exhibits oscillations, or repetitive motion, around an equilibrium position. It is a fundamental concept in physics that describes the behavior of various physical systems, including mechanical, electrical, and quantum-mechanical systems.
Hertz: Hertz is the unit of frequency, defined as one cycle per second. It is crucial in understanding oscillatory motion, where it quantifies how often an event occurs within a given time frame. This term is also important in various contexts, such as sound waves and electromagnetic waves, where frequency affects their properties and behaviors.
Hooke's Law: Hooke's law is a fundamental principle in physics that describes the relationship between the force applied to an object and the resulting deformation or displacement of that object. It states that the force required to stretch or compress a spring is proportional to the distance by which the spring is stretched or compressed, within the elastic limit of the material.
Internal kinetic energy: Internal kinetic energy is the sum of the kinetic energies of all particles within a system. It plays a crucial role in understanding how energy is distributed and conserved during elastic collisions.
Kinetic Energy: Kinetic energy is the energy of motion possessed by an object. It is the energy an object has by virtue of being in motion and is directly proportional to the mass of the object and the square of its velocity. Kinetic energy is a crucial concept in physics, as it relates to the work done on an object, the conservation of energy, and various other physical phenomena.
Mass-Spring System: A mass-spring system is a simple model used to describe the motion of an object attached to a spring and subjected to a force. It is a fundamental concept in the study of oscillatory motion and is central to the understanding of topics such as simple harmonic motion and damped harmonic motion.
Pendulum: A pendulum is a weight suspended from a fixed point that swings back and forth under the influence of gravity. Its motion is periodic, characterized by a constant period and frequency when displaced from its equilibrium position, making it an important example of oscillatory motion.
Period: The period is the time it takes for one complete cycle of an oscillation or wave to occur. It is typically measured in seconds.
Period: The period of an oscillation or wave is the time taken for one complete cycle to occur. It represents the time interval between successive repetitions of a particular state or event in a periodic motion or wave. This term is crucial in understanding various concepts related to oscillations, simple harmonic motion, pendulums, and waves.
Phase Constant: The phase constant is a parameter in the equation of motion for simple harmonic motion that determines the initial position and direction of an oscillating system at time zero. It plays a crucial role in defining the state of the system, affecting how the motion appears over time. The phase constant is typically represented by the symbol $$\phi$$ and is measured in radians.
Resonance: Resonance is a phenomenon that occurs when a system is driven by a periodic force at a frequency that matches the system's natural frequency of oscillation, resulting in a significant increase in the amplitude of the system's motion. This concept is fundamental in understanding various physical phenomena, including the behavior of oscillating systems, the propagation of waves, and the operation of electronic circuits.
Restoring force: A restoring force is a force that acts to bring a system back to its equilibrium position. It is directly proportional to the displacement and acts in the opposite direction.
Restoring Force: The restoring force is a force that acts to return a system to its equilibrium or resting state after it has been displaced or disturbed. This force arises from the inherent properties of the system and acts to counteract the external forces that caused the displacement, thereby restoring the system to its original position or configuration.
Robert Hooke: Robert Hooke was a 17th-century English scientist known for his foundational work in physics, particularly in the study of elasticity. He formulated Hooke's Law, which states that the strain in a solid is proportional to the stress applied to it, laying the groundwork for understanding materials under deformation. His contributions extend beyond elasticity to fields like biology and mechanics, establishing him as a pivotal figure in early scientific inquiry.
Simple Harmonic Motion: Simple harmonic motion is a type of periodic motion where an object oscillates back and forth around an equilibrium position, with a constant acceleration that is proportional to the displacement from the equilibrium point. This motion is characterized by a sinusoidal pattern and is the foundation for understanding many oscillatory phenomena in physics.
Simple Harmonic Motion (SHM): Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force acting on an object is directly proportional to its displacement from the equilibrium position. This motion is characterized by a sinusoidal pattern and is a fundamental concept in the study of oscillations and waves.
Simple harmonic oscillator: A simple harmonic oscillator is a system where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It exhibits periodic motion characterized by sinusoidal oscillations.
Spring Constant: The spring constant, also known as the force constant, is a measure of the stiffness of a spring. It represents the force required to stretch or compress a spring by a unit distance and is a fundamental property of the spring that determines its behavior in various physical contexts.
Wave Motion: Wave motion refers to the propagation of disturbances or oscillations through a medium, such as air, water, or a solid, without the actual transport of the medium itself. It involves the transmission of energy without the transfer of matter, as the wave travels through the medium.
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16.4 The Simple Pendulum