15.1 Simple Harmonic Motion
4 min read•june 24, 2024
is the backbone of oscillations and waves. It's everywhere, from pendulums swinging to springs bouncing. Understanding its key features like , , and helps us grasp how these motions work.
Energy plays a big role in harmonic motion, constantly shifting between kinetic and potential forms. Phase shifts and mass-spring systems add complexity, showing how these motions can interact and be described mathematically. It's all about patterns and predictability in motion.
Simple Harmonic Motion
Period and frequency in harmonic motion
Top images from around the web for Period and frequency in harmonic motion
16.6 Uniform Circular Motion and Simple Harmonic Motion – x-Douglas College Physics 1107 Fall ... View original
Is this image relevant?
The Simple Pendulum · Physics View original
Is this image relevant?
Simple Harmonic Motion: A Special Periodic Motion | Physics View original
Is this image relevant?
16.6 Uniform Circular Motion and Simple Harmonic Motion – x-Douglas College Physics 1107 Fall ... View original
Is this image relevant?
The Simple Pendulum · Physics View original
Is this image relevant?
1 of 3
Top images from around the web for Period and frequency in harmonic motion
16.6 Uniform Circular Motion and Simple Harmonic Motion – x-Douglas College Physics 1107 Fall ... View original
Is this image relevant?
The Simple Pendulum · Physics View original
Is this image relevant?
Simple Harmonic Motion: A Special Periodic Motion | Physics View original
Is this image relevant?
16.6 Uniform Circular Motion and Simple Harmonic Motion – x-Douglas College Physics 1107 Fall ... View original
Is this image relevant?
The Simple Pendulum · Physics View original
Is this image relevant?
1 of 3
- Period () time required for an oscillating object to complete one full cycle of motion
- Measured in seconds (s)
- Calculated using the equation , where is the frequency
- Examples: pendulum swing, vibrating string
- Frequency () number of oscillations or cycles per unit time
- Measured in (Hz) or cycles per second (s)
- Calculated using the equation , where is the period
- Examples: tuning fork vibration, alternating current
Characteristics of harmonic motion
- directly proportional to displacement and acts in opposite direction
- Described by the equation , where is the and is the displacement from equilibrium
- Examples: spring force, pendulum gravitational force
- Motion sinusoidal, with object oscillating back and forth about equilibrium position
- Follows a smooth, repetitive pattern
- Examples: wave motion, vibrating string
- () maximum displacement from equilibrium position
- Determines the energy of the oscillation
- Examples: wave height, pendulum swing angle
- Motion periodic, repeating itself at regular time intervals (period)
- Allows for predictable behavior
- Examples: clock pendulum, AC voltage
- Total energy (kinetic + potential) remains constant throughout motion
- Energy converts between kinetic and potential forms
- Examples: spring-mass system, pendulum swing
Phase shift in oscillations
- () difference in starting point of oscillation between two or more simple harmonic motions
- Measured in radians or degrees
- Represents a temporal offset between oscillations
- Examples: sound wave interference, coupled pendulums
- caused by differences in initial conditions or external factors
- Initial position, velocity, or applied force can affect phase
- Examples: two pendulums released at different times, driven oscillations
- Oscillations with same frequency and amplitude but different phase shifts have maximum and minimum displacements occurring at different times
- Results in constructive or destructive interference when combined
- Examples: sound waves from multiple sources, light wave interference
- Displacement with phase shift described by the equation
- is the
- determines the initial displacement and direction of motion
- Examples: shifted cosine wave, coupled harmonic oscillators
- (θ) represents the angle between the oscillating object and its equilibrium position
- Relates to linear displacement in circular motion and pendulums
Equations for mass-spring systems
- for derived using Newton's second law () and ()
- , where is mass and is displacement
- Examples: spring-mass oscillator, vibrating molecule
- Solution to for displacement:
- is the
- Examples: position of mass on spring, displacement of vibrating string
- Velocity found by differentiating position equation
- Represents the rate of change of displacement
- Examples: speed of mass on spring, velocity of vibrating particle
- Acceleration found by differentiating velocity equation or using original equation of motion
- Represents the rate of change of velocity
- Examples: acceleration of mass on spring, acceleration of vibrating object
Energy of vertical spring oscillations
- Mass oscillating on vertical spring experiences and gravity
- Equilibrium position slightly lower than relaxed spring length due to mass weight
- Examples: spring-mass system hanging vertically, bungee jumper
- () same as original spring constant ()
- Gravitational force does not affect the stiffness of the spring
- Examples: spring-mass system in vertical orientation, suspension bridge
- Equation of motion same as horizontal mass-spring system, with shifted equilibrium position
- Gravitational force accounted for in the new equilibrium position
- Examples: vertical spring-mass oscillator, diving board
- Potential energy includes elastic potential () and gravitational potential ()
- stored in the compressed or stretched spring
- Gravitational potential energy depends on the height of the mass relative to a reference point
- Examples: compressed spring, object raised above ground
- () same as in horizontal case
- Depends on the velocity of the oscillating mass
- Examples: moving mass on spring, bouncing object
- Total energy (kinetic + elastic potential + gravitational potential) remains constant throughout motion
- Energy converts between forms as mass oscillates
- Examples: energy conservation in spring-mass system, energy transfer in bouncing ball
Advanced oscillation concepts
- occurs when an oscillating system is driven at its , resulting in maximum amplitude
- Natural frequency is the frequency at which a system oscillates freely without external forces
- reduces the amplitude of oscillations over time due to energy dissipation
- Examples: shock absorbers, pendulum motion in air
- occurs when an external periodic force is applied to an oscillating system
- Can lead to if the driving frequency matches the system's natural frequency
Key Terms to Review (33)
Amplitude: Amplitude is the maximum displacement of a point on a wave from its equilibrium position. It is a measure of the energy carried by the wave.
Amplitude: Amplitude is the maximum displacement or extent of a periodic motion, such as a wave or an oscillation, from its equilibrium position. It represents the magnitude or size of the motion and is a fundamental characteristic of various physical phenomena described in the topics of 1.7 Solving Problems in Physics, 8.4 Potential Energy Diagrams and Stability, 15.1 Simple Harmonic Motion, and beyond.
Amplitude (A): Amplitude (A) is the maximum displacement of an oscillating object from its equilibrium position. It measures the extent of oscillation and is a key parameter in describing wave motion.
Angular Displacement: Angular displacement is a measure of the change in the angular position of an object or a system. It describes the rotation or the change in the orientation of an object around a fixed axis or point. This concept is fundamental in understanding rotational motion and its relationship with linear motion in various physics topics.
Angular frequency: Angular frequency, denoted by $\omega$, is the rate of change of angular displacement with time. It is commonly measured in radians per second (rad/s).
Angular Frequency: Angular frequency, often represented by the Greek letter $\omega$ (omega), is a fundamental concept that describes the rate of change of the angular position of an object undergoing rotational or oscillatory motion. It is a crucial parameter in understanding various physical phenomena, including simple harmonic motion, wave propagation, and the behavior of oscillating systems.
Damping: Damping refers to the process of reducing or controlling the amplitude or oscillation of a system over time. It is a phenomenon that occurs in various physical systems, including mechanical, electrical, and electronic systems, where it serves to dissipate energy and prevent excessive vibrations or oscillations.
Differential Equation: A differential equation is a mathematical equation that relates a function with its derivatives. These equations are used to model and analyze a wide range of physical, biological, and engineering systems that involve the rate of change of a quantity with respect to other variables.
Effective Spring Constant: The effective spring constant is a measure of the stiffness or elasticity of a system that exhibits simple harmonic motion. It represents the force required to produce a unit displacement in the system, and is a crucial parameter in understanding the dynamics of oscillating systems.
Elastic potential energy: Elastic potential energy is the energy stored in elastic materials as a result of their stretching or compressing. It is quantified by the equation $U = \frac{1}{2} k x^2$, where $k$ is the spring constant and $x$ is the displacement from equilibrium.
Elastic Potential Energy: Elastic potential energy is the potential energy stored in an object due to its deformation or compression. It is the energy that is stored in an elastic material when it is stretched or compressed and has the ability to do work as the material returns to its original shape.
Equation of Motion: The equation of motion is a fundamental concept in classical mechanics that describes the relationship between the position, velocity, acceleration, and time of an object undergoing motion. It is a mathematical expression that allows for the prediction and analysis of an object's movement under the influence of various forces.
Force constant (k): The force constant, often denoted as $k$, is a measure of the stiffness of a spring. It is defined in Hooke's Law, where the restoring force $F$ is proportional to the displacement $x$ from the equilibrium position: $F = -kx$.
Forced Oscillation: Forced oscillation refers to the phenomenon where an external force or driving force causes a system to oscillate at a specific frequency, even if the system's natural frequency is different. This concept is crucial in understanding both simple harmonic motion and the behavior of standing waves.
Frequency: Frequency is a fundamental concept in physics that describes the number of occurrences of a repeating event or phenomenon per unit of time. It is a crucial parameter in various areas of physics, including wave behavior, oscillations, and sound propagation.
Hertz: Hertz (Hz) is the unit of frequency, which measures the number of cycles or oscillations that occur per second. It is a fundamental concept in physics, particularly in the study of wave phenomena, such as sound waves and electromagnetic waves.
Hooke's Law: Hooke's law is a fundamental principle in physics that describes the linear relationship between the force applied to an elastic object and the resulting deformation or displacement of that object. It is a crucial concept that underpins the understanding of various physical phenomena, including work, conservative and non-conservative forces, potential energy diagrams and stability, stress, strain, and elasticity, as well as simple harmonic motion.
Kinetic energy: Kinetic energy is the energy possessed by an object due to its motion. It depends on the mass and velocity of the object.
Mass-Spring System: A mass-spring system is a physical model that consists of a mass attached to a spring, which represents a simple harmonic oscillator. This system is commonly used to study the principles of vibration, energy, and stability in various fields of physics.
Natural Frequency: Natural frequency is the inherent frequency at which a system tends to oscillate when it is not affected by external forces. It is a fundamental property of a system that depends on its physical characteristics and determines how the system will respond to various inputs or disturbances.
Orbital period: The orbital period is the time taken for a satellite or celestial body to complete one full orbit around another object. It is typically measured in seconds, minutes, hours, or years.
Period: The period of a periodic phenomenon is the time taken for one complete cycle or repetition of the event. This concept is fundamental in understanding various physics topics, including uniform circular motion, simple harmonic motion, and wave phenomena.
Periodic motion: Periodic motion is a type of motion that repeats itself at regular time intervals. Examples include oscillations of a pendulum or vibrations of a mass-spring system.
Phase shift: Phase shift is the amount by which a wave is shifted horizontally from a reference wave. It is typically measured in degrees or radians.
Phase Shift: Phase shift refers to the displacement or time difference between two periodic signals or waves. It describes the relative position of one wave's cycle compared to another, often measured in degrees or radians.
Resonance: Resonance occurs when a system is driven at its natural frequency, leading to a significant increase in amplitude. It is a crucial concept in oscillations and wave phenomena.
Resonance: Resonance is a phenomenon that occurs when a system is driven by a force that matches the system's natural frequency of oscillation, leading to a significant increase in the amplitude of the system's response. This concept is fundamental across various fields in physics, including mechanics, acoustics, and electromagnetism.
Restoring force: A restoring force is a force that gives rise to an equilibrium in a physical system. It acts in the direction opposite to the displacement of the object, aiming to bring it back to its equilibrium position.
Restoring Force: The restoring force is the force that acts to return an object or system to its equilibrium or original state after it has been displaced or disturbed from that state. This force plays a crucial role in understanding various physical phenomena, including the behavior of oscillating systems, the stability of structures, and the energy changes associated with different types of motion.
Simple Harmonic Motion: Simple harmonic motion 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 found in various physical systems, including pendulums, mass-spring systems, and vibrating molecules.
Simple harmonic motion (SHM): Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It is characterized by its sinusoidal oscillations in time.
Simple harmonic oscillator: A simple harmonic oscillator is a system where the force acting on an object is directly proportional to its displacement from equilibrium and acts in the opposite direction. This results in periodic motion, such as that of a mass on a spring or a pendulum.
Spring Constant: The spring constant, often denoted as 'k', is a measure of the stiffness of a spring. It quantifies the force required to stretch or compress a spring by a unit distance, and it is a fundamental property of a spring that is crucial in understanding its behavior in various physical contexts.