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Simple harmonic motion

Simple harmonic motion (SHM) is a type of periodic oscillatory motion in which a body moves back and forth about an equilibrium position such that its acceleration is always directed toward the equilibrium and is directly proportional to the displacement from it.^[1] This motion arises when the restoring force follows Hooke's law, F = -kx, where k is the force constant and x is the displacement.^[2] Mathematically, the position as a function of time is described by x(t) = A \cos(\omega t + \phi), where A is the amplitude, \omega = 2\pi / T is the angular frequency, T is the period, and \phi is the phase constant.^[3] The period of SHM for a mass-spring system is given by T = 2\pi \sqrt{m/k}, independent of amplitude, which is a key characteristic distinguishing it from other oscillations.^[4] Common examples include a mass attached to a spring oscillating horizontally or vertically, and a simple pendulum for small angular displacements less than about 15 degrees, where the motion approximates SHM.^[5] SHM serves as the foundational model for understanding more complex phenomena, such as vibrations in mechanical systems, sound waves, and atomic oscillations in molecules.^[6] The study of SHM originated with Galileo's observations of pendulum motion c. 1583, followed by developments from Christiaan Huygens and Isaac Newton in the late 17th century, who analyzed the isochronous properties and mathematical solutions of harmonic oscillators.^[7] These insights laid the groundwork for classical mechanics and continue to underpin fields like engineering, acoustics, and quantum physics, where the quantum harmonic oscillator models energy levels in atoms.^[6]

Definition and Characteristics

Definition

Simple harmonic motion (SHM) is a type of periodic motion in which an object oscillates about an equilibrium position such that the restoring force acting on it is directly proportional to its displacement from that position and directed opposite to the displacement.^[8] This proportionality results in the object's position varying sinusoidally with time, producing a smooth, repetitive back-and-forth trajectory.^[9] SHM represents an idealized form of oscillation commonly observed in systems where the net force follows Hooke's law, F = -kx, with k as the constant of proportionality and x as the displacement.^[10] The ideal assumptions underlying SHM include a frictionless environment to eliminate energy dissipation through damping, ensuring perpetual oscillation without amplitude decay.^[3] Additionally, the restoring force must be linear, adhering strictly to Hooke's law for springs or equivalent relations in other systems, and small-angle approximations are applied where necessary to maintain the proportionality, such as in pendulum motions./Book%3A_University_Physics_I_-Mechanics_Sound_Oscillations_and_Waves(OpenStax)/15%3A_Oscillations/15.02%3A_Simple_Harmonic_Motion) The foundational concepts of SHM emerged in the 17th century through the studies of Christiaan Huygens on pendulum oscillations and Robert Hooke on spring elasticity, laying the groundwork for understanding periodic restoring forces.^[11]^[12] In this motion, the period T is the time for one complete oscillation, during which the object returns to its initial position with the same direction of velocity, while the frequency f is the number of oscillations per unit time, given by f = 1/T./Book%3A_University_Physics_I_-Mechanics_Sound_Oscillations_and_Waves(OpenStax)/15%3A_Oscillations/15.02%3A_Simple_Harmonic_Motion)

Key Characteristics

Simple harmonic motion (SHM) exhibits symmetry about the equilibrium position, where the trajectory in phase space forms an ellipse symmetric with respect to both the displacement and velocity axes, ensuring that the motion mirrors itself across the equilibrium point.^[13] This symmetry implies that the time spent traversing from one extreme to the opposite extreme equals the time to return to the starting extreme, with each segment taking half the period T/2, reflecting the balanced nature of the oscillatory path.^[13] A defining feature of ideal SHM is isochronism, where the period of oscillation remains constant regardless of the amplitude, provided displacements are small enough to satisfy the linear approximation.^[14] This property, first noted by Galileo for pendulums, arises from the quadratic potential that governs the motion and holds approximately for small angles or extensions.^[14] The acceleration in SHM is directly proportional to the displacement from equilibrium but directed oppositely, expressed as

a = -\omega^2 x,

where \omega is the angular frequency and x is the displacement.^[1] In SHM, velocity reaches its maximum value at the equilibrium position, where displacement is zero, and drops to zero at the extreme positions of maximum displacement.^[1] Conversely, displacement is maximum at the extremes and zero at equilibrium. The state of motion evolves continuously through a phase angle, which parameterizes the sinusoidal functions describing position, velocity, and acceleration, capturing the progression through the cycle.^[1] This phase provides a unified way to describe the timing and orientation of the oscillation relative to a reference.^[1]

Mathematical Formulation

Differential Equation

The differential equation governing simple harmonic motion (SHM) arises from applying Newton's second law to a system where the restoring force is directly proportional to the displacement from equilibrium, such as a mass-spring system. The restoring force is given by Hooke's law, F = -kx, where k > 0 is the force constant and x is the displacement. Newton's second law states F = ma, with acceleration a = \frac{d^2x}{dt^2}, so for a mass m,

m \frac{d^2x}{dt^2} = -kx.

Dividing through by m yields the standard form of the differential equation:

\frac{d^2x}{dt^2} + \frac{k}{m} x = 0.

^[15]^[16] This equation is often expressed in a more general form as

\frac{d^2x}{dt^2} + \omega^2 x = 0,

where \omega = \sqrt{k/m} is the angular frequency of the motion, with units of radians per second.^[17] As a second-order linear homogeneous ordinary differential equation with constant coefficients, its solutions are inherently oscillatory and take the form of sinusoidal functions, reflecting the periodic nature of SHM.^[16] To confirm, substitute the trial solution x(t) = A \cos(\omega t + \phi), where A is the amplitude and \phi is the phase constant. The second derivative is \frac{d^2x}{dt^2} = -\omega^2 A \cos(\omega t + \phi) = -\omega^2 x(t). Plugging into the differential equation gives -\omega^2 x + \omega^2 x = 0, which is satisfied for \omega = \sqrt{k/m}.^[15]

General Solution

The general solution to the differential equation governing simple harmonic motion, \frac{d^2x}{dt^2} + \omega^2 x = 0, is given by

x(t) = A \cos(\omega t + \phi),

where A is the amplitude, representing the maximum displacement from the equilibrium position; \omega is the angular frequency, which determines the rate of oscillation; and \phi is the phase constant, which specifies the initial phase of the motion.^[1] This form arises from the characteristic equation r^2 + \omega^2 = 0, yielding roots r = \pm i \omega, and the real part of the exponential solution provides the cosine dependence.^[17] Equivalent alternative forms include

x(t) = A \sin(\omega t + \psi),

where \psi = \phi + \pi/2, or the linear combination

x(t) = C \cos(\omega t) + D \sin(\omega t),

with A = \sqrt{C^2 + D^2} and \tan \phi = -D/C (adjusting for the appropriate quadrant).^[1] These representations are interchangeable and chosen based on initial conditions for convenience.^[12] The constants A and \phi are determined from initial conditions, such as the position x(0) and velocity v(0) = \frac{dx}{dt}(0). Specifically, x(0) = A \cos \phi and v(0) = -A \omega \sin \phi, allowing solution for A = \sqrt{x(0)^2 + \left( \frac{v(0)}{\omega} \right)^2} and \phi = \tan^{-1} \left( -\frac{v(0)}{\omega x(0)} \right).^[1] The period of the motion is T = \frac{2\pi}{\omega}, the time for one complete cycle, and the frequency is f = \frac{\omega}{2\pi} = \frac{1}{T}, measured in hertz.^[17] In more advanced contexts, such as linear systems or wave mechanics, simple harmonic motion is often represented using complex exponentials for analytical convenience, where x(t) = \mathrm{Re} \left[ \tilde{A} e^{i \omega t} \right] and \tilde{A} is a complex amplitude incorporating the phase.^[18] This form facilitates superposition and Fourier analysis without altering the physical real-valued displacement.^[12]

Dynamics and Energy

Restoring Force and Acceleration

In simple harmonic motion (SHM), the restoring force acting on the oscillating object is directly proportional to its displacement from the equilibrium position and directed opposite to that displacement, given by Hooke's law as F = -kx, where k is the force constant and x is the displacement.^[3] This negative sign ensures the force always pulls or pushes the object back toward equilibrium, preventing unbounded motion and enabling periodic oscillation.^[19] The acceleration of the object follows from Newton's second law, a = F/m = -(k/m)x, where m is the mass, which simplifies to a = -\omega^2 x with \omega = \sqrt{k/m} as the angular frequency.^[9] Thus, acceleration is also proportional to displacement but opposite in direction, always directed toward the equilibrium point; for positive x, both force and acceleration are negative, causing the velocity to decrease and eventually reverse at the turning points where displacement is maximum.^[20] This phase opposition between displacement and acceleration—where acceleration peaks when displacement is zero and vice versa—characterizes the sinusoidal nature of SHM.^[21] Graphically, plotting acceleration against displacement yields a straight line passing through the origin with slope -\omega^2, confirming the linear relationship inherent to SHM.^[22] In real physical systems, this behavior requires the restoring force to be linearly dependent on displacement, a condition approximated in systems like springs or pendulums for small amplitudes but deviating at larger displacements due to nonlinear effects.^[3]

Kinetic and Potential Energy

In simple harmonic motion (SHM), the total mechanical energy E remains constant throughout the oscillation, assuming no dissipative forces are present, and is expressed as

E = \frac{1}{2} k A^2,

where k is the effective spring constant and A is the amplitude of the motion.^[23] This constancy arises from the conservative nature of the restoring force, ensuring that the system's energy is conserved as it oscillates.^[12] The total energy E is the sum of the kinetic energy K and the potential energy U, such that K + U = E at every point in the motion.^[23] The potential energy U, which represents the energy stored due to displacement from the equilibrium position, is given by

U = \frac{1}{2} k x^2,

where x is the instantaneous displacement.^[23] This potential energy reaches its minimum value of zero at the equilibrium position (x = 0) and its maximum value of E at the extremes of the motion (x = \pm A).^[23] The kinetic energy K, associated with the velocity of the oscillating mass, is expressed as K = \frac{1}{2} m v^2, where m is the mass and v is the instantaneous velocity.^[23] In the context of SHM, this can be rewritten using the relation for velocity derived from the general solution, yielding

K = \frac{1}{2} m \omega^2 (A^2 - x^2),

where \omega = \sqrt{k/m} is the angular frequency.^[23] Consequently, K achieves its maximum value of E at the equilibrium position (x = 0) and drops to zero at the displacement extremes (x = \pm A).^[23] As the system oscillates, energy continuously interconverts between its kinetic and potential forms while maintaining the total E invariant.^[23] At the points of maximum displacement, all energy is potential (K = 0, U = E); conversely, at equilibrium, all energy is kinetic (U = 0, K = E).^[23] This interconversion is driven by the restoring force, which performs negative work on the mass, thereby transferring energy from kinetic to potential (and vice versa) without net loss.^[23]

Examples

Mass-Spring System

The mass-spring system serves as the canonical example of simple harmonic motion (SHM), where a mass m is attached to one end of a spring with spring constant k, and the other end is fixed. In the horizontal configuration, the system oscillates along a frictionless surface, with displacement measured from the spring's unstretched equilibrium position. The restoring force follows Hooke's law, F = -kx, leading to the differential equation m \frac{d^2x}{dt^2} = -kx, which yields an angular frequency \omega = \sqrt{\frac{k}{m}} and period T = 2\pi \sqrt{\frac{m}{k}}.^[3]^[24] For the vertical configuration, gravity shifts the equilibrium position downward by \Delta x = \frac{mg}{k}, where the spring is stretched to balance the weight, but oscillations about this new equilibrium still exhibit SHM with the same \omega and T, independent of gravity. Displacement is measured from this stretched equilibrium, and the amplitude A represents the maximum deviation from it, ensuring the motion remains linear for small A. The effective restoring force remains F = -k(x - \Delta x), confirming the system's SHM behavior.^[21]^[3] This system is readily verified experimentally by first confirming Hooke's law through static measurements of force versus displacement to determine k, then observing the period's dependence on m and independence from A using timed oscillations. Such labs demonstrate the predicted T with high accuracy for moderate masses and small amplitudes.^[25]^[26] The analysis assumes an ideal spring: massless, perfectly elastic, and obeying Hooke's law linearly without damping or friction. In reality, the spring has negligible but non-zero mass, and for large amplitudes, deviations from linearity occur due to material nonlinearity, altering \omega and causing anharmonic motion.^[27]^[24]

Simple Pendulum

A simple pendulum consists of a mass m attached to a massless string or rod of length L, suspended from a fixed pivot and free to swing in a vertical plane under the influence of gravity.^[28] When displaced from its equilibrium position and released, the pendulum bob oscillates about the vertical.^[29] The restoring torque arises from the gravitational force component tangential to the arc of motion, given by \tau = -mg \sin \theta \, L, where \theta is the angular displacement from the vertical.^[28] For small angular displacements, the small-angle approximation \sin \theta \approx \theta (with \theta in radians) simplifies this to \tau \approx -mg \theta \, L.^[29] Applying Newton's second law for rotation, \tau = I \alpha with moment of inertia I = m L^2 and angular acceleration \alpha = d^2 \theta / dt^2, yields the differential equation

\frac{d^2 \theta}{dt^2} + \frac{g}{L} \theta = 0.

^[28] This is the standard form for simple harmonic motion, with angular frequency \omega = \sqrt{g/L} and period T = 2\pi \sqrt{L/g}.^[29] The period depends only on the length L and gravitational acceleration g, independent of mass m or amplitude under this approximation, illustrating the isochronous nature of small oscillations.^[28] The small-angle approximation holds well for initial displacements less than 15°, where \sin \theta and \theta differ by less than 1%, ensuring the period remains nearly constant.^[29] Beyond this range, the actual period increases slightly due to the nonlinearity of \sin \theta.^[28] Physically, the pendulum can be viewed as analogous to a mass-spring system, where gravity provides an effective spring constant k = mg / L, with the arc displacement s = L \theta serving as the linear extension.^[30] This gravitational restoring force mimics the linear proportionality of Hooke's law for small angles, enabling the harmonic behavior.^[30]

Projection of Uniform Circular Motion

Simple harmonic motion can be visualized as the projection of uniform circular motion onto a straight line. Imagine a particle moving at constant angular speed \omega around a circle of radius A, which represents the amplitude of the oscillation. If the particle's position in the plane is described by Cartesian coordinates, the x-component of its displacement from the center is x(t) = A \cos(\omega t + \phi), where \phi is the initial phase angle. This expression directly matches the general solution for the displacement in simple harmonic motion.^[31] The velocity and acceleration of this projected motion further confirm the harmonic nature. Differentiating the position gives the x-component of velocity:

v_x(t) = -A \omega \sin(\omega t + \phi).

The acceleration is then

a_x(t) = -\omega^2 A \cos(\omega t + \phi) = -\omega^2 x(t),

which satisfies the differential equation for simple harmonic motion, \frac{d^2 x}{dt^2} + \omega^2 x = 0. This kinematic derivation highlights how the centripetal acceleration in circular motion projects to a restoring acceleration proportional to displacement.^[32] The circular path, often called the reference circle, provides a geometric tool for representing the phase of the oscillation, with the angular position \theta = \omega t + \phi corresponding to the phase. The y-component of the motion, y(t) = A \sin(\omega t + \phi), executes simple harmonic motion that is shifted by 90 degrees relative to the x-component, a relationship known as quadrature. This phase difference arises naturally from the orthogonal projections and underscores the vectorial composition in two dimensions.^[33] This projection model finds applications in understanding more complex oscillatory systems. For instance, it facilitates the analysis of coupled oscillators by visualizing normal modes as synchronized circular projections, and it explains the formation of Lissajous figures, which emerge when two perpendicular simple harmonic motions with commensurate frequencies are combined.^[34] Notably, this analogy is purely mathematical and kinematic, without implying any physical force driving the circular motion; it serves solely to illustrate the sinusoidal time dependence and phase characteristics of simple harmonic motion.^[6]

Mechanical Linkages

Mechanical linkages are engineered mechanisms that generate or approximate simple harmonic motion (SHM) through the conversion of rotary to linear motion or vice versa, commonly employed in machinery to achieve precise oscillatory paths.^[35] The Scotch yoke mechanism exemplifies an ideal realization of SHM in a mechanical system, where a rotating crank pin slides within a slotted yoke, driving the yoke in pure linear oscillation. In this setup, the displacement of the yoke from its equilibrium position is described by the equation x = r \cos(\theta), with \theta = \omega t, where r is the crank radius, \omega is the angular frequency, and t is time; this directly mirrors the projection of uniform circular motion onto a diameter.^[36]^[37] Unlike more complex linkages, the Scotch yoke eliminates side loads on the sliding element in ideal frictionless conditions, ensuring the output follows exact sinusoidal motion.^[38] The piston-crank mechanism, also known as the slider-crank linkage, approximates SHM but deviates for larger angles due to the connecting rod's influence. For small crank angles relative to the connecting rod length, the piston's motion closely follows SHM, with displacement roughly x \approx r (1 - \cos(\theta)) near the extremes, making it suitable for applications like internal combustion engines where exact harmonicity is not critical.^[35] This approximation holds because the geometry reduces to a near-projection of circular motion when the rod is much longer than the crank, minimizing nonlinear effects.^[39] Other linkages, such as the double slider crank or elliptic trammel, generate SHM along specific paths by constraining two sliders in perpendicular grooves connected by a link with a marked point. In the elliptic trammel, the point traces an ellipse, but its projections onto the groove axes exhibit pure SHM with amplitude equal to the groove spacing, providing a practical tool for demonstrating orthogonal harmonic components.^[40] These mechanisms offer key advantages in converting rotary to linear motion exactly in ideal cases like the Scotch yoke, avoiding the approximations and secondary forces inherent in slider-crank designs, which enhances efficiency in low-friction environments.^[37] Historically, such linkages have been integral to steam engines and animation devices since the 18th century, with the Scotch yoke notably applied in early steam pumps to produce reliable reciprocating action for piston drives.^[41]

Extensions and Applications

Phasor Representation

In simple harmonic motion (SHM), the phasor representation models the oscillatory displacement as the projection of a rotating vector, offering a compact vector-based approach to analyze position, velocity, and acceleration. A phasor is a vector of fixed length equal to the amplitude A, rotating counterclockwise in the complex plane at constant angular speed \omega. The x-component (real part) of this vector at any time t yields the displacement x(t) = A \cos(\omega t + \phi), where \phi is the phase constant determining the initial position. This geometric interpretation links the linear SHM to uniform circular motion, with the phasor's tip tracing a circle of radius A. To represent associated quantities, additional phasors are introduced with specific phase shifts relative to the displacement phasor. The velocity phasor, which has magnitude A\omega, leads the displacement phasor by $90^\circ (or \pi/2 radians), corresponding to v(t) = -A\omega \sin(\omega t + \phi). The acceleration phasor, with magnitude A\omega^2, leads by $180^\circ (or \pi radians), aligning opposite to the displacement and matching a(t) = -A\omega^2 \cos(\omega t + \phi). Phasor diagrams visualize these relationships by drawing the vectors tail-to-origin: the displacement along the horizontal axis, velocity vertical and ahead, and acceleration horizontal but reversed, all rotating synchronously at \omega. The complex exponential notation formalizes this representation, expressing displacement as

x(t) = \operatorname{Re} \left[ A e^{i(\omega t + \phi)} \right],

where the complex amplitude A e^{i\phi} encodes magnitude and initial phase. Velocity follows as i\omega times the displacement phasor, yielding v(t) = \operatorname{Re} \left[ i\omega A e^{i(\omega t + \phi)} \right], while acceleration is -\omega^2 times it, simplifying time derivatives to algebraic multiplications by i\omega or -\omega^2. This method excels in handling superposition, where multiple SHMs of identical frequency add vectorially via phasor summation to find the resultant amplitude and phase directly. It also underpins Fourier analysis by decomposing complex periodic signals into harmonic phasors, enabling efficient computation of oscillatory systems.

Relation to Waves

Simple harmonic motion (SHM) forms the foundational mechanism underlying harmonic waves, which can be either transverse or longitudinal. In a harmonic wave, each particle of the medium executes SHM about its equilibrium position, with the disturbance propagating through the medium while the particles themselves do not travel net distances. This oscillatory behavior of individual particles gives rise to the wave's periodic nature, where the displacement varies sinusoidally with time at any fixed point.^[42]/12%3A_Waves_in_One_Dimension/12.01%3A_Traveling_Waves) The mathematical description of a traveling harmonic wave is captured by the one-dimensional wave equation,

\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2},

where y(x, t) is the displacement, v is the constant wave speed, x is position, and t is time. A general solution for a wave propagating in the positive x-direction is y(x, t) = A \cos(kx - \omega t + \phi), with A as the amplitude, k as the wave number, \omega as the angular frequency, and \phi as the phase constant. At a fixed position x, the equation simplifies to \frac{\partial^2 y}{\partial t^2} + \omega^2 y = 0, which is the standard differential equation for SHM, confirming the harmonic oscillation of each particle with frequency \omega / 2\pi. The wavelength \lambda relates to the wave number by \lambda = 2\pi / k, and the wave speed is given by v = \omega / k = f \lambda, where f is the frequency.^[42]/12%3A_Waves_in_One_Dimension/12.01%3A_Traveling_Waves) Standing waves emerge from the superposition of two identical harmonic waves traveling in opposite directions, such as along a string fixed at both ends. The resulting pattern features stationary nodes, where displacement is always zero, and antinodes, where displacement reaches maximum amplitude, with the wave profile oscillating in place rather than propagating. Phasor addition can represent this interference constructively at antinodes and destructively at nodes.^[43] These principles apply broadly in natural phenomena, including sound waves, where air molecules undergo longitudinal SHM to propagate pressure variations, and light, modeled as electromagnetic waves in which the electric and magnetic fields oscillate harmonically, effectively behaving as coupled SHM systems across vast collections of oscillators.^[44]

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[PDF] • Doppler effect • Standing waves from superposition - Physics - UMD
Standing Waves: graphical (I). • Two sine waves (same f, A, ) traveling in opposite directions: superposition is wave, but not moving λ. Page 10. • Nodes ...
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17.1 Sound Waves - University Physics Volume 1 | OpenStax
Sep 19, 2016 · The air molecules oscillate in simple harmonic motion about their equilibrium positions, as shown in part (b). Note that sound waves in air are ...<|control11|><|separator|>