simple harmonic motion (physics)

Simple harmonic motion

In physics, simple harmonic motion (SHM) is a periodic motion that is neither driven nor damped. An object in simple harmonic motion experiences a net force which obeys Hooke's law; that is, the force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction of the displacement.
A simple harmonic oscillator is a system which undergoes simple harmonic motion. The oscillator oscillates about an equilibrium position (or mean position) between two extreme positions of maximum displacement in a periodic manner. Mathematically, the motion of the oscillator can be described by means of a sinusoidal function such that the displacement x from the equilibrium position is given by:

$x(t) = A\cos\left(\omega t + \varphi\right),$

where A is the amplitude, ω is the angular frequency such that ω = 2πf where f is the frequency in units of hertz, and φ is the phase which is the elapsed fraction of wave cycle in radians.
The angular frequency of the motion is determined by the intrinsic properties of the system (often the mass of the object and the force constant), while the amplitude and phase are determined by the initial conditions (displacement and velocity) of the system. The kinetic and potential energies of the system are in turn determined by both intrinsic properties and initial conditions.

Simple harmonic motion. In this moving graph, the vertical axis represents the coordinate of the particle (x in the equation), and the horizontal axis represents time (t).

Simple harmonic motion can serve as a mathematical model of a variety of motions, such as the oscillation of a spring. Additionally, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum and molecular vibration.
Simple harmonic motion provides the basis of the characterization of more complicated motions through the techniques of Fourier analysis.

Introduction

Simple harmonic motion shown both in real space and phase space. The orbit is periodic. (Here the velocity and position axes have been reversed from the standard convention in order to align the two diagrams)

An idealized mass-spring system is typically a simple harmonic oscillator. A mass is attached to one end of the spring, and the other end of the spring is connected to a rigid support such as a wall. If the system is left at rest at the equilibrium position then there is no net force acting on the mass. However, if the mass is displaced from the equilibrium position, a restoring elastic force which obeys Hooke's law is exerted by the spring.
Mathematically, the restoring force F is given by

$F = - kx, \,$

where F is the restoring elastic force exerted by the spring (in SI units: N), k is the spring constant (N·m⁻¹), and x is the displacement from the equilibrium position (in m).
Every simple harmonic oscillator exhibits a characteristic feature.

When the system is displaced from equilibrium position, a restoring force which obeys Hooke's law exists and tend to restore the system to its equilibrium.

Once the mass is displaced from its equilibrium position, it experiences a net restoring force. Hence, it accelerates and starts going back to the equilibrium position. When the mass moves closer to the equilibrium position, the restoring force decreases. At the equilibrium position, the net restoring force vanishes. However, at x = 0, the momentum of the mass does not vanish due to the impulse of the restoring force that has acted on it.
Therefore, the mass shoots past the equilibrium position compressing the spring. A net restoring force then tends to slow it down, until its velocity vanishes, whereby it will attempt to reach equilibrium position again. As long as the system has no energy loss, the mass will continue to oscillate. Thus, the simple harmonic motion is known as one of the periodic motions.

Dynamics of simple harmonic motion

For one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential equation with constant coefficients, could be obtained by means of Newton's second law and Hooke's law.

$F_{net} = m\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = -kx,$

where m is the inertial mass of the oscillating body, x is its displacement from the equilibrium (or mean) position, and k is the spring constant constant.
Solving the differential equation above, a solution which is a sinusoidal function is obtained.

$x(t) = c_1\cos\left(\omega t\right) + c_2\sin\left(\omega t\right) = A\cos\left(\omega t + \varphi\right),$

where

$\omega = \sqrt{\frac{k}{m}},$

$A = \sqrt{{c_1}^2 + {c_2}^2},$

$\tan \varphi = \left(\frac{c_2}{c_1}\right),$

In the solution, c₁ and c₂ are two constants determined by the initial conditions, and the origin is set to be the equilibrium position.^[A] Each of these constants carries a physical meaning of the motion: A is the amplitude (maximum displacement from the equilibrium position) , ω = 2πf is the angular frequency, and φ is the phase.^[B]

Position, velocity and acceleration of a harmonic oscillator

Using the techniques of differential calculus, the velocity and acceleration as a function of time can be found:

$v(t) = \frac{\mathrm{d} x}{\mathrm{d} t} = - A\omega \sin(\omega t+\varphi),$

$a(t) = \frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = - A \omega^2 \cos( \omega t+\varphi).$

Position, velocity and acceleration of a SHM as phasors
Acceleration can also be expressed as a function of displacement:

$a(x) = -\omega^2 x.\!$

Then since ω = 2πf,

$f = \frac{1}{2\pi}\sqrt{\frac{k}{m}},$

and since T = 1/f where T is the time period,

$T = 2\pi\sqrt{\frac{m}{k}}.$

These equations demonstrate that the simple harmonic motion is isochronous (the period and frequency are independent of the amplitude and the initial phase of the motion).

Energy of simple harmonic motion

$K(t) = \frac{1}{2} mv(t)^2 = \frac{1}{2}m\omega^2A^2\sin^2(\omega t + \varphi) = \frac{1}{2}kA^2 \sin^2(\omega t + \varphi),$ The kinetic energy K of the system at time t is

$U(t) = \frac{1}{2} k x(t)^2 = \frac{1}{2} k A^2 \cos^2(\omega t + \varphi).$ and the potential energy is

The total mechanical energy of the system therefore examples
has $E = K + U = \frac{1}{2} k A^2.$ the constant value

An undamped spring-mass system undergoes simple harmonic motion.

The following physical systems are some examples of simple harmonic oscillator.

Mass on a spring

$T= 2 \pi \sqrt{\frac{m}{k}}$ A mass m attached to a spring of spring constant k exhibits simple harmonic motion in space. The equation

shows that the period of oscillation is independent of both the amplitude and gravitational acceleration.

Uniform circular motion

Simple harmonic motion can in some cases be considered to be the one-dimensional projection of uniform circular motion. If an object moves with angular velocity ω around a circle of radius r centered at the origin of the x-y plane, then its motion along each coordinate is simple harmonic motion with amplitude r and angular frequency ω.

Mass on a simple pendulum

The motion of an undamped Pendulum approximates to simple harmonic motion if the amplitude is very small relative to the length of the rod.

$T = 2 \pi \sqrt{\frac{\ell}{g}}$ In the small-angle approximation, the motion of a simple pendulum is approximated by simple harmonic motion. The period of a mass attached to a string of length ℓ with gravitational acceleration g is given by

This shows that the period of oscillation is independent of the amplitude and mass of the pendulum but not the acceleration due to gravity (g), therefore a pendulum of the same length on the Moon would swing more slowly due to the Moon's lower gravitational acceleration.
$m g \ell \sin(\theta)=I \alpha,$ This approximation is accurate only in small angles because of the expression for angular acceleration being proportional to the sine of position:
$m g \ell \theta=I \alpha$ where I is the moment of inertia; in this case I = m ℓ ². When θ is small, sin θ ≈ θ and therefore the expression becomes

which makes angular acceleration directly proportional to θ, satisfying the definition of simple harmonic motion.

Simple harmonic motion is typified by the motion of a mass on a spring when it

is subject to the linear elastic restoring force given by Hooke's Law. The motion

is sinusoidal in time and demonstrates a single resonant frequency.

The motion equation for simple harmonic motion contains a complete

description of the motion, and other parameters of the motion can be calculated

from it.

The velocity and acceleration are given by

The total energy for an undamped oscillator is the sum of its

kinetic energy and potential energy, which is constant at Simple Harmonic Motion

The motion equations for simple harmonic motion provide for calculating any

parameter of the motion if the others are known.

If the period is T = s

then the frequency is f = Hz and the angular frequency =

rad/s.

The motion is described by

Displacement = Amplitude x sin (angular frequency x

time)

Any of the parameters in the motion equation can be calculated by clicking on

the active word in the motion relationship above. Default values will be

entered for any missing data, but those values may be changed and the

calculation repeated. The angular frequency calculation assumes that the

motion is in its first period and therefore calculates the smallest value of

y = A x sin (

x t )

m = m x sin ( rad/s x s )

Any of the parameters in the motion equation can be calculated by clicking on

the active word in the motion relationship above. Default values will be

entered for any missing data, but those values may be changed and the

calculation repeated. The angular frequency calculation assumes that the

motion is in its first period and therefore calculates the smallest value of angular frequency which will match the other parameters. The time

calculation calculates the first time the motion reaches the specified

displacement, i.e., the time during the first period.

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