Characteristics of Simple Harmonic Motion

Characteristics of SHM

• It is a motion along a straight line.

• The body moves back and forth with respect to a mean position.

• The body returns to a given point in the path with the same velocity after regular intervals of time.

• The acceleration is directly proportional to its displacement from the mean position and is always directed towards it.

• The force acting on the body always tries to bring it back to its equilibrium position.

·         The maximum displacements on either side of the equilibrium position are equal.

What Is Simple Harmonic Motion?

Simple harmonic motion is any motion where a restoring force is applied that is proportional to the displacement and in the opposite direction of that displacement. Or in other words, the more you pull it one way, the more it wants to return to the middle. The classic example of this is a mass on a spring, because the more the mass stretches it, the more it feels a tug back towards the middle. A mass on a spring can be vertical, in which case gravity is involved, or horizontal on a smooth tabletop.

If you imagine pulling a mass on a spring and then letting go, it will bounce back and forth around an equilibrium position in the middle. Like with all simple harmonic motion, the velocity will be greatest in the middle, whereas the restoring force (and therefore acceleration) will be greatest at the outside edges (at the maximum displacement). Another example of simple harmonic motion is a pendulum, though only if it swings at small angles.

Equations

There are many equations to describe simple harmonic motion. The first we're going to look at, below, tells us that the time period of an oscillating spring, T, measured in seconds, is equal to 2pi times the square-root of m over k, where m is the mass of the object connected to the spring measured in kilograms, and k is the spring constant (a measure of elasticity) of the spring. The time period is the time it takes for an object to complete one full cycle of its periodic motion, such as the time it takes a pendulum to make one full back-and-forth swing.

Equation for Simple Harmonic Motion

All simple harmonic motion is sinusoidal. This can best be illustrated visually. As you can see from our animation (please see the video at 01:34), a mass on a spring undergoing simple harmonic motion slows down at the very top and bottom, before gradually increasing speed again as it approaches the center. It spends more time at the top and bottom than it does in the middle. Mathematically, any motion that has a restoring force proportional to the displacement from the equilibrium position will vary in this way.

Because of that, the main equation shown below is shaped like a sine curve. It says that the displacement is equal to the amplitude of the variation, A, otherwise known as the maximum displacement, multiplied by sine omega-t, where omega is the angular frequency of the variation, and t is the time. This displacement can be in the x-direction or the y-direction, depending on the situation. A vertical mass on a spring varies in the y-direction sinusoidally. A horizontal mass on a spring varies in the x-direction sinusoidally. A pendulum has such a variation in both directions.

Main Equation for Harmonic Motion

This equation has a sine in it, and a sine graph starts at zero. Using this equation is like starting your mathematical stopwatch in the middle of a pendulum swing: t = 0 is in the center of the oscillation. If, on the other hand, you replace sine with a cosine, then the equation is still correct; you're just starting to measure time at the maximum displacement instead.

But we also need to define angular frequency. Angular frequency is the number of radians of the oscillation that are completed each second. A full 360 degrees is 2pi radians, and that represents one complete oscillation: from the middle, to a fully stretched spring, back to the middle, to a fully compressed spring and then back to the middle again. You can convert angular frequency to regular frequency by dividing it by 2pi. Regular frequency, f, just tells you the number of full cycles per second, measured in

In mechanics and physics, simple harmonic motion is a type of periodic motion or oscillation motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.

Simple harmonic motion can serve as a mathematical model for a variety of motions, such as the oscillation of a spring. In addition, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum as well as molecular vibration. 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. For simple harmonic motion to be an accurate model for a pendulum, the net force on the object at the end of the pendulum must be proportional to the displacement. This is a good approximation when the angle of the swing is small.

Simple harmonic motion provides a basis for the characterization of more complicated motions through the techniques of Fourier analysis.

The motion of a particle moving along a straight line with an acceleration which is always towards a fixed point on the line and whose magnitude is proportional to the distance from the fixed point is called simple harmonic motion [SHM].[citation needed]

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 to align the two diagrams)

In the diagram, a simple harmonic oscillator, consisting of a weight attached to one end of a spring, is shown. 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, the spring exerts a restoring elastic force that obeys Hooke's law.

Mathematically, the restoring force F is given by

F = − k x , {\displaystyle \mathbf {F} =-k\mathbf {x} ,}

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 (m).

For any simple mechanical harmonic oscillator:

• When the system is displaced from its equilibrium position, a restoring force that obeys Hooke's law tends to restore the system to equilibrium.

Once the mass is displaced from its equilibrium position, it experiences a net restoring force. As a result, 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 mass has momentum because of the acceleration that the restoring force has imparted. Therefore, the mass continues past the equilibrium position, compressing the spring. A net restoring force then slows it down until its velocity reaches zero, whereupon it is accelerated back to the equilibrium position again.

As long as the system has no energy loss, the mass continues to oscillate. Thus simple harmonic motion is a type of periodic motion.

Dynamics

In Newtonian mechanics, for one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential equation with constant coefficients, can be obtained by means of Newton's second law and Hooke's law for a mass on a spring.

F n e t = m d 2 x d t 2 = − k x , {\displaystyle F_{\mathrm {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 a constant (the spring constant for a mass on a spring). (Note that in reality this is in fact an approximation, only valid for speeds that are small compared to the speed of light.)

Therefore,

d 2 x d t 2 = − k m x , {\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-{\frac {k}{m}}x,}

Solving the differential equation above produces a solution that is a sinusoidal function.

x ( t ) = x 0 cos ⁡ ( ω t ) + v 0 ω sin ⁡ ( ω t ) {\displaystyle x(t)=x_{0}\cos \left(\omega t\right)+{\frac {v_{0}}{\omega }}\sin \left(\omega t\right)}

This equation can be written in the form:

x ( t ) = A cos ⁡ ( ω t − φ ) , {\displaystyle x(t)=A\cos \left(\omega t-\varphi \right),}

where

ω = k m , A = c 1 2 + c 2 2 , tan ⁡ φ = c 2 c 1 , {\displaystyle \omega ={\sqrt {\frac {k}{m}}},\qquad A={\sqrt {{c_{1}}^{2}+{c_{2}}^{2}}},\qquad \tan \varphi ={\frac {c_{2}}{c_{1}}},}

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]

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

v ( t ) = d x d t = − A ω sin ⁡ ( ω t − φ ) , {\displaystyle v(t)={\frac {\mathrm {d} x}{\mathrm {d} t}}=-A\omega \sin(\omega t-\varphi ),}

Speed:

ω A 2 − x 2 {\displaystyle {\omega }{\sqrt {A^{2}-x^{2}}}}

Maximum speed: ωA (at equilibrium point)

a ( t ) = d 2 x d t 2 = − A ω 2 cos ⁡ ( ω t − φ ) . {\displaystyle a(t)={\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-A\omega ^{2}\cos(\omega t-\varphi ).}

Maximum acceleration: Aω2 (at extreme points)

By definition, if a mass m is under SHM its acceleration is directly proportional to displacement.

a ( x ) = − ω 2 x . {\displaystyle a(x)=-\omega ^{2}x.}

where

ω 2 = k m {\displaystyle \omega ^{2}={\frac {k}{m}}}

Since ω = 2πf,

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

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

T = 2 π m k . {\displaystyle 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

Substituting ω2 with k/m, the kinetic energy K of the system at time t is

K ( t ) = 1 2 m v 2 ( t ) = 1 2 m ω 2 A 2 sin 2 ⁡ ( ω t + φ ) = 1 2 k A 2 sin 2 ⁡ ( ω t + φ ) , {\displaystyle K(t)={\tfrac {1}{2}}mv^{2}(t)={\tfrac {1}{2}}m\omega ^{2}A^{2}\sin ^{2}(\omega t+\varphi )={\tfrac {1}{2}}kA^{2}\sin ^{2}(\omega t+\varphi ),}

and the potential energy is

U ( t ) = 1 2 k x 2 ( t ) = 1 2 k A 2 cos 2 ⁡ ( ω t + φ ) . {\displaystyle U(t)={\tfrac {1}{2}}kx^{2}(t)={\tfrac {1}{2}}kA^{2}\cos ^{2}(\omega t+\varphi ).}

In the absence of friction and other energy loss, the total mechanical energy has a constant value

E = K + U = 1 2 k A 2 . {\displaystyle E=K+U={\tfrac {1}{2}}kA^{2}.}

Examples

An undamped spring–mass system undergoes simple harmonic motion.

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

Mass on a spring

A mass m attached to a spring of spring constant k exhibits simple harmonic motion in closed space. The equation for describing the period

T = 2 π m k {\displaystyle T=2\pi {\sqrt {\frac {m}{k}}}}

shows the period of oscillation is independent of both the amplitude and gravitational acceleration. The above equation is also valid in the case when an additional constant force is being applied on the mass, i.e. the additional constant force cannot change the period of oscillation.

Uniform circular motion

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

Mass of a simple pendulum

The motion of an undamped pendulum approximates to simple harmonic motion if the angle of oscillation is small.

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 pendulum of length l with gravitational acceleration g {\displaystyle g} is given by

T = 2 π l g {\displaystyle T=2\pi {\sqrt {\frac {l}{g}}}}

This shows that the period of oscillation is independent of the amplitude and mass of the pendulum but not of the acceleration due to gravity, g {\displaystyle g} , therefore a pendulum of the same length on the Moon would swing more slowly due to the Moon's lower gravitational field strength. Because the value of g {\displaystyle g} varies slightly over the surface of the earth, the time period will vary slightly from place to place and will also vary with height above sea level.

This approximation is accurate only for small angles because of the expression for angular acceleration α being proportional to the sine of the displacement angle:

− m g l sin ⁡ θ = I α , {\displaystyle -mgl\sin \theta =I\alpha ,}

where I is the moment of inertia. When θ is small, sin θ ≈ θ and therefore the expression becomes

− m g l θ = I α {\displaystyle -mgl\theta =I\alpha }

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

Simple harmonic motion, in physics, repetitive movement back and forth through an equilibrium, or central, position, so that the maximum displacement on one side of this position is equal to the maximum displacement on the other side. The time interval of each complete vibration is the same, and the force responsible for the motion is always directed toward the equilibrium position and is directly proportional to the distance from it.

Many physical systems exhibit simple harmonic motion (assuming no energy loss): an oscillating pendulum, the electrons in a wire carrying alternating current, the vibrating particles of the medium in a sound wave, and other assemblages involving relatively small oscillations about a position of stable equilibrium.

A specific example of a simple harmonic oscillator is the vibration of a mass attached to a vertical spring, the other end of which is fixed in a ceiling. At the maximum displacement −x, the spring is under its greatest tension, which forces the mass upward. At the maximum displacement +x, the spring reaches its greatest compression, which forces the mass back downward again. At either position of maximum displacement, the force is greatest and is directed toward the equilibrium position, the velocity (v) of the mass is zero, its acceleration is at a maximum, and the mass changes direction. At the equilibrium position, the velocity is at its maximum and the acceleration (a) has fallen to zero. Simple harmonic motion is characterized by this changing acceleration that always is directed toward the equilibrium position and is proportional to the displacement from the equilibrium position. Furthermore, the interval of time for each complete vibration is constant and does not depend on the size of the maximum displacement. In some form, therefore, simple harmonic motion is at the heart of timekeeping.

The motion is called harmonic because musical instruments make such vibrations that in turn cause corresponding sound waves in air. Musical sounds are actually a combination of many simple harmonic waves corresponding to the many ways in which the vibrating parts of a musical instrument oscillate in sets of superimposed simple harmonic motions, the frequencies of which are multiples of a lowest fundamental frequency. In fact, any regularly repetitive motion and any wave, no matter how complicated its form, can be treated as the sum of a series of simple harmonic motions or waves, a discovery first published in 1822 by the French mathematician Baron Jean-Baptiste-Joseph Fourier.

Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a particular type of oscillation. It is useful because its time period stays the same even when its amplitude changes. We'll come to the full definition later!

Lets think about a simple example of shm to work out the relationship between displacement, velocity and acceleration: Now remember that displacement, velocity and acceleration are all vectors, and as a result, direction is important. Let's choose anything in the up-wards direction to be positive, anything downwards to be negative. (If you decide to do the opposite, it doesn't matter - just stick to your choice.)

If we set this system oscillating by lifting the mass and letting it go, then the system starts with:

Maximum positive displacement (because it's above the middle).

Zero velocity (it's not moving at the first instant).

Maximum negative acceleration (because it is about to start moving down).

The interaction below shows how velocity and acceleration change in simple harmonic motion. It shows the relationship between velocity and acceleration. Click "next" to see each part of the motion...

The displacement, velocity and acceleration of the mass are related as shown above. To draw these, think about what the object is doing at each point as it oscillates from the start position described above.

As it passes through the equilibrium position on the way down it's at maximum speed down (negative), its displacement is zero and because the spring is at its equilibrium position, there is no resultant force on the mass so it is not accelerating.

At the bottom, the mass stops momentarily as it changes direction, so velocity is zero. The displacement is a maximum in the negative direction, so the acceleration is a maximum in a positive direction as the spring tries to shorten again.

The important point to note is the phase difference between these three variables...

1. The velocity, v, is zero where there are stationary points at the peaks and troughs of the displacement graph and the velocity is a maximum when the displacement is zero. (Don't forget the gradient of the displacement graph will equal velocity.)

2. The displacement and acceleration graphs are 180 degrees out of phase and therefore look like a mirror image of each other in the time axis. (Don't forget the gradient of the velocity graph will equal acceleration.)

Definition of Simple Harmonic Motion:

All of the above leads us to the formal definition of shm:

A body is undergoing SHMwhen the acceleration on the body is proportional to its displacement, but acts in the opposite direction.

Acceleration a displacement

a α - s

It's also important to note that for SHM, the time period of the oscillations is constant and doesn't change even if the amplitude is changing.

There are two common examples of simple harmonic motion:
Where m = mass (kg) and k = spring constant (Nm-1)	Where L = length of pendulum (m) g = acceleration due to gravity (ms-2)

SHM is used to explain the behaviour of atoms in a lattice, which oscillate like masses on springs.