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Simple harmonic motion
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Simple harmonic motion (SHM) is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. The motion is periodic - as it repeats itself at standard intervals in a specific manner - and sinusoidal, with constant amplitude; the acceleration of a body executing SHM is directly proportional to the displacement of the body from the equilibrium position and is always directed towards the equilibrium position.
The motion is characterized by its amplitude (which is always positive), its period, the time for a single oscillation, its frequency, the reciprocal of the period (i.e.
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Simple harmonic motion (SHM) is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. The motion is periodic - as it repeats itself at standard intervals in a specific manner - and sinusoidal, with constant amplitude; the acceleration of a body executing SHM is directly proportional to the displacement of the body from the equilibrium position and is always directed towards the equilibrium position.
The motion is characterized by its amplitude (which is always positive), its period, the time for a single oscillation, its frequency, the reciprocal of the period (i.e. the number of cycles per unit time), and its phase, which determines the starting point on the sine wave. The period and frequency are constants determined by the overall system, while the amplitude and phase are determined by the initial conditions (position and velocity) of that system.
A general equation describing simple harmonic motion is , where x is the displacement, A is the amplitude of oscillation, f is the frequency, t is the elapsed time, and is the phase of oscillation. If there is no displacement at time t = 0, the phase .
Simple harmonic motion can serve as a mathematical model of a variety of motions and provides the basis of the characterization of more complicated motions through the techniques of Fourier analysis.
Introduction
. The orbit is periodic. (Here the velocity and position axes have been reversed from the standard convention in order to align the two diagrams)]]
A typical example of a system that undergoes simple harmonic motion is a spring-mass system, which is a mass attached to a spring.
If the spring is unstretched, there is no net force on the mass - in other words, the system is in equilibrium. However, if the mass is displaced from equilibrium, the spring will exert a restoring force, which is a force that tends to restore it to the equilibrium position. In the case of the spring-mass system, this force is the elastic force, which is given by Hooke's Law, , where F is the restoring force, x is the displacement, and k is the spring constant.
Any system that undergoes simple harmonic motion exhibits two key features.
- When the system is displaced from equilibrium there must exist a restoring force that tends to restore it to equilibrium.
- The restoring force must be proportional to the displacement, or approximately so.
The spring-mass system satisfies both.
Once the mass is displaced it experiences a restoring force, accelerating it, causing it to start going back to the equilibrium position. As it gets closer to equilibrium the restoring force decreases; at the equilibrium position the restoring force is 0. However, at x=0, the mass has some momentum due to the impulse of the force that has acted on it; this causes the mass to shoot past the equilibrium position, in this case, compressing the spring. The restoring force then tends to slow it down, until the velocity reaches 0, whereby it will attempt to reach equilibrium position again.
As long as the system does not lose energy, the mass will continue to oscillate like so; thus, the motion is termed periodic motion. Further analysis will show that in the case of the spring-mass system the motion is simple harmonic.
Mathematics
Simple harmonic motion is defined by the differential equation
where "k" is a positive constant, "m" is the mass of the body, and "x" is its displacement from the mean position.
It can be shown, by differentiation, exactly how the acceleration varies with time. Using the angular frequency , defined as
the displacement is given by the function
Differentiating once gives an expression for the velocity at any time.
And once again to get the acceleration at a given time.
These results can of course be simplified, giving us an expression for acceleration in terms of displacement.
which is equivalent to:
When and if total energy is constant and kinetic, the formula applies for simple harmonic motion, where E is considered the total energy while all energy is in its kinetic form and A represents the mean displacement of the spring from its rest position in MKS units.
Examples
Simple harmonic motion is exhibited in a variety of simple physical systems and below are some examples.
Mass on a spring
A mass M attached to a spring of spring constant k exhibits simple harmonic motion in space with
Alternately, if the other factors are known and the period is to be found, this equation can be used:
This shows that the period of oscillation is independent of both the amplitude and gravitational acceleration.
The total energy is constant, and given by where E is the total energy.
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 frequency around a circle of radius centered at the origin of the x-y plane, then its motion along the x and the y coordinates is simple harmonic with amplitude and angular speed .
Mass on a simple pendulum
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 with gravitational acceleration is given by
This shows that the period of oscillation is independent of both the amplitude and the acceleration due to gravity.
This approximation is accurate only in small angles because of the expression for angular acceleration being proportional to the sine of position:
where I is the moment of inertia; in this case . When is small, and therefore the expression becomes
which makes angular acceleration directly proportional to , satisfying the definition of Simple Harmonic Motion.
For a solution not relying on a small-angle approximation, see pendulum (mathematics).
Useful Formulas
Given mass attached to a spring/pendulum with amplitude with acceleration :
Where:
is the spring constant.
is the mass (usually in kilograms)
is the acceleration.
is the amplitude.
is the frequency (usually in hertz).
or is the time period of the spring or pendulum.
is the acceleration due to gravity (On Earth at sea level: 9.81 m/sē).
is the length of the pendulum.
is the total energy.
See also
External links
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