Difference between revisions of "Simple harmonic motion"
m (Fixed red link) |
m (Changed some equations so they aren't inline and fixed a red link) |
||
| Line 1: | Line 1: | ||
'''Simple harmonic motion''' is any motion caused by a [[force]] that acts to restore a body to [[equilibrium]] and has a magnitude proportional to the [[distance]] of the body from its equilibrium position. Mathematically, the [[acceleration]] of the body, <math>\ddot x</math>, may be related to the [[displacement]] of the body by: | '''Simple harmonic motion''' is any motion caused by a [[force]] that acts to restore a body to [[equilibrium]] and has a magnitude proportional to the [[distance]] of the body from its equilibrium position. Mathematically, the [[acceleration]] of the body, <math>\ddot x</math>, may be related to the [[displacement]] of the body by: | ||
| + | |||
<math> | <math> | ||
\ddot x(t) = -kx | \ddot x(t) = -kx | ||
| Line 9: | Line 10: | ||
==Mathematics== | ==Mathematics== | ||
Simple harmonic motion is described by a second order [[differential equation]]: | Simple harmonic motion is described by a second order [[differential equation]]: | ||
| + | |||
<math> | <math> | ||
\ddot x(t) = -kx | \ddot x(t) = -kx | ||
</math> | </math> | ||
| + | |||
The solution of this equation is: | The solution of this equation is: | ||
| + | |||
<math> | <math> | ||
x(t) = A \sin{\omega t} + B \cos{\omega t} | x(t) = A \sin{\omega t} + B \cos{\omega t} | ||
</math> | </math> | ||
| + | |||
where <math>\omega</math> is <math>\sqrt{k}</math> and called the angular frequency. <math>A</math> and <math>B</math> are arbitrary constants that can be determined from the initial conditions of the problem. Since the solution involves only sines and cosines which oscillate, the solution itself will oscillate. Simple harmonic oscillators also have the property that the frequency of the oscillation is independent to its amplitude. | where <math>\omega</math> is <math>\sqrt{k}</math> and called the angular frequency. <math>A</math> and <math>B</math> are arbitrary constants that can be determined from the initial conditions of the problem. Since the solution involves only sines and cosines which oscillate, the solution itself will oscillate. Simple harmonic oscillators also have the property that the frequency of the oscillation is independent to its amplitude. | ||
| Line 24: | Line 29: | ||
{{Math-h}} | {{Math-h}} | ||
Suppose there is a potential, <math>V(x)</math>, that has a equilibrium point (minima) at <math>x_0</math>. <math>V(x)</math> can be approximated around the minima using a [[Taylor series]] as: | Suppose there is a potential, <math>V(x)</math>, that has a equilibrium point (minima) at <math>x_0</math>. <math>V(x)</math> can be approximated around the minima using a [[Taylor series]] as: | ||
| + | |||
<math> | <math> | ||
V(x) = V(x_0) + V'(x_0)(x-x_0) + \frac{V''(x_0)(x-x_0)^2}{2!} + \frac{V^{(3)}(x_0)(x-x_0)^3}{3!}+... | V(x) = V(x_0) + V'(x_0)(x-x_0) + \frac{V''(x_0)(x-x_0)^2}{2!} + \frac{V^{(3)}(x_0)(x-x_0)^3}{3!}+... | ||
</math> | </math> | ||
| − | where <math>V'(x_0)</math>, <math> | + | |
| + | where <math>V'(x_0)</math>, <math>V''(x_0)</math> are the first and second derivatives of <math>V(x)</math> evaluated at <math>x_0</math>. As <math>x_0</math> is an equilibrium point, the gradient of the potential is 0 (at the equilibrium point, the net [[force]] is zero). Therefore the second term in the expansion is 0. Near to <math>x_0</math>, terms of cubic order or higher can be neglected and so the potential is approximately: | ||
| + | |||
<math> | <math> | ||
V(x) \approx V(x_0) + \frac{V''(x_0)(x-x_0)^2}{2!} | V(x) \approx V(x_0) + \frac{V''(x_0)(x-x_0)^2}{2!} | ||
</math> | </math> | ||
| − | in the vicinity of <math>x_0</math>. The constant term corresponds can be neglected as it only affects where the zero of the potential is. Hence the potential is approximately [[quadratic]] and behave like a simple harmonic oscillator. | + | |
| + | in the vicinity of <math>x_0</math>. The constant term corresponds can be neglected as it only affects where the zero of the potential is. Hence the potential is approximately [[quadratic equation|quadratic]] and behave like a simple harmonic oscillator. | ||
===Example=== | ===Example=== | ||
An example of this is that of a simple pendulum. A simple pendulum has the equation of motion: | An example of this is that of a simple pendulum. A simple pendulum has the equation of motion: | ||
| + | |||
<math> | <math> | ||
\ddot \theta = - \frac{g}{L} \sin{\theta} | \ddot \theta = - \frac{g}{L} \sin{\theta} | ||
</math> | </math> | ||
| + | |||
where: | where: | ||
:<math>\theta</math> is the angle of the pendulum to the vertical | :<math>\theta</math> is the angle of the pendulum to the vertical | ||
| Line 44: | Line 55: | ||
Here, the small angle approximation, that <math>\sin{\theta} \approx \theta</math> can be made, so that | Here, the small angle approximation, that <math>\sin{\theta} \approx \theta</math> can be made, so that | ||
| + | |||
<math> | <math> | ||
\ddot \theta \approx - \frac{g}{L} \theta | \ddot \theta \approx - \frac{g}{L} \theta | ||
| Line 49: | Line 61: | ||
This is of the form of simple harmonic motion above. Hence when the angle of oscillation is small, the period of the pendulum is | This is of the form of simple harmonic motion above. Hence when the angle of oscillation is small, the period of the pendulum is | ||
| + | |||
<math> | <math> | ||
P \approx 2 \pi \sqrt{\frac{g}{L}} | P \approx 2 \pi \sqrt{\frac{g}{L}} | ||
Revision as of 13:23, April 7, 2017
Simple harmonic motion is any motion caused by a force that acts to restore a body to equilibrium and has a magnitude proportional to the distance of the body from its equilibrium position. Mathematically, the acceleration of the body, <math>\ddot x</math>, may be related to the displacement of the body by:
<math> \ddot x(t) = -kx </math>
where <math>k</math> is a constant. This results in the body performing oscillations about its equilibrium position. A body undergoing simple harmonic motion is also known as a simple harmonic oscillator. A potential of <math>\frac{1}{2}kx^2</math> will cause a body to undergo simple harmonic motion.
Mathematics
Simple harmonic motion is described by a second order differential equation:
<math> \ddot x(t) = -kx </math>
The solution of this equation is:
<math> x(t) = A \sin{\omega t} + B \cos{\omega t} </math>
where <math>\omega</math> is <math>\sqrt{k}</math> and called the angular frequency. <math>A</math> and <math>B</math> are arbitrary constants that can be determined from the initial conditions of the problem. Since the solution involves only sines and cosines which oscillate, the solution itself will oscillate. Simple harmonic oscillators also have the property that the frequency of the oscillation is independent to its amplitude.
Uses
The simple harmonic oscillator is a very important potential as it can be used to approximate the equilibrium points other potentials. This is one of the most common approximations used in physics, and is used in quantum field theory.
Derivation
| <math>\frac{d}{dx} \sin x=?\,</math> | This article/section deals with mathematical concepts appropriate for late high school or early college. |
Suppose there is a potential, <math>V(x)</math>, that has a equilibrium point (minima) at <math>x_0</math>. <math>V(x)</math> can be approximated around the minima using a Taylor series as:
<math> V(x) = V(x_0) + V'(x_0)(x-x_0) + \frac{V(x_0)(x-x_0)^2}{2!} + \frac{V^{(3)}(x_0)(x-x_0)^3}{3!}+... </math>
where <math>V'(x_0)</math>, <math>V(x_0)</math> are the first and second derivatives of <math>V(x)</math> evaluated at <math>x_0</math>. As <math>x_0</math> is an equilibrium point, the gradient of the potential is 0 (at the equilibrium point, the net force is zero). Therefore the second term in the expansion is 0. Near to <math>x_0</math>, terms of cubic order or higher can be neglected and so the potential is approximately:
<math> V(x) \approx V(x_0) + \frac{V(x_0)(x-x_0)^2}{2!} </math>
in the vicinity of <math>x_0</math>. The constant term corresponds can be neglected as it only affects where the zero of the potential is. Hence the potential is approximately quadratic and behave like a simple harmonic oscillator.
Example
An example of this is that of a simple pendulum. A simple pendulum has the equation of motion:
<math> \ddot \theta = - \frac{g}{L} \sin{\theta} </math>
where:
- <math>\theta</math> is the angle of the pendulum to the vertical
- <math>g</math> is the acceleration due to gravity
- <math>L</math> is the length of the pendulum[1]
Here, the small angle approximation, that <math>\sin{\theta} \approx \theta</math> can be made, so that
<math> \ddot \theta \approx - \frac{g}{L} \theta </math>
This is of the form of simple harmonic motion above. Hence when the angle of oscillation is small, the period of the pendulum is
<math> P \approx 2 \pi \sqrt{\frac{g}{L}} </math>