Simple harmonic motion

A thing that is moving back and forth is said to be vibrating. Another word for vibration is oscillation. A special way of vibrating or oscillating is called simple harmonic motion. When measuring motion, it is normal to make a graph with time on one axis and distance on the other. Sometimes, when something moves its distance from a fixed point it looks like a sine wave in that kind of graph. In mathematics and physics, this is called simple harmonic motion (sometimes shortened as SHM). This movement will happen when the force towards the fixed point is proportional to the distance from the point (the force goes down as much as the distance goes up) and always acts towards that point. Some examples are a weight on a spring and a simple pendulum (for small oscillations). These are not perfect examples, but they are close to having simple harmonic motion.

Introduction

In simple harmonic motion, a particle moves along a straight line. Its acceleration will always point towards fixed point on this line. The magnitude (or amount) of acceleration changes proportionally to the particle's displacement from the fixed point. This means that the acceleration of the particle is bigger when the particle is farther away from the fixed point. This also means that the acceleration of the particle will be smaller if the particle is closer to the fixed point.^[1]

A simple harmonic oscillator can be made in many ways. One common way is to put a weighted object at the end of a spring . The other end of the spring is connected to a support that does not move. Two things can happen to the weight. If the spring puts a force on the mass, then the weight is in "equilibrium". This means that the weight is not moving at all (or is moving, but has no acceleration). In equilibrium, the weight would be at an "equilibrium position". If the spring is compressed (pushed in) or extended (pushed out), the weight will not be at the equilibrium position. If the weight is not in the equilibrium position, the spring puts a force on the weight. This spring is called a "restoring elastic force". This means that the spring tries to push the weight back to the equilibrium position. The amount of push, or force that the weight puts back to the equilibrium can be found with Hooke's law.

Hooke's law is written as: $\mathbf {F} =-k\mathbf {x} ,$ $F$ is the "restoring elastic force" (in SI units: N). This is the force that moves a spring back to the equilibrium position.

$k$ is the spring constant (N·m⁻¹). This constant is different depending on the spring that is used.

$x$ is the displacement from the equilibrium position (in metres).

Simple harmonic motion happens in a few steps. A spring puts a force on the weight when the weight is not at the equilibrium position. This causes the weight to accelerate to the equilibrium position. As the weight gets closer to the equilibrium position, the amount of restoring force becomes lower. When the weight reaches the equilibrium position, there is no force on the weight at all. At the equilibrium position, the weight is still moving because it has momentum. This means that the weight will still move past the equilibrium position. The weight will then compress the spring again. This causes another restoring force. The restoring force will slow the weight. Soon, the weight will stop moving. There is still a force on the weight, however. This means that it will move back to the equilibrium position.

These steps will happen forever as long as no energy is lost. If energy is "lost" (for example, if the spring has friction), then the spring will eventually stop vibrating. When energy is lost, the spring has "damped oscillation".

Dynamics

Simple harmonic motion of a particle can be explained using Newtonian mechanics and a second-order differential equations. This equation combines Newton's second law and Hooke's law. $F_{\mathrm {net} }=m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-kx,$ $m$ is mass

$x$ is the displacement from the equilibrium position (in metres). It can also be called the "mean position"

$k$ is the spring constant (N·m⁻¹). This constant is different depending on the spring that is used.

In this equation, you can use algebra to divide mass from both sides of the equation. ${\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-{\frac {k}{m}}x$ This differential equation can be solved using calculus. The solved equation is an equation that uses sine waves: $x(t)=c_{1}\cos \left(\omega t\right)+c_{2}\sin \left(\omega t\right)$ . In this equation, ω is angular frequency. In this equation, ${\textstyle {\omega }={\sqrt {{k}/{m}}}.}$ In this equation, $c_{1}$ and $c_{2}$ are constants. These constants can be found using algebra. When $t=0$ , $x(0)=c_{1}$ . The original placement, or position of this particle can be shown as $c_{1}=x_{0}$ This equation is now $x(t)=x_{0}\cos \left(\omega t\right)+c_{2}\sin \left(\omega t\right)$ . The derivative of this equation shows that ${\dot {x}}(0)=\omega c_{2}$ . This means that the original speed of the particle divided by angular frequency is c₂. Or, $c_{2}={\frac {v_{0}}{\omega }}$ . This equation can now be written as: $x(t)=x_{0}\cos \left({\sqrt {\frac {k}{m}}}t\right)+{\frac {v_{0}}{\sqrt {\frac {k}{m}}}}\sin \left({\sqrt {\frac {k}{m}}}t\right).$

This equation can also be written in another form: $x(t)=A\cos \left(\omega t-\varphi \right),$ In this equation,

$A={\sqrt {{c_{1}}^{2}+{c_{2}}^{2}}}$ , or the amplitude of the particle.
$\tan \varphi ={\frac {c_{2}}{c_{1}}},$
$\sin \varphi ={\frac {c_{2}}{A}},\;\cos \varphi ={\frac {c_{1}}{A}}$

$c 1$ is the original (or initial) position. $c 2 ω$ is also the original (or initial) velocity.^[A] The amplitude A is the amount of displacement the particle can be from the equilibrium point. The angular frequency is used as $ω = 2 πf$ . In this equation, $φ$ is the original (or initial) phase.

Calculus can be used to find velocity and acceleration equations. $v(t)={\frac {\mathrm {d} x}{\mathrm {d} t}}=-A\omega \sin(\omega t-\varphi ),$

Its speed is ${\omega }{\sqrt {A^{2}-x^{2}}}$
The highest speed a particle will have is at the equilibrium point. It is found by $v = ωA$

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

The highest acceleration a particle will have is at the amplitude points. It is found by $Aω 2$

The frequency and period can also be found. $f={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}},$ $T=2\pi {\sqrt {\frac {m}{k}}}.$

These equations show that simple harmonic motion is isochronous. This means that the amplitude of a particle will not change its period and frequency.

Related pages

Newtonian mechanics

Notes

^ Cosine does not have to be used in this equation. In this equation, cosine can also be written as: $x(t)=A\sin \left(\omega t+\varphi '\right),$ where $\tan \varphi '={\frac {c_{1}}{c_{2}}},$ since $cos θ = sin(.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}π/2 − θ)$ .

References

↑ "Simple Harmonic Motion – Concepts".

Fowles, Grant R.; Cassiday, George L. (2005). Analytical Mechanics (7th ed.). Thomson Brooks/Cole. ISBN 0-534-49492-7.
Taylor, John R. (2005). Classical Mechanics. University Science Books. ISBN 1-891389-22-X.
Thornton, Stephen T.; Marion, Jerry B. (2003). Classical Dynamics of Particles and Systems (5th ed.). Brooks Cole. ISBN 0-534-40896-6.
Walker, Jearl (2011). Principles of Physics (9th ed.). Hoboken, New Jersey: Wiley. ISBN 978-0-470-56158-4.

Other websites

[cnote_A] Cosine does not have to be used in this equation. In this equation, cosine can also be written as: $x(t)=A\sin \left(\omega t+\varphi '\right),$ where $\tan \varphi '={\frac {c_{1}}{c_{2}}},$ since $cos θ = sin(.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}π/2 − θ)$ .

[1] "Simple Harmonic Motion – Concepts".

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