Simple harmonic 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.

$${\displaystyle F_{\mathrm{n}\mathrm{e}\mathrm{t}}=m\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}=-kx,}$$ 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).

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

Solving the differential equation above produces a solution that is a sinusoidal function: ${\displaystyle x(t)=c_1\cos \left(\omega t\right)+c_2\sin \left(\omega t\right),}$ where $\omega =\sqrt{k/m}.$ The meaning of the constants ${\displaystyle c_1}$ and ${\displaystyle c_2}$ can be easily found: setting ${\displaystyle t=0}$ on the equation above we see that ${\displaystyle x(0)=c_1}$, so that ${\displaystyle c_1}$ is the initial position of the particle, ${\displaystyle c_1=x_0}$; taking the derivative of that equation and evaluating at zero we get that ${\displaystyle \dot{x}(0)=\omega c_2}$, so that ${\displaystyle c_2}$ is the initial speed of the particle divided by the angular frequency, ${\displaystyle c_2=\frac{v_0}{\omega }}$. Thus we can write: $${\displaystyle x(t)=x_0\cos \left(\sqrt{\frac{k}{m}}t\right)+\frac{v_0}{\sqrt{\frac{k}{m}}}\sin \left(\sqrt{\frac{k}{m}}t\right).}$$

This equation can also be written in the form: $${\displaystyle x(t)=A\cos \left(\omega t-\phi \right),}$$ where

• ${\displaystyle A=\sqrt{{c_1}^2+{c_2}^2}}$
• ${\displaystyle \tan \phi =\frac{c_2}{c_1},}$
• ${\displaystyle \sin \phi =\frac{c_2}{A},\cos \phi =\frac{c_1}{A}}$

or equivalently

• ${\displaystyle A=|c_1+c_{2}i|,}$
• ${\displaystyle \phi =\arg (c_1+c_{2}i)}$

In the solution, c₁ and c₂ are two constants determined by the initial conditions (specifically, the initial position at time t = 0 is c₁, while the initial velocity is c₂ω), and the origin is set to be the equilibrium position. 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 initial phase.

Using the techniques of calculus, the velocity and acceleration as a function of time can be found: $${\displaystyle v(t)=\frac{\mathrm{d}x}{\mathrm{d}t}=-A\omega \sin (\omega t-\phi ),}$$

• Speed: ${\displaystyle \omega \sqrt{A^2-x^2}}$
• Maximum speed: v = ωA (at equilibrium point)

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

• Maximum acceleration: Aω² (at extreme points)

By definition, if a mass m is under SHM its acceleration is directly proportional to displacement. $${\displaystyle a(x)=-{\omega }^{2}x.}$$ where $${\displaystyle {\omega }^2=\frac{k}{m}}$$

Since ω = 2πf, $${\displaystyle f=\frac{1}{2\pi }\sqrt{\frac{k}{m}},}$$ and, since T = ⁠1/f⁠ where T is the time period, $${\displaystyle T=2\pi \sqrt{\frac{m}{k}}.}$$

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 $${\displaystyle K(t)=\frac{1}{2}m{v}^2(t)=\frac{1}{2}m{\omega }^2{A}^2{\sin }^2(\omega t-\phi )=\frac{1}{2}k{A}^2{\sin }^2(\omega t-\phi ),}$$ and the potential energy is $${\displaystyle U(t)=\frac{1}{2}k{x}^2(t)=\frac{1}{2}k{A}^2{\cos }^2(\omega t-\phi ).}$$ In the absence of friction and other energy loss, the total mechanical energy has a constant value $${\displaystyle E=K+U=\frac{1}{2}k{A}^2.}$$

Examples

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: $${\displaystyle T=2\pi \sqrt{\frac{m}{k}}}$$ shows the period of oscillation is independent of the amplitude, though in practice the amplitude should be small. 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.

Oscillatory motion

The motion of a body in which it moves to-and-fro about a definite point is also called oscillatory motion or vibratory motion. The time period is able to be calculated by $${\displaystyle T=2\pi \sqrt{\frac{l}{g}}}$$ where l is the distance from rotation to the object's center of mass undergoing SHM and g is acceleration due to gravity. This is analogous to the mass-spring system.

Mass of 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 pendulum of length l with gravitational acceleration ${\displaystyle g}$ is given by $${\displaystyle T=2\pi \sqrt{\frac{l}{g}}}$$

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, ${\displaystyle g}$, 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 ${\displaystyle g}$ 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:

$${\displaystyle -mgl\sin \theta =I\alpha ,}$$

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

$${\displaystyle -mgl\theta =I\alpha }$$

which makes angular acceleration directly proportional and opposite to θ, satisfying the definition of simple harmonic motion (that net force is directly proportional to the displacement from the mean position and is directed towards the mean position).

Scotch yoke

A Scotch yoke mechanism can be used to convert between rotational motion and linear reciprocating motion. The linear motion can take various forms depending on the shape of the slot, but the basic yoke with a constant rotation speed produces a linear motion that is simple harmonic in form.