## Simple Harmonic Motion and Orbits

As you can (hopefully) see from the above (rather clunky) graph, two of the quantities that characterise simple harmonic motion are the maximum value *A* of the displacement *x*, and the period of the oscillation *T*. The period is the time taken for one complete oscillation or cycle of the motion, so its reciprocal 1/T determines the rate of the occurrence of cycles (the number per second) and is known as the **frequency**,* f = 1/T*.

The displacement from equilibrium of a simple harmonic oscillator may be described by the equation,

*x(t) = A sin(ωt + ϕ)
*

the quantities *A*, ω, and ϕ are parameters that characterise the motion. Their values may vary from one case of SHM to another, but for any particular case, they will be constants. The quantity *A* is known as the **Amplitude** of the oscillation. It represents the magnitude of the maximum displacement from the equilibrium position and therefore cannot be negative. ω is known as the **angular frequency **of the oscillation. It is related to the frequency *f* and the period *T* by the relations,

ω = 2π*f* = 2π/*T*

The angular frequency of SHM is represented by the same symbol ω as the angular speed of circular motion. However the units of angular frequency are s^{-1}, not rad s^{-1}.

ϕ is called the phase constant or the initial phase of the oscillation. It represents the value of ω*t* + ϕ (which is called the **phase**) at time t = 0s, and determines the initial displacement of the oscillator, *x*(0) = *A* sin (ϕ).

The velocity *v* at any time *t* is the derivative of *x(t)*,

*v*_{x}(*t*) = *dx*/*dt* = *d/dt* [*A sin( ωt + ϕ)*] = *A*ω cos(ω*t* + ϕ)

Similarily, the acceleration is,

*a _{x}(t)* =

*dv*=

_{x}/dt*d*=

^{2}x/dt^{2}*d*[

^{2}/dt^{2}*A sin (ωt + ϕ)*] = –

*A*ω

^{2}sin(ωt + ϕ)

All three of the quantities (displacement, velocity and acceleration) vary** sinusoidally** with time. All three quantities are characterised by the same angular frequency ω. They therefore all have the same period, *T* = 2π/ω. The maximum value of the displacement is the amplitude A, the maximum velocity is *A*ω, and the maximum acceleration is Aω^{2}. When the magnitude of the displacement is a maximum (*A*), the velocity is momentarily zero, and the acceleration has its maximum magnitude of Aω^{2}. This is because the velocity reaches its maximum value a quarter a cycle ahead of the displacement, and the acceleration reaches its maximum a quarter of a cycle ahead of the velocity.

It can be seen that the displacement and the acceleration have the same general shape but they differ in maximum and in sign. *Ax(t)* = – ω^{2}*x(t) *(the **simple harmonic motion equation**).

In SHM, the acceleration of the oscillator is proportional to its displacement from the equilibrium position, and the constant of proportionality is the negative quantity –ω^{2}, where ω is the angular frequency of the motion.

Any kind of motion in which the acceleration is, at all times, proportional to the displacement, and the constant of proportionality is negative, will be simple harmonic motion.

The SHM equation often appears as,

*d ^{2}x(t)/dt^{2} = -ω^{2}x(t)*

An equation like this which relates a variable quantity to one or more of its derivatives is called a **differential equation**. The solution to a differential equation is not simply a number, but is itself a function. The SHM equation is a **second-order differential equation** because it involves a second derivative.

Orbital Motion

The equation of an **ellipse** is *x ^{2}/a^{2} + y^{2}/b^{2 }*= 1. Such an ellipse has a

**semimajor**axis of length

*a*and a

**semiminor**axis of length

*b*. A measure of how much an ellipse differs from a circle is given by its eccentricity,

*e = 1/a (a2-b2)*An ellipse has two special point on its major axes, each called a

^{1/2}**focus**. These are located at the points (

*ae*,0) and (-

*ae*,0).

**Kepler’s first law**: The orbit of each planet in the solar system is an ellipse with the sun at one focus.

**Kepler’s second law**: A radial line from the sun to a planet sweeps out equal areas in equal intervals of time.

**Kepler’s third law**: The Square of the orbital period of each planet is proportional to the cube of its semi major axis.

The period of a planetary orbit is the time taken for a planet to complete one full circuit of the sun. Kepler’s third law tells us that there is a single constant such that for any body that orbits the sun with a period *T* and a semimajor axis *a*, *T*^{2}/*a*^{3} = *K
*

It turns out that the most efficient way of getting directly from earth to another planet is to follow what is known as a Hohmann Transfer Orbit The Hohmann transfer orbit is part of an ellipse with the sun at one focus and with the orbits of the two planets just touching the ellipse. The space vehicle leaves the Earth at a tangent and arrives at the other planet tangentially to its orbit.