A Quantitative Evaluation of Natural Systems, Volume II A Quantitative Evaluation of Natural Systems

Volume II

by Dr Profidious Vile

A continuation of Volume I

The reader of this volume is assumed to have read and fully understood the concepts described in "A Quantitative Evaluation of Natural Systems, Volume I", including force, momentum, energy and the conservation thereof. In this volume, we will explore the mechanics of circular motion and waves, and how we use mathematics to predict circular motion and wave behavior.

Circular Motion and Centripetal Force

Consider a ball suspended from a length of string, about a meter in length. It is possible to manipulate the ball in such a way that it whirls around in wide circles on the string. Yet, clearly there is no lateral force when the ball is set in motion. The string only permits a pulling force inward. So why is it, then, that the ball manages to accelerate in a circular fashion?

The answer is that the ball isn't accelerating in a circular fashion, but rather is constantly accelerating inward, due to a force towards the center of its circular path. Conversely, the velocity of the ball is always at a tangent to this circular path (and therefore at a tangent to this inward force). Thus, when there is an acceleration inward, the ball adopts a more oblique velocity comprised of the vectors of its initial velocity, and the additional velocity caused by the acceleration towards the center. As this occurs, the direction in which acceleration occurs immediately changes to become tangential to the new resultant velocity. This continues for as long as this inward force is applied. I like to call this force "Centripetal Force".

Let's quantify circular motion. We know that there are 2π radii in the circumference of a circle. We also know that the velocity of an object is equal to its change in displacement, over the time for which the displacement occurs. If we substitute displacement, s for 2πr, and replace time, t with the period of the circular motion (i.e. the time it takes to perform one full revolution), T, we can obtain the velocity of rotation:

v = 2πr/T

Let's now consider a π/2 phase shift. The change in velocity is the initial velocity, subtracted from the final velocity. The result is a pair of vectors forming a right-angle trangle. The time is T/4, and the magnitude of the resultant is given by (8π^2r^2/T^2)^.5 = 8^.5 * πr / T (since a^2 = b^2 + c^2 - 2bc.cosA)

When considering centripetal acceleration, we must remember one of the kinematic equations from Volume I: s = ut + at^2/2 ut and at^2/2 are in different directions. ut is at a tangent to the circle, and at^2/2 is towards the center of the circle, o. The radius, ut and at^2/2 + r make up a right-angled triangle.

We know that u = v, as the magnitude of v is constant in circular motion. We also know that acceleration is perfectly perpendicular to the vector vt, so the arclength subtended by vt and at^2/2 must be very small.

Let's say that x = at^2/2

v^2t^2 + r^2 = (r + x)^2 v^2t^2 + r^2 = r^2 + 2rx + x^2 v^2t^2 = 2rx + x^2 v^2t^2 = x(2r + x)

As the arclength is very small, x is negligible. Thus: v^2t^2 = 2rx

Now we substitute at^2/2 back into x: v^2t^2 = 2at^2r/2 v^2 = ar a = v^2/r

Which is the acceleration experienced by the object undergoing circular motion. We can further develop this to produce centripetal force.

F = ma a = v^2/r F = mv^2/r

A new, but familiar system

It must be apparent to the reader by now that attempting to describe circular motion in terms of our linear equations of velocity quickly gets very complicated.

We need a new system - an angular system of kinematics; particularly one which takes angular displacement as an angle in radians, which is easily determined, manipulated and translated between systems.

Let's take this angular displacement and call it θ. As after T seconds, θ = 2π (there are 2π radians in a revolution), and linear displacement after the same time, s = 2πr, θ and s are related thusly:

s = rθ Where θ is measured in radians

Now: v = Δθr/Δt (as v = s/t) Thus, dθ/dt, which is angular velocity (we will call this ω) can be derived from linear velocity: v/r = dθ/dt ω = v/r

v = rω Which is the relationship between linear and angular velocity. ω is measured in rad/s

Let's refer to linear systems again. This time: a = Δv/t a = rΔω/t a/r = Δω/t

And since we know that angular acceleration, α = dω/dt:

a/r = α a = rα Where α is measured in rad/s^2

It is very simple now to construct a set of equations for angular kinematics. We simply substitute values of a for α, v for ω(f), u for ω(i) and s for θ into the linear kinematic equations from the last volume (these hold, as the basic premise for kinematics is preserved. All we have changed is the quantification of displacement from meters to radians):

ω(f) = ω(i) + αt θ = (ω(f) + ω(i))t/2 (ω(f))^2 = (ω(i))^2 + 2αθ θ = ω(i).t + αt^2/2

Angular momentum

The following emblem is stamped prominently on the spine of this book:

Dr Vile leaves the book partially completed on the shelves. Inserted into the pages is a small, paper note which reads: "Complete angular mechanics by Flamerule 7th"