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That integral gives us a dilemma because the forms of the derivatives of the trigonometric functions give us two possible solutions:
Which of those solutions must we keep and why?
We find a straightforward answer to that question in the double angle formulae of trigonometry;
We thus have our integral as
So we could have simply guessed that we could obtain the correct solution by taking the average value of the sum of the two solutions obtained in Equations 1 & 2. Of course, we don= t allow guessing in mathematics: we require strict logical analysis. Ultimately, we believe, rigorous logic will interconnect all of the parts of mathematics into a single structure. In resolving the dilemma that this integral gives us we see one way in which that will happen.
Kinetic Energy of a Body Oscillating on a Spring
Now I want to move this integral out of the realm of pure mathematics and into the realm of describing Reality. To what extent does this weird Štherial game that we play with pure logic mimic the realm that we apprehend through our senses? We will only find out through our efforts to apply its results to various features of that outer realm. And we achieve that exploration through the use of similes.
In this case the product in the intengrand represents twice the area of the right triangle used to define the basic trigonometric functions. The triangle=s hypotenuse has a length of one unit and makes an angle x with the horizontal axis of our coordinate frame. The length of the triangle=s base thus corresponds to the cosine of that angle and the length of the triangle=s remaining side corresponds to the sine of the angle. Multiplying that product by the differential of the angle and then integrating gives us a result analogous to something accumulating on the area of the triangle with respect to some parameter that corresponds to a change in the angle between the hypotenuse and the base.
The example that I chose here may seem obvious, but it actually took some time for me to find it. I started with trying to imagine some phenomenon accumulating something on a triangular area, taking time to be the obvious parameter corresponding to the angle. That seemed too contrived, but then I noticed anew that with Cosxdx = dSinx I had an integrand in the form fdf. Further, the function f was sinusoidal in time, so the thought of an oscillating motion came up out of memory along with the fact that the product of velocity and the differential of velocity corresponds to the differential of kinetic energy. So now I can use Equation 5 to represent the kinetic energy in a body of mass m attached to the free end of an anchored spring.
In those equations the Greek letter omega represents the angular frequency of the body=s oscillation. So we get for the body=s kinetic energy
in which I reversed the normal order of the limits on the integral in order that the calculation of the energy not take on negative values. As we expect, that equation gives us E=0 when t=0 and
its maximum value, whenωt=π/2.
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