Applied Maths - Simple Harmonic Motion

alan4cult

Just a little confused on the results of the a=-w^2x equation on page 301/302 (Fundamental Applied Maths)

Firstly, how is x=ACos(wt+e) a solution to the above equation? I keep getting just x=ASin(wt+e)

Secondly, how do you get the period of the function ASin(wt+e) to be equal to 2pi/w. I'm not aware of how to find the period.

Any help?

MathsManiac

Well, for a start, you can verify that x=ACos(wt+e) is a solution by differentiating it twice with respect to t.

Regarding the other question, you need to become a bit more familiar with the graphs of sine and cosine functions, and then you'll see how the period works. Have you ever drawn a graph of y=sin(x) and similar?

You could try drawing graphs like sin(x), sin(2x), sin(x+1), 3sin(x), by hand, or using a graphics calculator.

Or you could check out some web resources:

Check out the following link, for notes on such graphs:
http://faculty.ed.umuc.edu/~swalsh/Math%20Articles/Graphing%20Sin%20Cos.html

Also very useful, if your browser is java-enabled is the fact that there are a number of java-based interactive explorations of the behaviour of the sin and cos graphs, including the following two:
http://www.ies.co.jp/math/java/trig/ABCsinX/ABCsinX.html

http://members.shaw.ca/ron.blond/sc.APPLET/index.html


Hope this helps.

alan4cult

I think I get the graphs
y=Sin(X) Standard Graph
y=Sin(X+1) Standard Graph moves 1 unit to the left
y=Sin(X-1) Standard Graph moves 1 unit to the right
I need to move my graph to where the sine function repeats itself
The sine function repeats every 2pi units.

Basically I've got that far now how to I get 2pi/w since I've got the 2pi part can somebody tell me the rest? I'm not trying to cheat I'm trying to understand!

MathsManiac

Multiplying the x by a number will stretch or squash the graph horizontally. (Did you check out the links?) So, if you graph sin(2x) you'lll see that its period is half that of sin(x). Thinks about it: as x ranges from 0 to Pi, 2x will go from 0 to 2Pi, so you get the complete cycle twice as quickly. similarly, multiplying x by any number k will make the cycle happen k times faster. In other words, the period is 1/k times the basic period.

Using the letters you originally asked the question with: sin(wt) has period 1/w times the period of sin(t); that is, 2*Pi/w. And adding the e makes no difference to the period, as it's just shifting the graph left or right (as you've already observed with sin(x+1)).

Any use?

alan4cult

Perfecct thanks for your help. Thanks for explaining about the constant "e". I didn't actually know whether it altered the period or not, but you've explained that it just translates the graph.

Thanks again.

jaycummins

fundamental applied maths. what a book. my applied maths teacher wrote that book. (Ollie Murphy) what a legend. i love applied maths. especially statics (ladders friction etc.)