The discussion revolves around the equation of sinusoidal motion, specifically x = A sin(wt), where A represents amplitude, w is angular velocity, and t denotes time. Participants express confusion regarding the derivation of this equation, particularly in relation to its connection with circular motion and the choice of sine versus cosine functions.
There is an active exchange of ideas, with some participants providing insights into the relationship between sine and cosine functions in the context of harmonic motion. Multiple interpretations of the diagram and the underlying principles are being explored, although no consensus has been reached.
Participants note potential confusion stemming from the textbook's explanation and the graphical representation of the motion. The discussion highlights the importance of understanding the initial conditions and the implications of starting from different positions in the motion.
rock.freak667 said:Well what in the equation is confusing you exactly?
Sinusoidal motion is basically a periodic motion. Every fixed amount of radians or time, the cycle repeats. This is seen if the graph is plotted.
mlostrac said:Umm, well the way they get the equation. The text says "we can derive a formula for the period of simple harmonic motion by comparing it to an object rotating in a circle"
How they get their variables, using the object rotating in a circle, is what I find confusing. For example: why choose "sin" when the object is at the equilibrium position (and t = 0)?
rock.freak667 said:If you inspect the graph of x=sint, you will see that at x=0, t= 0. Meaning that they are starting off with the object not moving from it's equilibrium position.
Had the stretched it to its maximum value, they'd use x=cos(t).
So essentially, starting from equilibrium position, use sine.
Starting from maximum position, use cosine.
mlostrac said:http://www.freeimagehosting.net/image.php?73b50a3632.jpg
I uploaded the picture from my text.
So what I was wondering, if the object is directly in the middle (x = 0), the "A" line would be straight up and down. Therefore, where does sin come in if there is no "x" to form a triangle like that in picture (a)?
rock.freak667 said:Normally, you displace the mass at an angle θ from the equilibrium position. This is done so that you can arrive at an equation of motion for the mass. It should be noted that, 'x' is a horizontal displacement, so 'x' will not be a sine but a cosine.
x=Acosθ
'A' is a constant radius is in the circle so if viewed from the side, the maximum distance the mass moves on either side of x=0 is A.
When x=0, you get Acosθ or θ=π/2. So if you view from the top (circle), you will see the mass directly above x=0. If you view from the side (harmonic motion) you will see the mass at x=0.
mlostrac said:Ok, but if you start at x = 0 (the equilibrium position) and then push the object, you use sine?
mlostrac said:My book says, "If at t = 0 the object is at the equilibrium position and the oscillations are begun by giving the object a push to the right (+x), the equation would be:
x = A(sin)wt = A(sin)[2(pi)t/T]"
I understand the cosine; but just don't know why they use sine?
Ok, I think I can do good just memorizing that.rock.freak667 said:Yes will it is just like how your book says, since t=0 should correspond to x=0, the only functions you have to use are sin(t) and cos(t), sin(t) fits this best.
mlostrac said:But Looking at the diagram I attached, if there is no "x" (because the "A" is perpendicular to the x-axis --> "A" is at the 12 o'clock position on the circle) where does sine come from? There should be no angle because there's no x value?
rock.freak667 said:In your diagram, you are measuring θ anti-clockwise, so at that position, x=0 when θ=π/2 radians or 90°.
But given how that diagram is drawn with the arrows as is, it looks like they started at x=A and then displaced (since the angle is drawn how it is), so for that motion, the equation would be x=Acosθ or x=Acosωt.
If they wanted to show the correct diagram for the sine motion, what they should have done was draw the angle measured from the 12 o'clock position. (as that would show x=Asinωt). OR they should have drawn the observer at the adjacent side of the table.