The discussion centers on the equation for sinusoidal motion, x = A sin(wt), where A represents amplitude, w is angular velocity, and t is time. Participants clarify that sinusoidal motion is periodic and can be derived from circular motion principles. The choice of sine over cosine is explained: sine is used when starting from the equilibrium position (x = 0), while cosine is used when starting from the maximum displacement. The conversation emphasizes the importance of understanding the relationship between circular motion and harmonic motion in deriving these equations.
PREREQUISITESStudents of physics, educators explaining harmonic motion, and anyone interested in the mathematical foundations of periodic 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.