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.
Sinusoidal motion is a type of periodic motion where an object moves back and forth in a repetitive pattern, similar to a wave. This type of motion is often described using a mathematical function, such as the equation x = A sin(wt), where x represents the position of the object, A is the amplitude, t is time, and w is the angular frequency.
The equation x = A sin(wt) represents the position of an object undergoing sinusoidal motion. The variable A represents the amplitude, which is the maximum displacement of the object from its equilibrium position. The variable w represents the angular frequency, which determines the speed at which the object oscillates. The variable t represents time.
Sinusoidal motion is different from other types of motion because it follows a specific pattern of movement. The position of the object can be described using a mathematical function, and the object oscillates between a maximum and minimum position with a constant period. Other types of motion, such as linear or circular motion, do not follow this pattern.
The motion described by x = A sin(wt) can be affected by the amplitude (A) and the angular frequency (w). A larger amplitude will result in a greater maximum displacement of the object, while a larger angular frequency will result in a faster oscillation. The medium in which the object is moving and any external forces acting on the object can also affect its motion.
Sinusoidal motion is used in many real-world applications, such as in the motion of a pendulum, the vibration of a guitar string, and the sound waves produced by a speaker. It is also commonly used in electrical engineering to describe alternating current (AC) circuits. Understanding sinusoidal motion is essential in fields such as physics, engineering, and mathematics.