Trouble with sinusoidal waves

[tj.]

Homework Statement

A continuous succession of sinusoidal wave pulses are produced at one end of a very long string and travel along the length of the string. The wave has frequency 36.0Hz, amplitude 5.50mm, and wavelength 0.635m.
a) How long does it take the wave to travel a distance of 8.50m along the length of the string?
b) How long does it take a point on the string to travel a total accumulated transverse distance of 8.50m, once the wave train has reached the point and set it into motion?

Homework Equations

So I've been able to do part a) with ease but the trouble I'm having is with part b). It just completely confuses me!

Do I need to use the equation $y(x,t) = Acos(kx - \omega t)$ or $\frac{\partial^{2}y(x,t)}{\partial x ^{2}} = \frac{1}{v^{2}} \frac{\partial ^{2}y(x,t)}{\partial t^{2}}$ or $v_{y}(x,t) = \omega Asin(kx - \omega t)$ or something different??

The Attempt at a Solution

So for part a) I got t=0.372 seconds using $v=f\lambda$ and b) is an unknown to me.

Thanks in advance if anyone can help!

 

[kuruman]

First find how much distance the point travels in one oscillation, then find how many oscillations correspond to 8.50 m.

 

[tj.]

So I got the exact same answer as part a). Would that be correct??

 

[Redbelly98]

Moderator's note: thread moved to Introductory Physics, as this appears to be college sophomore level physics.

So I got the exact same answer as part a). Would that be correct??

Uh, no. First answer this: how much transverse distance does the point travel in one oscillation? Hint: it is related to the amplitude.

(Back to you, kuruman ... )

 

[rwisz]

I can understand your confusion and frustration with the equations and concepts involved in this problem. Let me try to break it down for you.

First, for part a), you have correctly used the equation v = fλ to find the time it takes for the wave to travel a distance of 8.50m. This is the time it takes for the wave to physically travel through the medium.

For part b), we need to consider the motion of a single point on the string. When the wave reaches this point, it begins to oscillate up and down, creating a transverse motion. This motion can be described by the equation y(x,t) = Acos(kx - ωt), where A is the amplitude, k is the wave number, and ω is the angular frequency. This equation shows us that the position of the point on the string (y) depends on both the position along the string (x) and the time (t). In this case, we are interested in the total accumulated transverse distance, so we need to integrate this equation over time from 0 to t (the time it takes for the wave to travel 8.50m). This will give us the total displacement of the point on the string.

To do this, we need to use the equation ∫ y(x,t) dt = A∫ cos(kx - ωt) dt. This integral will give us the total accumulated transverse distance traveled by the point on the string. You can then set this equal to 8.50m and solve for t to find the time it takes for the point to travel this distance.

I hope this helps clarify things for you. Remember, as a scientist, it is important to break down complex problems into smaller, more manageable parts and use the appropriate equations to solve each part. Good luck!