What is the Coordinate of Point B in a Sinusoidal Wave?

In summary, the problem is asking for the coordinate of point B on a cosine graph with a wavelength of 40 cm, a frequency of 8.00 Hz, and a phase constant of 0.157 radians per cm. The wave function given is y = (15.0 cm) cos(0.157x - 50.3t), and the question specifies that at a certain instant, point B is 60.0° out of phase with point A at the origin. To find the coordinate of point B, we need to determine the value of x that would cause a phase change of 60 degrees, given that t is constant. After solving, the correct value for x is 2.12 cm.
  • #1
jpond89
3
0

Homework Statement



Consider the sinusoidal wave in the figure, with the wave function below.

At a certain instant, let point A be at the origin and point B be the first point along the x-axis where the wave is 60.0° out of phase with point A. What is the coordinate of point B?

y = (15.0 cm) cos(0.157x - 50.3t)

and I am not allowed to post the image but here's some helpful information:
its a cosine graph, and [tex]\lambda[/tex] = 40 cm, T = 0.125 s, f = 8.00 hz, k = 0.157 rads/cm

Homework Equations



Some equations that I have are:
k=2[tex]\pi[/tex]/[tex]\lambda[/tex]
v=[tex]\lambda[/tex]*f
y=Asin(kx-[tex]\omega[/tex]t)

The Attempt at a Solution


What I have tried so far is:
y=15*sin((.157)x-(50.3)t + [tex]\pi[/tex]/3)
The example says " The vertical position of an element of the medium at t = 0 and x = 0 is also 15.0 cm "
The only problem is that I am really stuck and I cannot seem what to set the 60.0[tex]\circ[/tex] equal to. Help would be much appreciated! Thank you!
 
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  • #2
Everything in the argument of the cosine is the phase of the wave. So the phase is a function of both x and t. However, we're looking at this wave at a particular "instant", which means that t is constant. So, the only thing that would cause a phase change is x. So how far from the origin would the x coordinate of point B have to be in order for 0.157x - 50.3t to differ from 0.157*(0) - 50.3t by 60 degrees, given that t is constant?
 
  • #3
Would x = 6.37 * [tex]\pi[/tex] because that would make it ([tex]\pi[/tex]/3 -50.3t)
 
  • #4
I think you might be off by a factor of 3. Check your work again.
 

What is a sinusoidal wave?

A sinusoidal wave, also known as a sine wave, is a type of wave that has a repeating pattern of a single frequency and amplitude. It is a smooth, continuous wave that can be described by a mathematical function, the sine function.

What are the characteristics of a sinusoidal wave?

The characteristics of a sinusoidal wave include its amplitude, wavelength, frequency, and period. The amplitude is the height of the wave, the wavelength is the distance between two consecutive peaks or troughs, the frequency is the number of cycles per unit time, and the period is the time it takes for one complete cycle to occur.

How is a sinusoidal wave represented mathematically?

A sinusoidal wave is represented mathematically by the sine function, which is y = A sin (ωt + φ), where A is the amplitude, ω is the angular frequency, t is the time, and φ is the phase angle. The sine function produces a wave that has a smooth, curved shape.

What are some real-life applications of sinusoidal waves?

Sinusoidal waves have many real-life applications, including in sound and light waves, electrical circuits, and mechanical systems. They are also used in fields such as acoustics, electronics, and optics to analyze and design systems and devices.

What are some common problems involving sinusoidal waves?

Common problems involving sinusoidal waves include finding the amplitude, frequency, and period of a given wave, determining the phase shift or phase difference between two waves, and analyzing the interference and superposition of multiple waves. These problems are often encountered in physics, engineering, and mathematics courses.

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