Answer to a Question Recently Posed

Someone asked this question recently:

"Two particles are simultaneously performing SHM along the same path with same time period and equal amplitude A. If the maximum separation b/w particles if (root3)A, then find their phase difference and positions when they cross each other."

This question involves some interesting mathematics. Consider the complex representation of a sinusoid ("Re" denotes "real part" and "j" is the imaginary unit):

Acos(ωt)=Re[Aejωt]A \, cos(\omega t) = Re[A e^{j \omega t}]

Notice that the coefficient on the exponential term is the magnitude of the sinusoid. That will be important to remember. Write general expressions for the two sinusoids under consideration:

f1=Acos(ωt)f2=Acos(ωt+θ)f_1 = A \, cos(\omega t) \\ f_2 = A \, cos(\omega t + \theta)

Take the difference:

f1f2=Acos(ωt)Acos(ωt+θ)=Re[Aejωt]Re[Aej(ωt+θ)]=Re[AejωtAej(ωt+θ)]=Re[ejωt(AAejθ)]f_1 - f_2 = A \, cos(\omega t) - A \, cos(\omega t + \theta) \\ = Re[A e^{j \omega t}] - Re[A e^{j (\omega t + \theta)}] \\ = Re[A e^{j \omega t} - A e^{j (\omega t + \theta)} ] \\ = Re[e^{j \omega t} (A - A e^{j \theta}) ]

We can therefore infer that the magnitude (peak value) of the resultant sine wave is the magnitude of the complex quantity AAejθA - A e^{j \theta}, which we also know to be 3A\sqrt{3} A. The peak value of the resultant sine wave will occur when the complex exponential has a phase angle that is the negation of the phase angle of AAejθA - A e^{j \theta}, making the argument of the Re() operation simply equal to the length of AAejθA - A e^{j \theta}. The following therefore must be true:

1ejθ2=3(1cosθ)2+(sinθ)2=32cosθ=1    cosθ=12|1 - e^{j \theta}|^2 = 3 \\ (1- cos\theta)^2 + (sin\theta)^2 = 3 \\ -2 cos\theta = 1 \\ \implies cos\theta = -\frac{1}{2}

The two sinusoids are therefore separated by plus or minus 120 degrees. The other timing information should be fairly easy to figure out from here.

Note by Steven Chase
3 years, 7 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link]( link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


There are no comments in this discussion.


Problem Loading...

Note Loading...

Set Loading...