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Description of a wave

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  1. Sep 4, 2015 #1
    1. The problem statement, all variables and given/known data
    A simple harmonic wave train of amplitude 3 cm and frequency 200 Hz travels in the +ve direction of x-axis with a velocity of 20 m/s. Calculate the displacement, velocity, and acceleration of a particle situated at 50 cm from the origin at t = 2 s.

    2. Relevant equations
    I used [itex]y(x, t) = Acos(2\pi f(\frac{x}{v}-t))[/itex]

    3. The attempt at a solution
    Plugging in the values into the above equation, I got [itex]y(0.5, 2) = 0.03cos(400\pi (\frac{.5}{20} - 2))[/itex], which evaluates to 0.0236 m. However, the book says the answer is 0.02523 m.

    This is marked as an easy question and is one of the first ones, so I think that I'm missing something basic?
     
  2. jcsd
  3. Sep 4, 2015 #2

    haruspex

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    You need to be a bit careful evaluating trig functions of large angles. The series expansions used by calculators get rather inaccurate. Instead, first reduce the angle to something less than 2 pi. I think you'll find both your answer and the given answer rather inaccurate!
     
  4. Sep 4, 2015 #3
    Part of my confusion lies in whether I should be in radians or degrees - the examples in the book all use radians, but when I evaluated the above, I got [itex]cos(-790\pi)[/itex], or 1. Changing the angle to something less than 2[itex]\pi[/itex] still gets me 1.

    When I use degrees, I get 0.0236, which is closer to 0.02523. I plugged the expression into Wolfram-Alpha, which got me the same as the one on my calculator. Changing the angle to something less than 360 degrees got me 0.0271.
     
  5. Sep 4, 2015 #4

    haruspex

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    Definitely radians. The standard equation you quoted assumes radians.
    It gives you 1 for the value of the cos function, but you still have to multiply by A.
     
  6. Sep 5, 2015 #5

    rude man

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    I got the same answer as you.
    BTW the problem should state that the wave is inded ~ cos(kx - wt) and not something like cos(kx - wt + φ), φ ≠ 0.
    Always assume radians. And always assume natural instead of base-10 logs. Calculus falls apart otherwise!
     
  7. Sep 5, 2015 #6

    haruspex

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    Then you must be using a calculator that truncates the precision of pi at the same point. The right answer is clearly 0.03m.
     
  8. Sep 5, 2015 #7

    rude man

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    Not truncate. Round off.
    But yes, score one for the Aussies. Again, only if the wave is cos(kx - wt + φ), φ = 0 assumed. The problem is not clearly stated.
     
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