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Linear & Angular Velocity Related Rates

  1. Dec 24, 2015 #1
    1. The problem statement, all variables and given/known data
    A rotating beacon is located 1 kilometer off a straight shoreline (see figure). If the beacon rotates at a rate of 3 revolutions per minute, how fast (in kilometers per hour) does the beam of light appear to be moving to a viewer who is 1/2 kilometer down the shoreline?

    2. Relevant equations


    3. The attempt at a solution
    Okay so I used the equation w = 2πf where w is the angular velocity and f is the frequency of rotation. I implicitly differentiated with respect to time, then plugged in 3 rev/min for df/dt, then converted to km/hr, and got 360π km/hr. The answer is 450π km/hr. I think my units are off somewhere or I'm using the wrong equation but I'm not sure.
     

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  3. Dec 24, 2015 #2

    Student100

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    What was the distance you used in the conversion? Can you type out your work step by step?
     
  4. Dec 24, 2015 #3
    d/dt[w] = d/dt[2πf]
    dw/dt = 2π(df/dt)
    dw/dt = 2π(3 rev/min) = 6π km/min = 360π km/hr
     
  5. Dec 24, 2015 #4

    Student100

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    What about the distance from the shore to the light house? How are you going from revs to km?
     
  6. Dec 24, 2015 #5
    I honestly don't know how to relate the units of revolutions to kilometers, that's what I'm not understanding, like they give me the frequency of revolutions of the beacon but I don't know what to do with that to end up getting kilometers/hour.
     
  7. Dec 24, 2015 #6

    Student100

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    First use your trig relationship to find the distance from the observer on the shore to the beacon house.
     
  8. Dec 24, 2015 #7
    Okay it's sqrt(5)/2 but what do I do with it? I've looked up angular velocity equations and they all have like 1 too many variables that I don't have enough information on.
     
  9. Dec 24, 2015 #8

    LCKurtz

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    Let the lighthouse be ##Q##, let ##P## be the point on the shore nearest the lighthouse, let ##x## be the distance from ##P## to where the spotlight hits the shore, and ##\theta## be the angle of rotation from ##QP##. What you need to calculate is ##\frac{dx}{dt}##. You don't need the distance from the observer on shore to the beacon house to work this problem.
     
  10. Dec 24, 2015 #9
    Okay I got it now, thank you!
     
  11. Dec 24, 2015 #10
    If you don't mind, could you tell me how you ended up solving this problem? I was trying it myself but I'm not sure if I have the right equation to differentiate or if I'm using the rate of the rotating beacon light correctly.
     
  12. Dec 24, 2015 #11

    LCKurtz

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    Show us what you did if you want advice or help.
     
  13. Dec 25, 2015 #12
    Okay so based off your guidance in the last post, I basically sorted out what was given and what we want. You mentioned that the hypotenuse or the length of the beam isn't necessary for this problem so I decided to use the equation tan(Θ) = (x)/1
    From implicitly differentiating with respect to time, I got the equation sec^2(Θ)*dΘ/dt = dx/dt
    Since we know that dr/dt(rate of change of revolution of beacon) = 3 revs/min, I used dimensional analysis to get that dΘ/dt at that moment is 6π rad/min, but then I realized I definitely went wrong somewhere with my related equation since I would still need to know the length of the beam of light and even when I did, I got a pretty small number. I'm stuck on exactly how you have to deal with the dr/dt and what the best equation to relate these rates would be for this specific problem.

    Thanks in advance!
     
  14. Dec 25, 2015 #13

    Samy_A

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    You are almost there.
    Hint: ##1+\tan^²(\theta)=\sec²(\theta)##
     
  15. Dec 25, 2015 #14

    LCKurtz

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    That would be ##\frac{d\theta}{dt}##, and it is the rate of revolution, not the "rate of change" of it. And don't forget the final units requested in the problem are kilometers/hour for ##\frac{dx}{dt}##.

     
    Last edited: Dec 26, 2015
  16. Dec 25, 2015 #15
    Wow I can't believe I haven't thought of using this identity for these types of related rates problems! Thanks for the tip.

    Ah so I just forgot to have ##\frac{d\theta}{dt}## in the proper units before plugging in the numbers into the differentiated equation! Thank you for pointing that out! I haven't seen a related rates problem like this so I guess I was overthinking about it.
     
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