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Math Help - Phase difference

  1. #1
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    Phase difference

    Hi
    Please, I am not sure about this problem. It reads this way:
    Two identical pendulums X and Y each consist of a small metal sphere attached to a thread of a certain length. Each pendulum makes 20 complete cycles of oscillation in 16s. State the phase difference in radians, between them motion of X and that of Y if

    (a) X reaches maximum displacement at the same time as Y reaches maximum displacement in the opposite direction.

    I know the phase difference is calculated by
    Phase difference =360 X Change in t/Time period. (Or 2pi in radians)

    I am able to establish the time period as 0.8s. (16/20). So, for the equilibrium positions to maximum displacement (Both negative and positive directions ) it will take 0.8s. But I got stuck from here. I am not sure how to calculate the change in time. I am tempted to say 0.8s. But that seems to imply the phase difference is zero.

    Please comments
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  2. #2
    MHF Contributor Unknown008's Avatar
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    Well, if you can visualise the two oscillating pendulum, how are you measuring 1 oscillation?

    It can be taken, in this problem, as when the bob starts from one maximum, goes to the opposite maximum and back to the initial maximum.

    In your problem, Y is at the opposite maximum.

    Can you deduce the phase difference?


    Or... let's say if you were to release the two bobs so that they oscillate as in the problem, but starting from the same point. You would obviously hold both at the same maximum point, let Y go first, and when Y reaches the other maximum, you let go X. How much of a cycle did Y go through before X is released?
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  3. #3
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    Quote Originally Posted by Unknown008 View Post
    Well, if you can visualise the two oscillating pendulum, how are you measuring 1 oscillation?

    It can be taken, in this problem, as when the bob starts from one maximum, goes to the opposite maximum and back to the initial maximum.

    In your problem, Y is at the opposite maximum.

    To the latter part of yout question, Y has gone through half a cycle when I released X. Regarding the first part, Y would have completed half a cycle when X is at the maxim opposite. In terms to phase difference, It is looking like pi/2 + pi/2. They have both respectively gone through half a cycles at the opposite maximums(Negative and positive directions)Therefore the total phase difference will be Pi. Is that correct?

    Can you deduce the phase difference?


    Or... let's say if you were to release the two bobs so that they oscillate as in the problem, but starting from the same point. You would obviously hold both at the same maximum point, let Y go first, and when Y reaches the other maximum, you let go X. How much of a cycle did Y go through before X is released?
    They would have half cylcles at their respective maximums. So, they will be out of phase by pi/2 + pi/2. The total phase difference will be pi. Is that correct?
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  4. #4
    MHF Contributor Unknown008's Avatar
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    No, you consider only one relative to the other.

    So, the phase difference between X and Y is pi/2

    There is no total phase difference.

    EDIT: Change pi/2 to pi.
    Last edited by Unknown008; January 18th 2011 at 08:14 PM.
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    Quote Originally Posted by Unknown008 View Post
    No, you consider only one relative to the other.

    So, the phase difference between X and Y is pi/2

    There is no total phase difference.
    Hmm! But the question says they are both at maximum displacement. If X is at the equilibrium position and Y is at maximum? What is the phase difference? I thought that was pi/2.
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  6. #6
    MHF Contributor Unknown008's Avatar
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    No, there are two maximum displacements, one positive and one negative. Which is which does not matter here, it's just that they are different maxima.

    If now X were to be at equilibrium, it will depend on the velocity of X at equilibrium. If X is following Y, then Y is ahead of X by a phase difference of pi/4, otherwise, if instead Y is following X, then X is ahead of Y by pi/4.

    EDIT: Change pi/4 to pi/2
    Last edited by Unknown008; January 18th 2011 at 08:14 PM.
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  7. #7
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    Quote Originally Posted by Unknown008 View Post
    No, there are two maximum displacements, one positive and one negative. Which is which does not matter here, it's just that they are different maxima.

    If now X were to be at equilibrium, it will depend on the velocity of X at equilibrium. If X is following Y, then Y is ahead of X by a phase difference of pi/4, otherwise, if instead Y is following X, then X is ahead of Y by pi/4.
    Okay. Thanks. Let me chew on this.
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  8. #8
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    Is a full cycle 2pi? I am looking at the sine curve. If that is correct, how can half a cycle be pi/2? Please explain.
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  9. #9
    MHF Contributor Unknown008's Avatar
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    Oh, sorry for my earlier post, I got myself messed up into cycles and radians, yes, the phase difference is pi.

    Again, sorry for the confusion, I was initially thinking 1 cycle -> pi

    You see, when you released the sphere from a maximum, you aren't forced to see it as a sine curve, but you can see it also as a cos curve, and from there, you see immediately that there is a half cycle difference between the first maximum and the second maximum, no need to find quarter of a cycle and add to another quarter of a cycle.
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  10. #10
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    Quote Originally Posted by Unknown008 View Post
    Oh, sorry for my earlier post, I got myself messed up into cycles and radians, yes, the phase difference is pi.

    Again, sorry for the confusion, I was initially thinking 1 cycle -> pi

    You see, when you released the sphere from a maximum, you aren't forced to see it as a sine curve, but you can see it also as a cos curve, and from there, you see immediately that there is a half cycle difference between the first maximum and the second maximum, no need to find quarter of a cycle and add to another quarter of a cycle.
    Thats okay. PHEW! I was getting worried.(lol) Thanks a lot.
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