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Math Help - Simple Harmonic Motion

  1. #1
    DBA
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    Simple Harmonic Motion

    Hello, I need help with the following:

    The motion of a spring-mass system is described by the equation y=2sin(pi*t - pi/2) where y is the distance in feet from equilibrium position and t is time in seconds.

    a. If the weight is 24 feet from the ceiling in a state of equilibrium, find the starting position of the weight (time=0).

    b. Find the position of the mass at 3 seconds.

    c. What is the first time the mass will reach equilibrium position?

    Thanks
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  2. #2
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    Hello DBA

    Welcome to Math Help Forum!
    Quote Originally Posted by DBA View Post
    Hello, I need help with the following:

    The motion of a spring-mass system is described by the equation y=2sin(pi*t - pi/2) where y is the distance in feet from equilibrium position and t is time in seconds.

    a. If the weight is 24 feet from the ceiling in a state of equilibrium, find the starting position of the weight (time=0).

    b. Find the position of the mass at 3 seconds.

    c. What is the first time the mass will reach equilibrium position?

    Thanks
    I assume that y is measured positive upwards, so y represents the height above the equilibrium position, which is 24 feet below the ceiling. (Your statement of the question says that y is the distance from the equilibrium position, which is not the same thing!)

    (a) When t = 0, y = 2\sin(0 - \pi/2) = -2. On the assumption I've made above, this means that the mass is 2 feet below the equilibrium position. So the starting position of the mass is 26 feet below the ceiling.

    (b) When t = 3, y = 2\sin(3\pi - \pi/2) = 2\sin(5\pi/2) = 2. So the mass is 24-2=22 feet below the ceiling at t=3.

    (c) When y = 0, \sin(\pi t - \pi/2) = 0

    \Rightarrow \pi t - \pi/2 = 0, \pi, 2\pi, ...

    \Rightarrow t = \tfrac12, \tfrac32, \tfrac52, ...

    \Rightarrow the first time the mass is in the equilibrium position is after \tfrac12 sec.

    Grandad
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  3. #3
    DBA
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    Hello, thank you very much.
    The solution for part a must be 22 feet. Can you explain what the different between the two statements is (the one you assumed and the one given in the text)?

    It seems that I have to substract the -2 from the 24feet and I do not understand why.

    Can you please help with that.
    Thanks
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  4. #4
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    Hello DBA
    Quote Originally Posted by DBA View Post
    Hello, thank you very much.
    The solution for part a must be 22 feet. Can you explain what the different between the two statements is (the one you assumed and the one given in the text)?

    It seems that I have to substract the -2 from the 24feet and I do not understand why.

    Can you please help with that.
    Thanks
    Yes, as I said at the beginning, I am assuming that y is measured positive upwards. If it's positive downwards, then the value of y must be added to the distance of 24 feet to get the total distance of the mass from the ceiling.

    So in part (a) the distance initially is 24 + (-2) = 22 feet, and in part (b) the distance when t = 3 is 24 + 2 = 26 feet.

    Read the question you were given again, and study the wording carefully. It should make it clear which way y is to be measured.

    Grandad
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