did you guys get this : The x-component of motion of an object is given by x(t) = Axcos(ωxt + φx) and the y-component of motion of the objec

Question

did you guys get this : The x-component of motion of an object is given by x(t) = Axcos(ωxt + φx) and the y-component of motion of the object is given by y(t) = Aycos(ωyt + φy). What relationships between the A,ω, and φ parameters must be true so that the motion of the object is on a circle?

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Ladonna 4 years 2021-09-03T17:49:05+00:00 1 Answers 19 views 0

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  1. Philomena
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    2021-09-03T17:50:23+00:00
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    Answer:

    Aₓ = A_y = A

              wₓ = w_y = w

              Фₓ = Ф_y = Ф

    Explanation:

    For a movement to be circular it must meet the exception of the circle

              R² = x² + y²

    in the exercise indicate the expressions of the movement in the two axes

            x (t) = Aₓ cos (wₓ t + Фₓ)

            y (t) = A_y cos (w_y t + Ф_y)

    we substitute

             R² = Aₓ² cos²  (wₓ t + Фₓ) + A_y² sin²  w_y t + Ф_y)

             

    for this expression to be a circle it must meet

              Aₓ = A_y = A

              wₓ = w_y = w

              Фₓ = Ф_y = Ф

    with these expressions

               R² = A² [cos² (w t + Ф) + sin² (wₓ t + Фₓ) ]

    use the trigonometry relationship

              cos² θ + sin² θ = 1

               

              R² = A²

    Therefore, it is fulfilled that it is a circle whose radius is equal to the amplitude of the movement

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