The motion of a particle is described by x = 10 sin (πt + π/3), where x is in meters and t is in seconds. At what time in seconds is the pot

Question

The motion of a particle is described by x = 10 sin (πt + π/3), where x is in meters and t is in seconds. At what time in seconds is the potential energy equal to the kinetic energy?

in progress 0
Neala 5 years 2021-08-16T12:57:40+00:00 1 Answers 21 views 0

Answers ( )

  1. thucuc
    0
    2021-08-16T12:59:08+00:00
    Reply

    Answer:

    5/12

    Explanation:

    Given

    x = 10 sin (πt + π/3)

    v = distance/time

    So, v = dx/dt

    Differentiating x with respect to t

    v = 10π cos(πt + π/3)

    Also,

    ½kx² = ½mv²

    Substituting values for x and v in the above equation

    ½k(10sin (πt + π/3))² = ½m(10πcos(πt + π/3))²

    Divide through by ½

    k(10sin (πt + π/3))² = m(10πcos(πt + π/3))²

    Open both bracket

    100ksin²(πt + π/3) = 100mπ²cos²(πt + π/3)

    Divide through by 100

    ksin²(πt + π/3) = mπ²cos²(πt + π/3)

    Divide through by kcos²(πt + π/3)

    ksin²(πt + π/3) ÷ kcos²(πt + π/3) = mπ²cos²(πt + π/3) ÷ kcos²(πt + π/3)

    tan²(πt + π/3) = mπ²/k

    tan²(πt + π/3) = (m/k)π²

    But w² = k/m and w = 2π/T

    (2π/T)² = k/m

    (2π)²/T² = k/m

    1/T² = k/m ÷ (2π)²

    1/T² = k/m*(2π)²

    T² = m(2π)²/k

    From the Question, T is when πt = 2π or T = 2

    Substitute 2 for T in the above equation

    2² = m(2π)²/k

    4 = m(2π)²/k

    4 = 4π²m/k

    m/k = 1/π²

    (m/k)π² = 1

    Remember that tan²(πt + π/3) = (m/k)π²

    So, tan²(πt + π/3) = 1

    This gives

    πt + π/3 = 45° = π/4

    πt + π/3 = π/4

    Divide through by π

    t + ⅓ = ¼

    t = ¼ – ⅓

    t = -1/12 — Negative

    Using the second quadrant

    πt + π/3 = 3π/4

    Divide through by π

    t + ⅓ = ¾

    t = ¾ – ⅓

    t = 5/12

Leave an answer

Browse

Giải phương trình 1 ẩn: x + 2 - 2(x + 1) = -x . Hỏi x = ? ( )