A mass weighing 32 pounds stretches a spring 2 feet. Determine the amplitude and period of motion if the mass is initially released from a p

Question

A mass weighing 32 pounds stretches a spring 2 feet. Determine the amplitude and period of motion if the mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of 6 ft/s.

in progress 0
RuslanHeatt 5 months 2021-08-22T05:58:17+00:00 1 Answers 67 views 0

Answers ( )

    0
    2021-08-22T06:00:05+00:00

    A mass weighing 32 pounds stretches a spring 2 feet.

    (a) Determine the amplitude and period of motion if the mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of 6 ft/s.

    (b) How many complete cycles will the mass have completed at the end of 4 seconds?

    Answer:

    A = 1.803 ft

    Period = \frac{\pi}{2} seconds

    8 cycles

    Explanation:

    A mass weighing 32 pounds stretches a spring 2 feet;

    it implies that the mass (m) = \frac{w}{g}

    m= \frac{32}{32}

    = 1 slug

    Also from Hooke’s Law

    2 k = 32

    k = \frac{32}{2}

    k = 16 lb/ft

    Using the function:

    \frac{d^2x}{dt} = - 16x\\\frac{d^2x}{dt} + 16x =0

    x(0) = -1        (because of the initial position being above the equilibrium position)

    x(0) = -6          ( as a result of upward velocity)

    NOW, we have:

    x(t)=c_1cos4t+c_2sin4t\\x^{'}(t) = 4(-c_1sin4t+c_2cos4t)

    However;

    x(0) = -1 means

    -1 =c_1\\c_1 = -1

    x(0) =-6 also implies that:

    -6 =4(c_2)\\c_2 = - \frac{6}{4}

    c_2 = -\frac{3}{2}

    Hence, x(t) =-cos4t-\frac{3}{2} sin 4t

    A = \sqrt{C_1^2+C_2^2}

    A = \sqrt{(-1)^2+(\frac{3}{2})^2 }

    A=\sqrt{\frac{13}{4} }

    A= \frac{1}{2}\sqrt{13}

    A = 1.803 ft

    Period can be calculated as follows:

    = \frac{2 \pi}{4}

    = \frac{\pi}{2} seconds

    How many complete cycles will the mass have completed at the end of 4 seconds?

    At the end of 4 seconds, we have:

    x* \frac{\pi}{2} = 4 \pi

    x \pi = 8 \pi

    x=8 cycles

Leave an answer

Browse

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