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10 August, 12:05

A hollow steel ball weighing 4 pounds is suspended from a spring. This stretches the spring feet. The ball is started in motion from the equilibrium position with a downward velocity of feet per second. The air resistance (in pounds) of the moving ball numerically equals 4 times its velocity (in feet per second). Suppose that after t seconds the ball is y feet below its rest position. Find y in terms of t.

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  1. 10 August, 12:29
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    The solution when after t seconds the ball is y feet below its rest position is y (t) = 3 et ^⁻16

    Step-by-step explanation:

    Solution

    Given that:

    The hollow steel ball weight (W) = 4 pounds

    We take acceleration, g. = 32 ft/sec

    The mass m = W/g = 4/32

    =1/8 slug

    The damping constant β = 4

    The starting displacement, y (0) = 0

    The initial velocity y' (0) = 3

    The ball stretches the spring = 1/8 feet

    Now

    From Hooke's Law, F = ks

    here,

    k = constant spring

    so,

    4 = k (1/8)

    k = 32

    Now from Newton's law,

    my'' = - ky - βy'

    = (1/8) y'' = - 32y - 4y'

    y'' = - 256 y - 32 y'

    y'' + 32y' + 256 = 0 this is the equation (1)

    Thus,

    we solve for DE (1)

    The equation (auxiliary) is m² + 32m + 256 = 0

    so,

    m² + 16m + 16 m _ 256 = 0

    m (m + 16) + 16 (m + 16) = 0

    (m + 16) (m + 16) = 0

    m₁ = - 16,

    m₂ = - 16

    In this case, the root is known as repeated roots

    Now,

    The solution is y (t) = c₁e^⁻16 t + c₂ te^⁻16 t

    Now, we make use of the initial condition y (0) = 0

    y (0) = c₁e^⁻16 (0) + c₂ (0) e^⁻16 (0)

    0 = c₁ e^0 + 0

    c₁ = 0

    Again, we apply the initial condition y' (0) = 3

    y' (t) c₁e^⁻16 t (-16 ( + c₂ (te^⁻16 t (-16) + e^-16)

    y' (t) = - 16c₁e^⁻16 t + c₂ (-16te^ ⁻16 + e^-16)

    y' (0) = - 16c₁e^⁻16 (0) + c₂ (-16 (0) e^ ⁻16 (0) + e^-16 (0))

    3 = 16c₁ + c₂ (1)

    So,

    3 = - 16 (0) + c₂ (1)

    c₂ = 3

    Therefore, the required solution is y (t) = (0) e^⁻16 + 3 et ^⁻16

    y (t) = 3 et ^⁻16
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