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3 July, 10:09

A brick of mass 8 kg hangs from the end of a spring. When the brick is at rest, the spring is stretched by 3 cm. The spring is then stretched an additional 4 cm and released. Assume there is no air resistance. Note that the acceleration due to gravity, g, is g=980 cm/s2. Set up a differential equation with initial conditions describing the motion and solve it for the displacement s (t) of the mass from its equilibrium position (with the spring stretched 3 cm). s (t) = 4 cos (sqrt (980/3) t) cm

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  1. 3 July, 10:28
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    s (t) = 0.04cos (sqrt (980/3))

    Step-by-step explanation:

    Find spring constant. Use balance force between weight of brick and spring elastic force when brick at rest:

    Weight of brick = force of spring

    mg = kx

    Mass brick, m = 8kg

    Gravity constant, g = 9.8 m/s2

    Spring elongation, x = 0.03m

    Hence, spring constant, k = mg/x

    = 8*9.8/0.03

    = 7840/3

    Since there is no other external forces, spring acts in simple harmonic motion

    -kx = ma

    a = - kx/m

    note that a is the acceleration which is the double derivative of distance over time. Hence

    d2y/dx2 = - kx/m

    d2y/dx2 + kx/m = 0

    Note that this equation is similar to simple harmonic motion:

    d2y/dx2 + (w2) x = 0

    Comparing these two equations we found:

    w2 = k/m

    Using the values obtained earlier:

    w2 = (7840/3) / 8 = 980/3

    w = sqrt (980/3) = 18.07 rad/s

    Since the movement of the spring will be sinusoidal, similar to the movement of pendulum, we use the general equation for oscillating motion:

    s (t) = A sin (wt + c)

    Note that in initial condition when t=0,

    displacement of spring s = A = 0.04m

    Hence 0.04 = 0.04 sin (0+c)

    sin c = 1

    c = π/2 - hence indicating that the motion is π/2 further than the equation.

    Using trigonometry identity, we know that cos (theta) = sin (π/2 + theta)

    So, we can change sin (wt+π/2) to cos (wt)

    Updating the equation, we'll get

    s (t) = 0.04 cos (wt).

    w = sqrt (980/3)

    Finally,

    s (t) = 0.04cos (sqrt (980/3)) (in m)

    Or

    s (t) = 4cos (sqrt (980/3)) (in cm)
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