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7 October, 22:42

An engineer is designing a spring to be placed at the bottom of an elevator shaft. If the elevator cable should happen to break when the elevator is at a height h above the top of the spring, calculate the value that the spring constant k should have so that passangers undergo an acceleration of no more than 5.0G when brought to rest. Let M be the total mass of the elevator and passengers.

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  1. 7 October, 22:57
    0
    Value that the spring constant k = 12Mg / h

    Explanation:

    According to 2nd law of Newton:

    upward force of the spring = F

    The weight of the elevator W = mg

    F = Mg = M (5g)

    ==> F = 6Mg.

    As the spring is compressed to its maximum distance ie s, the maximum upward acceleration comes just, Hence

    F = ks = 6Mg

    ==> s = 6Mg/k

    We have gravitational potential energy turning into elastic potential of the spring as the elevator starts at the top some distance h from the spring, and undergoes a total change in height equal to h + s, so:

    Mg (h+s) = 1/2ks2

    And plugging in our expression for s:

    Mg (h+6Mg/k) = 1/2k (6Mg / k) 2

    gh + 6M2g2/k = 1/2k (36M2g2 / k2)

    Mgh + 6M2g2/k = 1/2k (36M2g2 / k2)

    gh + 6Mg2/k = 18Mg2 / k

    gh = 12Mg2 / k

    h = 12Mg / k

    k = 12Mg / h
  2. 7 October, 23:07
    0
    k = 18.0Mg/h

    Explanation:

    Considering the forces acting on the elevator, the weight of the elevator and the restoring force of the spring

    From Newtown's second law of motion,

    Kx - Mg = Ma

    Given that a = 5.0g

    Kx - Mg = M (5.0g)

    Kx = 5.0Mg + Mg

    Kx = 6.0Mg ... (1)

    From the energy consideration

    The potential energy change in falling through the height of h is converted in to the elastic potential energy of the spring

    Elastic potential energy = Gravitational potential energy

    1/2Kx² = Mgh

    Kx² = 2Mgh

    x² = 2Mgh/K

    x = √ (2Mgh/K)

    So substituting x in the equation (1) above

    K * √ (2Mgh/K) = 6.0Mg

    Squaring both sides

    K² * 2Mgh/K = 36.0M²g²

    K * 2Mgh = 36.0M²g²

    Dividing through by 2Mgh

    We have that

    K = 18.0Mg/h
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