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.

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

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

i just did an experiment like this in my science class. also the answer is yes, it will get separated because they are two opposites meaning if they are opposite they can not attract.

hope this .!