The cable of an elevator of mass M = 1900 kg snaps when the elevator is at rest at one of the floors of a skyscraper. At this point the elevator is a distance d = 10.6 m above a cushioning spring whose spring constant is k = 8500 N/m. A safety device clamps the elevator against the guide rails so that a constant frictional force of f = 14499 N opposes the motion of the elevator. Find the maximum distance by which the cushioning spring will be compressed.

net force down = m g - 14499

= 1900 (9.81) - 14499
= 4140 Newtons

Work done by downward force = 4140(10.6+x)Joules

Energy stored in spring = (1/2) k x^2
so
4140(10.6+x) = 4250 x^2

4250 x^2 - 4140 x - 43884 = 0

x^2 - .974 x - 10.3 = 0

x = [ .974 +/- sqrt (.949+41.2) /2
= [ .974+/-6.49 ]/2
= 3.73

Thank you sooo much!

To find the maximum distance by which the cushioning spring will be compressed, we can use the principle of conservation of mechanical energy.

First, let's calculate the gravitational potential energy stored in the elevator when it is at a height of 10.6 m above the cushioning spring. The formula for gravitational potential energy is given by:

PE_gravity = m * g * h

where m is the mass of the elevator, g is the acceleration due to gravity, and h is the height above the cushioning spring.

Substituting the given values, we have:

PE_gravity = 1900 kg * 9.8 m/s^2 * 10.6 m

Next, let's calculate the work done by friction in opposing the motion of the elevator. The work done by friction can be calculated using the formula:

W_friction = f * d

where f is the frictional force and d is the distance the elevator is displaced.

Substituting the given values, we have:

W_friction = 14499 N * 10.6 m

Now, let's calculate the potential energy stored in the cushioning spring when it is compressed. The formula for potential energy stored in a spring is given by:

PE_spring = 0.5 * k * x^2

where k is the spring constant and x is the displacement of the spring from its equilibrium position (maximum compression in this case).

Finally, since the elevator is at rest, the total mechanical energy is conserved. Therefore, the sum of the potential energies of the elevator and the spring should be equal to the work done by friction:

PE_gravity + PE_spring = W_friction

Solving for x:

0.5 * k * x^2 = PE_gravity + W_friction

Substituting the known values for k, PE_gravity, and W_friction, we can solve for x.