The cable of an elevator of mass = 5320 kg snaps when the elevator is a rest at one of the floors of a skyscraper. At this point the elevator is a distance = 31.6 m above a cushioning spring whose spring constant is = 11700 N/m. A safety device clamps the elevator against the guide rails so that a constant frictional force of = 18158 N opposes the motion of the elevator. Find the maximum distance by which the cushioning spring will be compressed.

Question

The cable of an elevator of mass  = 5320 kg snaps when the elevator is a rest at one of the floors of a skyscraper. At this point the elevator is a distance  = 31.6 m above a cushioning spring whose spring constant is  = 11700 N/m. A safety device clamps the elevator against the guide rails so that a constant frictional force of  = 18158 N opposes the motion of the elevator. Find the maximum distance by which the cushioning spring will be compressed.

Damon · Accepted Answer

Kinetic energy that the elevator has when it hits the spring is:
KE = m*g*h-work done by friction so far

the work done by friction so far =18158*h

where
m = 5320 kg
g = 9.81 m/s^2
h - 31.6 m

Now we compress the spring a distance x

Work done by gravity in falling additional distance x = m g x
So at the bottom we have
gained m g x additional
so we have a total available of KE + m g x
That amount goes into compressing the spring and doing further work against friction
(1/2) k x^2 + 18158 x

so in the end

m*g*h-18158*h + m g x = (1/2) k x^2 + 18158 x
or
m g (h+x) - 18158 (h+x) = (1/2) k x^2

find the two solutions to the quadratic. Probably only one will make sense.