The cable of the 1800 kg elevator in Figure 8-56 snaps when the elevator is at rest at the first floor, where the cab bottom is a distance d = 3.7 m above a cushioning spring whose spring constant is k = 0.15 MN/m. A safety device clamps the elevator against guide rails so that a constant frictional force of 4.4 kN opposes the motion of the elevator.
Using conservation of energy, find the approximate total distance that the elevator will move before coming to rest.
The Attempt at a Solution
In the beginning, the moment the cable snaps, the total energy of the elevator is its gravitational potential energy. If I want the total energy of the elevator to be only the elastic potential energy when the elevator has compressed the spring as much as it can the first time it falls onto the spring, then I would define the zero point for gravitational potential energy as that point of maximum compression.
With this setup, the total energy that the elevator has is mg(h+x), where h is the 3.7 meters above the equilibrium position and x is the maximum compression of the spring, known by calculation to be 0.901 meters.
All this gravitational potential energy is converted into elastic potential energy and heat energy due to friction. Then all that elastic potential energy is converted into gravitational potential energy and heat energy due to friction. Then all THAT gravitational potential energy is converted into elastic potential energy and heat energy due to friction... I want to find when the elevator is not moving, which is to say, when it has no more energy, except for the gravitational and elastic potential energies it would have from resting on the spring, which would be compressed some but not to the zero gravitational point. So, I figure I could just equate the total energy at the beginning to some gravitational and elastic potential energy and heat energy lost to friction over some distance d.
So I get
mg(h+x) = fd + mgc +(1/2)k(c^2)
where h is the initial height from the equilibrium position of the spring, x is the maximum compression of the spring, f is the force due to friction, d is the total distance traveled by the elevator, and c is the compression of the spring when the elevator is at rest.
SO, I think if I could find c, I could solve the problem. But how could I find c, considering that the elevator never actually is at rest, but continuously oscillates?
Sorry I'm so long-winded, but I wanted to describe my train of logic well. Thanks for reading this!