An elevator cable breaks when a 925 kg elevator is 285 m...

An elevator cable breaks when a 925-kg elevator is 28.5 m above the top of a huge spring $(k = 8.00 \times 10^4 N/m)$ at the bottom of the shaft. Calculate ($a$) the work done by gravity on the elevator before it hits the spring; ($b$) the speed of the elevator just before striking the spring; ($c$) the amount the spring compresses (note that here work is done by both the spring and gravity).

Transcript

00:01 Okay, everyone, this is question number 76 from chapter 6.

00:03 This problem is about an elevator falling, and there's a spring at the bottom of the elevator shaft.

00:10 All right, so part a asks us to find the work done by gravity.

00:14 So work equals force times distance, force of gravity equals m g and times d, and then d is going to have an angle of cosine, 0.

00:35 Gravity's pulling down, the motion is down, cosine 0 is 1, so that makes sense.

00:40 So then the work of gravity equals mass, which is 925 kilograms, times g, 9 .8 meters per second squared, times distance, which we're given is 28 .5 meters.

01:00 Plug all that in and you get work equals 2 .583 times 10 to the fifth joules which we can round or we can use sigfigs with to get 2 .58 times 10 to the fifth joules.

01:31 That's part a, work done by gravity.

01:33 All right, part b asks the speed just before the elevator hits the spring.

01:41 So to do that, work equals kinetic, change of kinetic energy.

01:48 And we're going to assume that this, that the elevator dropped from rest.

01:57 So we have work equals final minus initial, one half, and the, 2 squared minus 1 half mv1 squared saying initial velocity is 0 it's dropping from rest so we have work we have all the rest of this we can solve for v2 or for v so v so v right before it's the spring equals a square root of 2 w over m plug in numbers square root 2 2 times 2 0 .585 835 once a three times 10 to 5th all over 925 kilograms plug all that into your calculator and you get 23 .6 meters per second alright so the last part of this question asks us to find the amount that the spring is compressed.

03:19 So in order to do that, we need to think about what two things are doing work here.

03:24 All right? so work in this equation is done by gravity, which is down in the direction of motion, which is down in the direction that the spring is going to compress.

03:36 Okay.

03:37 And then we also have work done by the spring, and the spring is going to be pushing back up against.

03:45 Back up against the spring is going to be pushing back up against the elevator in the opposite direction of motion.

03:55 So it's going to be minus work of the spring.

04:02 And by work energy theorem, the net work done has to be equal to zero because the object, the elevator that we're talking about ends at rest.

04:17 Okay, so zero equals wg minus ws.

04:24 All right? so we can go ahead and start solving here.

04:30 Wg we just found is going to be equal to mg.

04:38 So zero equals mg...