In a worst case design scenario a 2000 kg elevator with...

In a "worst-case" design scenario, a 2000 kg elevator with broken cables is falling at 4.00 m/s when it first contacts a cushioning spring at the bottom of the shaft. The spring is supposed to stop the elevator, compressing 2.00 m as it does so. The spring coefficient is 10.6 kN/m. During the motion, a safety clamp applies a constant 17000-N frictional force to the elevator. 1. What is the speed of the elevator after it has moved downward 1.00 m from the point where it first contacts a spring? 2. When the elevator is 1.00 m below the point where it first contacts a spring, what is its acceleration?

Transcript

0:00 Hi there.

00:01 So for this problem, we are told that in a worst case design a scenario, a 2 ,000 kilograms elevator with a broken cable is falling at a speed of 4 meters per second.

00:13 So we are given its mass 2 ,000 kilograms and the speed is also given.

00:20 This corresponds to the initial speed, is 4 meters per second.

00:25 And when it first contacts cohesion in a spring at the bottom of the shaft.

00:34 The spring is supposed to stop the elevator compressing two meters as it does so.

00:40 So the distance is going to be two meters first.

00:45 So the spring is supposed to stop the elevator compressing two meters.

00:50 Okay, so that sets.

00:52 So the spring coefficient is also given and that is equal to 10.

00:56 10 .6, kilo -neutoms per meter, where kilo, you know, you means 10 to the 3.

01:04 During the motion of state if calum applies a force, a frictional force.

01:10 And that frictional force is equal to 17 ,000 newtoms to this elevator.

01:18 So for part one of this problem, we are asked about what is the speed of the elevator after it has moved downward 1 meters from the point where, it first contacts the spree.

01:32 So with that said, we need to find the speed when the spring is compressed by a distance of one meter.

01:43 So what we need to do is to apply the work energy theorem, so we will have that the word done by the gravitational force plus the war done by the frictional force plus the war done by the spring is equal to the change in the kinetic energy.

01:57 Where we know the change in the kinetic energy is 1 divided by 2 times the mass, times the final speed square minus the initial speed square.

02:04 Now we know that the final speed is, is, oh sorry, the final speed is the one that we need to determine.

02:13 And yeah, so what we need to do is to simply substitute all of the values and then solve for the, well, let's put the definitions in here.

02:27 So the work done by the gravitational force, that is the mass times the acceleration due to gravity times the distance adds.

02:34 The word -done by the frictional force is just simply, well, in this case, it is negative because it opposes the direction of motion, and that is the frictional force given times the distance x.

02:47 This minus the force of the spring that also poses the motion of this, which is 1 divided by 2 times the springs constant, times the distance adds to the square, and then this is equal to just simply this in here.

03:02 So we can simplify this further by just dividing everything by the mass and also multiplied everything by 2.

03:12 So we will have 2 times the acceleration due to the gravity times the distance x minus the frictional force and times the distance x divided by the mass.

03:24 This minus the spring constant divided by the mass times the distance x to the square.

03:31 So that is equal.

03:33 So, yeah, so that is the difference between these speeds right here.

03:41 But we can pass this initial speed to the other side now.

03:46 So we will have plus the initial speed square.

03:49 So that will be the final speed square.

03:51 So let's first calculate this value to the square.

03:54 So that will be two times acceleration 9 .8 meters per second square times the distance x, that is 2 meters...