00:01
Okay, we have a mass that's resting on a frictionless table and it's accelerated by a spring.
00:08
So let's draw this real quick.
00:11
Here's our table, there's our spring, and our mass m.
00:18
There's like a rough patch or something over here.
00:20
We'll color it blue.
00:23
Um, mass is 11 kilograms.
00:26
The spring constant is 4 ,119 newton's per meter.
00:38
Mass is 11 kilograms.
00:43
Okay, the coefficient of kinetic friction is 0 .53.
00:53
And we are told that the mass leaves the spring at 3 .9 meters per second.
00:59
How much work is done by the spring as it accelerates the mass? okay, so what kind of principles are we going to use here? what we're dealing with work, so anything to do with energy is probably a good bet for this kind of problem.
01:20
Work and energy go hand in hand.
01:24
Kind of the same thing.
01:26
That's a way to think about it.
01:27
So let's use the work kinetic energy theorem, which tells us that the network on something equal to its change in kinetic energy.
01:41
In other words, 1 .5m times its final velocity squared, minus 1 .5m times its initial velocity squared.
01:54
Okay.
01:55
Now, we're just looking at the block being accelerated by the spring.
02:00
So the initial velocity is going to be where, when the block has been compressed.
02:11
And the final velocity is the velocity that it leaves the spring with.
02:17
So in the problem, they give you a final speed, v -final is 2 .1 meters per second, this is for part 3.
02:23
That is not the speed we want.
02:25
This is the speed it leaves the block with.
02:29
Sorry, this is the speed it leaves the spring with.
02:32
So that's what our final velocity there is.
02:34
Okay, it has no initial velocity because it's been compressed.
02:39
So the initial velocity is talking about when the block is being compressed by the spring.
02:50
Or the spring is being compressed by the block, sorry.
02:54
And the final velocity is when it's leaving the spring.
02:58
So let me just draw this out to make sure i didn't confuse you.
03:05
Okay, so here's the block.
03:08
It's compressed the spring.
03:12
And now here's the block, it's leaving the spring.
03:17
So this is the initial case, this is the final case.
03:22
So initially what's the velocity? zero.
03:24
And for the final, what's the velocity? 3 .9 meters per second.
03:30
Okay, so the network is just going to be the work done by the spring.
03:41
So that's the only thing doing work here, since we have a frictionless surface.
03:47
So the work done by the spring is just one half m times v final squared.
03:56
V final being the speed it leaves the spring.
04:01
Okay, so this is 83 .7 joules.
04:12
So 83 .7 joules of work.
04:16
Okay.
04:21
How far was the spring stretched from its unstretched length? and i think, unless i'm just looking at this problem incorrectly, i think when it means stretched, it's talking like about how far it's been compressed, right? so you have the natural length of a spring.
04:41
So here's our spring.
04:43
No one's pulling it.
04:44
It's this long.
04:45
But if you compress it, that distance x, that's the distance we're looking for.
04:53
How far has this spring been displaced from its unstretched length? um, yeah.
05:07
Okay, so to solve this one, we can use the fact that the work done by a spring is equal to one half kx squared.
05:26
So that's how much work it takes to compress the spring.
05:32
Right? so if you want to take this spring, it's unstretched, then you want to compress it...