00:01
You've given two problems to answer here, both dealing with energy conservation and non -conservative forces and conservative forces.
00:10
So let's look at the first question here.
00:12
You have a box going down incline on rollers.
00:16
The rollers, there's static friction there, but does not work on the box.
00:21
That's why they show it like that.
00:24
So we don't have to worry about frictional force doing work.
00:30
But between what i marked c and b, there is kinetic friction.
00:34
So there will be work being done here.
00:36
So we'll take each interval one at a time.
00:40
In problems like this, we have change in height.
00:42
You always want to mark your gravitational potential energy zero.
00:45
So i made it to ground.
00:48
Our goal in this problem is to find the distance it travels on the horizontal till it's at rest.
00:55
So let's look at the different intervals, a to c.
01:02
E is equal to a constant.
01:10
Since w and c, ac, the work done by non -conservative forces, nc, non -conservative, is zero.
01:16
Like i said, you'll understand if you've not talked about rolling yet, you'll understand better about that concept when you talk about rolling and friction, static friction and so on later on in your course.
01:29
So they can be non -conservative forces present, but as long as they don't do any work, the mechanical energy is constant.
01:38
So let's look at this realm.
01:40
1 half mva squared, that's kinetic energy at a, mgya.
01:46
1 half mvc squared plus mgyc.
01:54
Yc is zero, so at the ground, and we can cancel out the mass.
02:02
So i get from this, vc squared is equal to va squared plus 2gya.
02:10
Could you calculate that if you wanted to? certainly.
02:13
Remember, g, they're giving things in feet.
02:15
G is 32 .2 feet per second squared, not 9 .8 meters per second squared.
02:20
But i'm going to leave that to the final formula for my calculation.
02:25
C to, but you do whatever you're comfortable with.
02:27
C to b, now there is kinetic friction.
02:30
So e is not equal to a constant, since wnccb not equal to zero.
02:47
So wnccb is equal to change in mechanical energy, eb minus ec.
02:55
You might say, wait a second, i thought work was equal to change in kinetic energy.
03:00
That's the total work.
03:02
Here i've already segmented off.
03:05
Inside of e is already the potential energy, which is the negative of the work done by, the change in potential energy is the negative of the work done for conservative force.
03:18
So it's already in here.
03:20
So i'm left with just a non -conservative work term.
03:24
That's how we get to this.
03:25
It's one step farther from the work -kinetic -energy theorem, which is the total work equals change in kinetic energy.
03:30
So we can put this all in, 1 half m vb squared plus mgyb minus 1 half m vc squared plus mgyc.
03:49
Ok, what's zero? vb is zero.
03:52
Yb is zero.
03:54
Yc is zero.
03:55
So this just becomes minus 1 half m vc squared, which is minus 1 half m va squared plus 2gya.
04:09
That's our formula for the work, i mean what the work is equal to.
04:18
So now let's calculate the left -hand side.
04:22
Wnccb, work done by kinetic friction, the magnitude of it, magnitude of the displacement, cosine of the angle between them when they are tail to tail, which is 180 degrees.
04:36
Here is fk, here is the displacement, this angle is 180 degrees.
04:45
So this is equal to minus fkd, which is equal to minus mu knd.
04:57
Now remember the free body diagram on the flat, n is straight up, mg straight down, and kinetic friction is to the left.
05:09
So n is just equal to mg in this case.
05:12
The net force in y is zero, so n is equal to mg, has to be.
05:16
So this becomes minus mu kmgd.
05:22
Now be careful, they did not give you, if you were going to try to calculate anything, they did not give you m, they gave you the weight of the box.
05:29
If you wanted m, you got to do the weight divided by, 5 pounds is the weight, 5 pounds divided by 32 .2 to get the mass.
05:37
But we don't need to as you're going to see.
05:40
But be aware of that, you'd have to do m divided by g to get the actual mass.
05:46
But we do know mg, but over here you wouldn't have to have m by itself.
05:51
You're going to see what happens.
05:52
Let's put it all together, let's put the left and the right together.
05:57
Minus mu kmgd minus 1 half m, va squared plus 2 gya.
06:08
Okay, solving for d, but notice m's go away.
06:15
Never needed it, never needed it.
06:18
So d is equal to 1 over 2 mu kgva squared plus 2 gya.
06:27
And we know everything on the right, so let's put everything in.
06:31
1 over 2, 0 .4 is mu k, 32 .2 feet per second squared is g, 10 feet per second is va, got to square that though, plus 2, 32 .2 feet per second squared times 4 feet.
06:55
And this works out to be 13 .88 feet.
07:04
And that's what you actually, in your poster of your answer, you actually listed that as your, as the answer to this question.
07:13
And there it is.
07:14
That's how we get, how you get that.
07:17
Okay, that's 14 .8, 14 .9.
07:20
I got it written down here.
07:22
We have kinetic friction again on the surface, throughout the whole surface.
07:28
And this time though we have a spring and i've, we're given the vastic constant, the spring constant of the spring.
07:36
So in this picture here, i'll call this picture a, frame a, because now you got not only a mass, but you also got a spring.
07:43
So it's really a frame.
07:45
We have the mass moving 8 meters per second.
07:50
The spring is 8 meters away.
07:53
It's unstretched, uncompressed.
07:56
In frame b, you've now compressed the spring and you're at, and the mass is at rest.
08:04
The amount you're compressing is s sub b.
08:07
I don't want to use x because x is what we're looking for, the total distance it travels from, from this, in frame a to when it's at rest in frame b.
08:17
So i don't want to use x in two places.
08:21
Okay, so can we use mechanical energy conservation? we cannot.
08:28
It is not equal to a constant since wnc is not equal to zero.
08:40
So we got to do what we just did.
08:42
Wnc is equal to eb minus ea.
08:49
And put everything in, 1 half m vb squared plus 1 half k sb squared minus mechanical energy in this frame, 1 half m va squared plus 1 half k sa squared.
09:12
Now what's zero? vb is zero.
09:16
Va is not zero.
09:17
Sa is.
09:18
The spring is not compressed or stretched in frame a...