00:01
Once again welcome to a new problem.
00:04
This time we have an inclined plane that is twenty four degrees from the horizontal.
00:12
There's a package on top of the incline on the pocket.
00:16
Has a weight mg.
00:19
The components of the way include mg co signed ada.
00:25
Remember, data is the angle between the weight on this component and then also mg sign off data, the packages sliding downwards.
00:40
So there's a frictional force, which you're going to call f k opposing the motion.
00:48
The normal force is equivalent to mg call sign ada.
00:53
The normal force is based on action and reaction forces.
00:56
Newton's third law that's a zach shin in reaction force is equal and opposite, so the frictional forces mu k time's end.
01:10
Remember mu k? he's given us zero point three one zero.
01:17
Also, the moss off the object happens to be five point zero zero kilograms.
01:24
No, that's the mass of the object that's gonna slide down once on inclines slow that happens to be in a land of displacement.
01:37
Two point eight meters.
01:39
I can call that d d equals to two point eight readers.
01:44
So these are the facts of the problem and we need to find the work done by friction.
01:51
So that's the fast part.
01:53
We're doing the work done by fiction.
01:56
Uh, remember, in general, from your previous energy foundations, walk has given us the dark product ofthe force times displacement, which is the same as force times displacement co.
02:13
Sign of alfa alfa happens to be the angle between so alfa is the hungry between the force in question and the direction of motion.
02:46
That's what alpha is in.
02:48
Alpha is going to show up in multiple aspects ofthe the problem.
02:51
So the fast part is we want to get the work done by friction.
02:56
And so that's simply the frictional force times the dot product displacement, the frictional forces mu k and the cosine alpha.
03:13
Then, from this part, you can see what emi annie's, annie's mg co sign data.
03:19
So this becomes mu k uh m g call sign data times the displaced mench times cause i now, for the last part of this problem right here is with yourself to plug in the numbers zero point three one times the mass, which is five point zero zero kilograms times acceleration due to gravity, which is nine point eight meters per second squared times co.
03:55
Sign off data data was given us twenty fours.
03:57
You can recall this was data co sign off twenty four degrees times the displacement, which is two point eight meters and then times co sign off alfa alfa happens to be one aide in this case, and the reason for that is think about friction is facing that way upwards the incline on dh, then the packages moving down with.
04:25
So this angle between these two happens to be one eighty degrees on.
04:30
Remember, co sign off one eighty is just one.
04:34
It's going to make our life easier on that.
04:37
If you plug in these numbers, you get actually co sign off one eighty is negative one.
04:43
Not one negative wants.
04:45
Your final answer will be negative.
04:48
Eighty eight point eight nine jewels.
04:52
No, that's the walk down or the energy expanded in moving the package energy expended by friction.
04:59
The second part of the problem, which i'm going to do in the next page, eyes asking for the walk, none by gravity again.
05:08
We have the incline, not the pocket.
05:11
Jj, this is gravity mg.
05:14
We could still put the components of gravity.
05:18
The vertical component is mg ho.
05:22
Sign off data and the supposed by the normal force is the horizontal component that's moving.
05:30
The pakistan was that's empty science data, which is being opposed by the frictional force f k.
05:37
The angle is twenty four degrees.
05:41
The other critical aspect ofthe part b is that these an angle between the weight and the direction of motion on that's our fight happens to be ninety minus the the twenty four degrees, which is the angle that the incline relative to the horizontal that sixty six degrees so going back to the energy work formula.
06:11
So what done by gravity becomes the force of gravity, which is empty times the distance, that the object is going to move all the way down up until the slope to point aged to the bottom of the smoke, which is d and then times co sign of ar for because it's the dot product to plug in the numbers.
06:31
The masses five point zero zero kilograms, the displacement, or the acceleration of gravity, is nine point eight meters per second.
06:40
The displacement is two point eight meters on then we have courts and of our for alfa happens to be sixty six degrees.
06:49
If you multiply all these numbers, you end up getting fifty five point eight.
06:56
Jules, that's the energy expanded in moving the park.
07:00
It's all the way down up until the base by gravity.
07:04
Odie thie.
07:05
Other energy that we have to compute all the other work that we have to compute.
07:10
I'm going to do that in the next section.
07:12
Remember, we have the incline.
07:14
This is gravity mg.
07:17
We need to find the work done by the normal force, which is end, so walk down by the normal force.
07:24
This angle right here, it's twenty four degrees.
07:27
The angle between the horizontal and then kind is a component vertical component off the weight, which is mg co sign off data the angle between that component and the way it is data.
07:39
There's a horizontal component that that's moving the package down ones, which is mg sign off data and also the frictional forces opposing the motion.
07:51
So the work done by the normal force remember the normal force is zack with same as mg cause, and a based on newton's, that law would say is action and reaction force is equal and opposite.
08:08
And so that's why the normal forces mg cause and data.
08:12
We apply that as a force or mg ko.
08:17
Sign off data.
08:19
That's our force times the displacement, which is ah to point eight meters.
08:31
Okay, time's the displacement and then also times the coastline of alpha.
08:39
Okay, time's costa navarre, for our far in this case, is going to be a different our for the offer between the normal force and the direction of motion happens to be ninety degrees.
08:51
So that also happens to be ninety degrees.
08:54
So we plug in the numbers.
08:56
We get the masses five kilograms times co sign off ada, which is twenty four degrees.
09:05
Ah, and also we have the acceleration due to gravity right here, which is nine point eight meters for second squared times.
09:15
The displacement, which is two point eight meters time's coastline of alfalfa, is ninety.
09:22
But since co sign of nineties era, it means the whole thing will become zero.
09:26
And so we have zero jewels right there.
09:30
Input.
09:30
D you want to find the total work done by all the these forces? that's the work done by friction, plus the work done by gravity, plus the work done by the normal force with those other three things you're looking at, the work done by friction.
09:50
If you go back, happens to be negative.
09:52
Thirty eight point eight, nine point eight nine jewels plus the work done by gravity is fifty five point eight jules.
10:05
And then finally, the work done by the normal force zero jules, if you son those up together, you find that the total walk done is gonna be so.
10:15
This is the work total.
10:18
It's going to be some those two.
10:21
Together you get sixteen point nine jewels, sixteen point nine jewels.
10:27
The final part of the problem is about finding the final velocity at the base.
10:32
So this is twenty four degrees...