00:01
In this prime, you have a solid sphere.
00:03
It's going to go down an incline and then go up and launch vertically into the air.
00:11
And we have a few things we have to calculate along the way to get to the end, the maximum height.
00:22
Now, let's understand what's going to happen here.
00:25
I made my zero line for gravitational potential energy, the ground, nice and simple.
00:32
We're given all the particular, you know, ha and hb are given to us.
00:37
Like i said, hc is one of our, is our last thing we're looking for.
00:43
Now, can we, what's the approach to a problem like this? and it is rolling without slipping.
00:51
This is an energy conservation problem.
00:52
We have, there's no work being done by non -conservative forces.
00:58
So all this, there is friction here.
01:00
It has to be.
01:01
Otherwise, this can't go from rest to roll it.
01:03
But no work is done.
01:06
No work is being done.
01:08
Wnc is equal to zero.
01:10
So we can use mechanical energy conservation throughout all this from a to b to c.
01:16
But we have some work to talk about as we get a little after b.
01:21
So let's, though, find out what the launching speed will be the translational, linear.
01:29
Same thing.
01:30
Speed at b, and also then we'll be able to get the angular speed at b.
01:37
So let's write our mechanical energy conservation equation.
01:42
So let's talk about, let's just do it in general.
01:45
We'll figure out what we'll put in what zero later.
01:48
Let's talk, an object rolling has translational kinetic energy.
01:57
So this is the translational kinetic energy of the center of mass, plus the rotational kinetic energy.
02:07
That's at point a, but also it has gravitational, potential energy.
02:16
So that's what we have at any point.
02:20
And for point b, it's the same thing just with a replaced by b.
02:29
Com stands for center of mass.
02:31
And they remember that i should mention when you read this, they told you that the center of mass is 1 .9 meters.
02:37
That's where we think of the mass of the objects is acting.
02:40
So that's where the gravitational potential energy height has to be measured two.
02:46
So this is the square there plus.
02:56
Okay.
02:57
So that's our chemical energy conservation equation between points a and b.
03:06
Now let's put in what is zero.
03:10
Starts from rest.
03:11
So there's no translation and no rotation.
03:16
We're looking for vb.
03:19
That's the first thing we're asked for.
03:22
None of these are zero.
03:25
Or at least no, well, none of them are clearly zero.
03:31
Could something be zero through calculation? i'm not saying it will be, but we'll leave that possibility open.
03:39
Okay, so let us do a little work.
03:41
Can't cross out any m's yet.
03:43
Got an i in there.
03:44
There's no m yet.
03:45
In a couple minutes, we'll see.
03:47
Now, let me bring all the m -g terms to left.
03:52
M -g -h -a minus h -b.
03:57
It's equal to one -half -m -v -b, c -o -m squared, plus one -half -i -e.
04:07
Omega b squared.
04:09
Now remember this, when we talk about i here, it's going to have to, is the moment of inertia through the center of mass of the sphere.
04:19
Okay, so we have that.
04:21
Now, now we can make use of the no slipping.
04:27
So it rolls without slipping between a and b.
04:31
So the condition for that, the relationship you find is that the speed of the center of mass is equal to the angle of speed times the radius of the object.
04:46
It's the radius of the sphere.
04:55
So that's no slipping gives us that.
04:56
If it was slipping, that's not true.
04:58
It's not true.
05:02
So we can put that in, and also the moment of inertia for a solid sphere.
05:06
Let me put all that in now.
05:07
And the moment of inertia, you don't have to calculate it or anything.
05:11
Obviously, you could.
05:12
But every textbook in physics, doesn't matter if it's college physics, algebra -based physics, or calculus -based physics, they all have a table of some size in them with all different shapes.
05:31
Solid spheres, spherical shells, discs, plates, hoops, all different types of things, rods, things of that nature.
05:43
One half m vb, center of mass squared, plus, now here's the i from the table.
05:52
Solid sphere, two -fifths, m -r -squared, v -b, center of mass squared, over r squared.
06:05
So that's our equation at the moment.
06:07
And we can clean this up.
06:09
2s go away.
06:11
R squared go away.
06:13
So this becomes mvb, com squared.
06:21
We got a one -half left over from the first one, and a 1 -5th left over from the second term.
06:30
And making ourselves a common denominator, 10.
06:37
So i've got to have 5 plus 2 over 10.
06:40
So 7 tenths.
06:43
M.
06:45
Vb.
06:47
Center, mass square.
06:49
That's what we have.
06:52
That cleaner than what it looks like up here.
06:54
All right, now the mass can go away.
06:58
Now the mass can go away.
06:59
So i get solving for vb center mass, multiply both sides by 10, and divide both sides by 7, take the square root.
07:12
So square root, 10 over 7, g, 8 ,000.
07:18
H .a minus hb.
07:22
And putting in our numbers, 10 over 7, 9 .8 meters per second squared, 1 .9 meters minus 0 .4 meters.
07:38
And this works out to be 4 .58 meters per second.
07:47
That is the translational speed at b.
07:54
Now, point b is the last time you are rolling.
07:58
That you're rolling on a surface, rolling without slipping on a surface.
08:03
So that's the last time there can be an adjustment, whereas v -center mass changes and mega changes.
08:12
And we'll talk more about that in part c.
08:17
So using my relationship here for the rolling with no slipping.
08:24
And i should mention some books will just use different terms.
08:28
They'll actually write it rolling with no slipping, or they're just, they'll say, pure rolling, or when they say rolling, they mean no slipping.
08:36
So it depends on your textbook and your professor, what is common.
08:44
But in doubt, you know, write it out.
08:47
Megab is equal to vb center mass.
08:49
We're just using this over r.
08:54
4 .58 meters per second over 0 .2 meters.
08:59
They gave us the radius.
09:01
And this is 22 .9 meters per second.
09:06
Not meters, radiance per second.
09:10
It's an angular quantity, radiance, radiance per second.
09:17
Okay, so that's the angular speed at b.
09:22
Next thing i'd ask is what is the angular speed at c? hmm.
09:30
Do we have to do something of energy again? no.
09:36
Why should it change? their only force is acting at the center of mass, gravitational force, the weight.
09:44
Does that generate any torque around the center of mass? none.
09:49
There is no way for the angular velocity to be precise, because that's when we're talking about conservation law, we're talking about velocities.
09:58
There is no way for the angular velocity to change.
10:03
Angular momentum is conserved from b to c.
10:06
It will not change...