AP Physics C: Mechanics Energy & Conservative Forces

Hello, everyone, and welcome back to www.educator.com.0000

I'm Dan Fullerton and in this lesson we are going to talk about Energy and Conservative forces.0003

Our objectives include defining energy, describing various types of energy,0008

talking about some alternative definitions of conservative force,0014

describing some examples of conservative and non conservative forces.0019

Finally using a relationship between force and potential energy to find forces and potential energy functions.0023

Let us dive right in by talking about what is energy.0030

Energy is the ability or capacity to do work.0033

Work is the process of moving object.0037

If we put those together energy is the ability or capacity to move on the object.0040

Energy can be converted to different types.0047

It can be transformed from one type to another.0049

You see that all the time.0051

You transfer energy from one object to another by doing work.0053

We talked about the work energy theorem in our last lesson.0057

The work done on a system by an external force changes the energy of that system.0060

We are going to be talking about lots of energy transformations in the next couple lessons0066

but one of those is going to come up again and again is kinetic energy.0071

Kinetic energy is energy of motion or if energy is the ability or capacity to move an object.0075

Kinetic energy that is the ability or capacity of a moving object to move another object.0081

Something that is moving has the ability to move something else.0088

The baseball coming toward your nose with a bunch of Velocity has kinetic energy.0091

It has kinetic energy because it has the ability to move something else.0096

Basically the bones in your nose.0100

If that hits your nose it is going to squish it up.0101

It is going to cause something else to move.0103

Kind of a messy example but that is the idea.0106

Kinetic energy is the capacity or ability of that moving object to move something else.0108

The formula for kinetic energy is ½ mass × the square of your speed.0116

Units of course energy Joules.0122

Let us take a look at a quick example here.0126

A frog speeds along a frog o cycle at a constant 30 m/s.0128

If the mass of the frog and motorcycle is 5 kg find the kinetic energy of the frog cycle system.0133

Kinetic energy is ½ MV² to be ½ × the mass 5 kg × our speed 30 m/s² 900 × 5=4500.0141

½ of that is going to be 2250 joules.0155

Let us talk now about potential energy.0166

Potential energy which gets the symbol capital U is energy an object possesses based on its position or its speed of being.0168

There are lots of different types of potential energy.0177

Gravitational potential energy will talk about quite a bit in this course.0180

It is the energy an object possesses because of its position in the gravitational field.0183

Elastic potential energy we talked about a little bit with Hooke's law already.0189

The energy that you have from some sort of elastic displacement something like a spring or an elastic band.0193

Chemical potential energy, electric potential energy, nuclear potential energy, all different forms of potential energy.0202

It is important to point out that a single object in isolation can only have kinetic energy.0210

In order to have some from a potential energy you need to have an interaction between objects.0218

You need to have at least two objects in your system or something they have potential energy.0222

As we talk about these we are also going to run across the term internal energy every now and then.0230

The internal energy of the system includes the kinetic energy of the objects that make up that system0235

and potential energy of the configuration of the objects that make up the system.0241

For example if you think of the temperature, the heat and temperature of an object that0246

is based on the speed with which the molecules inside are moving around.0253

That is an internal energy because it includes the kinetic energy of the objects comprising that system.0258

We can make changes internally that system.0264

The changed the internal energy of the object.0267

Change in a systems internal structure can result in changes in the internal energy.0271

Let us see if we can put some of us together as we talk about different types of energy.0277

I like to break energy up in the two main types potential and kinetic.0281

Some of these cross the boundary between each one.0285

If we talk about electrical energy depending on how you are looking at moving charges create current.0288

Moving there is kinetic energy.0296

Light moving photons as they do not have mass but they do have energy.0299

Light is another goofy one.0304

We could talk about is kinetic energy.0305

Wind is moving air molecules.0308

Thermal energy the motion of molecules and atoms making up of an object.0310

Sound vibrating air or vibrating molecules and waves.0316

Potential side we have chemical potential energy, gravitational potential energy, we can talk about electrical potential energy,0322

in terms of voltage charges held at different levels.0329

Nuclear potential energy, elastic potential energy, and tons of others.0335

Let us start with gravitational potential energy Ug.0343

If we have some object we will start here at some arbitrary point we will call y = 00347

and we take it across a meandering path until eventually gets to another position.0354

If this is our +y direction well we have changed its gravitational potential energy.0360

How much if we change it?0366

Well let us take a look.0368

The work done in moving it there which should be the amount of potential energy that has its final point.0370

As we are dealing with the conservative force we will talk about that in a minute.0377

As we go from y=0 to Y equals some final value let us call that the h level.0381

So that is h.0389

F • dr define their work which would be the integral from 0 to h are force.0391

To lift that up we have to overcome the force of gravity so that is going to be MG and DR.0398

We are worried about the Dy position.0405

MG should be constant as long as we are in the same relative level, the same gravitational field if we are going miles and miles up.0409

A very big distance into the atmosphere G might change.0417

As long as we are relatively close to use pretty constant.0420

We will pull those out MG we should say work equals MG integral from 0 to H.0423

Dy which is just going to be MGH.0430

As long as are the constant gravitational field gravitational potential energy can be written as MGH you have probably seen that before.0434

Let us take a look and example here.0447

In the diagram we have 155 N box on a ramp.0449

Applied force F causes the box to slide from point A to point B.0453

What is the total amount of gravitational potential energy gained by the box?0458

As I look at this change in gravitational potential energy is going to be MG Δ h or change in height.0464

Which is going to be 155 N that describes its weight or the force of gravity MG on it.0475

Δ H and goes up 1.8 m.0481

That would be 279 joules pretty straightforward.0486

Let us take a look at graph in question.0496

The hippopotamus is throwing vertically upward.0498

I do not know why and I really do not know how.0500

Which pair of graphs best represent the hippos kinetic energy in gravitational potential energy as functions of its displacement while it rises?0503

As it goes higher and higher as it rises, it is going to slow down so we have to expect that we are going to see kinetic energy decreasing.0513

On the other hand as it gets higher and higher its gravitational potential energy should be increasing.0521

The kinetic energy is being transformed into gravitational potential energy.0527

Which graph we have that displays that the best?0532

That looks like one as we get further and further displacements on its way up kinetic energy goes to 0 at its highest point where it stops.0535

That is where we have our maximum of gravitational potential energy.0541

The answer must be 1.0545

Looking at a slightly more involved question.0549

A pendulum of mass M swings on the light string of length L.0551

If the mass hanging directly down is set to 0. Of gravitational potential energy0556

find a gravitational potential energy the pendulum as a function of θ and L.0561

It looks like if I break this up a little bit or really doing is we are changing the height of our pendulum as it swings from there to there.0568

There is our change the Δ Y.0586

We have to figure out what that is.0591

To do that I am going to take a look at my triangle here and realize that our hypotenuse here is going to be equaled L.0594

We got the adjacent side of our angle right here and this would be the opposite side.0601

To help figure out that height let us say that the cos θ SOHCAHTOA.0607

Cos is adjacent over hypotenuse.0614

That is a adjacent over hypotenuse which implies that this adjacent side, this piece right here from there to there is going to be the hypotenuse × cos θ.0618

Since our hypotenuse is the length of our string L that is going to be L cos θ.0638

Our Δ y is going to be the entire length L - adjacent side which is L cos θ.0646

I'm going to find that Δ y is going to be L × (1 - cos θ).0660

If I wanted a change in potential energy I can write Δ Ug = Mg Δ Y.0667

Which would be Mg × L 1 - cos θ.0677

We have our gravitational potential energy as a function of θ and L for our swinging pendulum.0689

Let us talk for a minute about conservative forces.0698

A force in which the working on an object is independent of the path is known as a conservative force.0701

Gravity for example.0706

If I left a pen straight up or go to the side and run it around all over the place, the only thing that matters0707

for the total change in the energy or the total work done is the initial and final points.0714

You can also write that define a conservative forces of force in which the work done moving along the close path is 0.0721

As long as you come back to where you start whoever you get there and net work done must be 0.0727

Or it is a force in which the work that is directly related to a negative change in the potential energy.0733

The work that is equal to the opposite of the change in the potential energy.0738

Some examples of these.0743

Let us talk about some conservative forces and some non conservative forces.0746

Conservative forces would be things like gravity.0762

We are only worry about the initial and final points or elastic forces.0767

When we talk about electricity in the following course on ENM columbic or electric force.0775

Non conservative forces are one you typically think of as the law C forces for example.0783

Friction the energy is ever destroy your loss that can be converted less useful forms.0789

Friction drag forces, retarding forces, air resistance, fluid resistance, non conservative forces.0795

There are some properties of conservative forces.0807

For the work done by conservative forces, let us say it is a work done by a conservative force0811

is equal to the opposite of the change in its potential energy.0817

Which implies then that Δ U equals the opposite of the work done by the conservative force.0822

Which implies that the Δ U equals the opposite of the integral from some initial position to some final position of F•Dr.0833

And we can use this in a bunch of different ways.0849

We are going to apply it with a couple different types of conservative forces in the next few slides.0851

Let us take a look at in the context of newton's law the universal gravitation.0857

You have probably seen this before and other courses.0861

I do not know if it is formally introduced here yet but the gravitational force between two objects is - G universal gravitational constant.0863

It is a fudge factor to make the unit is work out equal to 6.67 × 10⁻¹¹ Nm² / kg².0872

A constant × first mass × second mass divided by the square of the distance between their centers.0882

The r hat just tells you the direction of r hat in the direction of the unit vector from the first object0888

to the second in the negative tells you that it is going to be attractive there.0896

We can look at it as here we have mass 1.0901

Somewhere over here we have a mass 2.0905

The distance between their centers we can draw that right in nice and quick.0908

There it is that would be our r vector.0916

R hat would just be a unit vector in that direction.0919

If we start with a Newton’s law of universal gravitation we can use what we just found out about conservative forces to find a potential energy.0926

The universal gravitational potential energy.0935

Change in potential energy is minus the work done by conservative forces is going to be - the work done by gravity0941

or - the integral as we go from infinity to r some point of - GM 1 M2 /r² with respect to r.0950

As I do this integration I can pull my constant out.0964

Potential energy due to gravity is going to be GM 1 M 2 should not change for the purposes of this problem.0969

This will be GM1 M2 our negative and negative cancel out.0977

Integral from infinity to some r of DR/ r² or r⁻² Dr.0982

Let us write it that way just to make it a little easier to see.0991

GM 1 M2 integral from infinity to r of r⁻² DR is going to be GM 1M2.0993

The integral of r⁻² is going to be 1/r.1003

Evaluated from infinity to r so potential energy due to gravity is just going to be GM 1M2 with 1.1008

We will plug that in - GM 1M21 1/ infinity will be 0.1024

I'm going to come up with - GM 1 M2 /r.1029

And that is our formula for gravitational potential energy between two objects separated by some distance r.1041

We are able to derive it knowing the force because it is a conservative force.1050

We can do the same thing with the elastic potential energy.1055

Remember the force in the spring by Hooke’s law – KX.1060

The work done is negative change in potential energy and that is a conservative force or - the integral from 0 to X of F• DR1065

as we are finding the work done which is - the integral from 0 to X of – KXDX.1081

The potential energy stored in the spring is the integral from 0 to X of KXDX which is K plot a constant integral from 0 to X.1092

0 to X of XDX or K × X²/2 evaluate from 0 to X which is just going to end up being ½ KX 2.1105

There is a potential energy stored in the spring.1120

If we can go from force to potential energy we should be able to get force from potential energy.1125

Let us take a look at that.1132

As soon as we have an object on some path dr or we could also call this DL that might be just as common dl.1134

As it travels along this path some forces acting on a net force can be changing.1141

If we want to find V potential energy to this force along the path what we can do is break it up1147

and these little tiny pieces DL and find the potential energy for each one of those by finding the work done through each one of those.1155

The differential potential energy that tiny amount of potential energy is the opposite of that little tiny bit of work done by the conservative force.1166

Or - F dotted with will use dl there which if we want to force in the direction DL this is going to be - F cos θ DL.1168

If you remember our dot product definition.1192

What we are going to do is we are going to call this F cos θ we are going to call that the force in the direction of L.1196

We will call that fl.1202

Once we define that the little tiny bit of energy, potential energy is - F cos θ DL which we could also write as – FLDL.1205

That means that FL the force in the direction of that displacement is just going to be - du DL.1224

The opposite of the derivative of the potential energy function with respect to L gives you that force.1234

We can go from potential energy to force.1241

Let us do that with gravity.1246

FL equals - du DL and their potential energy function for gravity is - GM 1M2/r.1250

Let us say we are looking for force in the direction r.1263

Force r equals - the derivative with respect to R of our potential energy function - GM 1M2/r.1267

We can pull out our constants GM 1M2 × the derivative with respect to r of 1/r1277

which means that the force in the direction of R is going to be GM 1M2.1288

The derivative of 1/ R is -1/ r² or - GM 1M2 /r² which is what we had initially force = - GM 1M2/r² in that direction of r hat.1294

We have gone from force to potential energy.1315

We can go back the other way using this formula.1318

Let us take a look at Hooke’s law.1323

Here we are going to take a look at a spring and we are going to give its potential energy function as ½ KX².1326

As you extend it to the right you have more potential energy.1335

The area under the graph from Ug and no kinetic.1340

If you are let it go with the object, it gets to its center pointed it no longer has elastic kinetic energy instead it so elastic potential energy.1346

It is all kinetic.1354

We get to the other side it slows down and stops with all potential again it stretch out or compressed spring.1355

And then you come back.1364

You have this constant back and forth potential kinetic energy.1366

We can look at this in terms of the force realizing that - du DL =Fl.1372

The force in the direction of the L is – d/dx for spring of ½ KX² which is just going to be – KX.1380

We have gone the other way again.1392

To put a quick summary together.1396

If we start with potential energy something like gravitational potential energy is MGH.1398

We can look at the force - Du DL.1407

A force the derivative of this with respect to h is going to be – MG.1415

There is our formula for weight.1425

Assuming you are in a constant gravitational field.1428

If you want to get more general with Newton’s universal gravitation, potential energy due to gravity is - GM 1M2/r.1431

The force using - dudl is going to be - GM 1M2/r².1441

Newton’s law of universal gravitation.1449

Or if we start with the spring where the stored potential energy this spring is ½ KX² in a linear spring1452

that follows Hooke’s law we can find the force by taking the opposite of the derivative that with respect to x which is – KX.1460

We can go that direction.1470

We previously went that direction.1474

Let us do a couple examples.1479

5 kg sphere’s position is given by the function x of T = 3T³ -2T.1482

The term in the kinetic energy of the sphere at time T = 3s.1488

Kinetic energy is ½ MV².1496

It would be helpful to know the velocity.1500

If we know x (t) is t3³- 2t the velocity is the first derivative of x which will be 9t² -2.1503

The velocity of t = 3s is just going to be 9 × 9=81 – 2 = 79 m/s.1519

We can go back to our kinetic energy function.1528

Kinetic energy is ½ × 5kg × speed 79 m/s or about 15600 joules.1531

An example where we find force from potential energy.1547

Another 5kg sphere’s potential energy U is described by this function U(x) = 4x² + 3x – 2.1550

Determine the force on the particle at x = 2m.1559

Force = -du dl in this case it is going to be – d /dx of 4x² + 3x – 2= -derivative of 4x² 8x.1564

The derivative of +3x is 3, -2 is 0.1581

Which is -8x – 3.1587

If we want the force at x = 2m that is going to be -8 × 2m -3.1591

-16 – 3 = -19 N.1604

One last problem here.1614

Work on a spinning disc.1616

A 2kg disc moves in uniform circular motion on a frictionless horizontal table.1618

Attach to the point of rotation by a 10cm spring but the spring constant of 50 N/m.1624

When stationary the spring has a length of 8cm.1631

While it is turning it is extended 2cm.1634

How much work is performed on the disc by the spring as the disc moves through 1 full revolution?1638

The force in order to have it moving in a circle must be that way.1647

The velocity at any given point in time is that way.1651

They are perpendicular.1654

If we do not have any force in the direction of the displacement you cannot do any work.1657

F cos θ cos 90 is 0°.1663

Therefore the work done is 0.1666

A tricky question there.1673

Hopefully that gets you a good start on energy and conservative forces.1675

Thank you so much for joining us here at www.educator.com.1679

I hope to see you again real soon.1682

Make it a great day everybody.1683