Chapter 7: Potential Energy and Energy Conservation

Welcome to Last Minute Lecture.

This free chapter overview is designed to help students review and understand key concepts.

These summaries supplement, not replace the original textbook and may not be redistributed or resold.

For complete coverage, always consult the official text.

Have you ever watched a diver up on that high platform, you know, just standing there before they jump?

Right.

Frozen.

You know, they haven't even started moving yet, but you can just feel it.

Oh yeah.

There's all that energy.

Waiting to explode when they finally hit the water.

Totally.

We usually think about the big splash, the speed, all that kinetic energy.

Yeah, the motion.

But where does it all come from?

What's the secret ingredient before they even take a step?

That's what we call potential energy.

That's like stored energy.

Even when they're completely still way up there.

The diver has energy just because of where they are.

Okay, stored energy.

I like that.

So it's not just about the height.

Like it's a long way down.

That height itself is a type of energy just waiting to be unleashed.

Exactly.

And that's gravitational potential energy.

The higher they are, the more potential energy they have.

And then boom, it all gets converted into that incredible speed as they fall.

That makes sense.

So we're talking about gravitational potential energy here.

The higher, the more.

Okay.

Exactly.

Gravitational potential energy is the energy an object has because of its position in the gravitational field.

But let's go back to that diving board for a second.

Think about a diver using the springiness of the board.

They bounce.

Oh, right.

The board bends way down.

Then it snaps back, giving them that extra boost.

Precisely.

It's like stretching a rubber band, storing energy in the shape.

So is that what elastic potential energy is all about then?

You got it.

It's potential energy that's stored in something that can change shape, like that diving board or a spring.

The chapter we're looking at today actually mentions a third type too.

Electrical potential energy.

But we're gonna focus on those first two for this deep dive.

The gravitational and elastic kinds of potential energy.

What gets me is how all these different types of stored energy can switch back and forth with the energy of motion.

It's like they're trading places constantly.

And that's exactly where we're gonna unravel today.

How kinetic energy and potential energy, both the gravitational and the elastic flavors,

can transform into each other.

And one of the most important things to understand is that in certain situations, the total mechanical energy that's the sum of the kinetic and potential energies together in the system stays the same.

It's like a balanced budget that never changes its bottom line.

This is one of the most fundamental laws in all of physics, the law of conservation of energy.

The total amount stays the same.

It just switches forms.

I hope so.

Got it.

So this chapter breaks it down into a few different parts to help us get a handle on it.

Right.

First, it covers gravitational potential energy.

Then it dives into elastic potential energy.

And after that, it gets into this really important difference between forces that are conservative and those that are non -conservative.

Then it gets kind of mathy, looking at how force and potential energy are connected through equations.

And finally, it shows us how to use these things called energy diagrams to actually picture all of this happening.

Sounds like we've got our work cut out for us.

It's quite a journey.

But these concepts are foundational to physics.

Understanding them opens up a whole new way of looking at tons of physical phenomena.

Yeah.

What we wanna do in this deep dive is to break down these ideas in a way that makes sense.

Right.

Pull out the key principles, why they matter,

without getting bogged down in all the nitty gritty details.

Sounds good.

Let's do it.

Okay.

So first up, gravitational potential energy.

I thought it was interesting how the chapter presented two different ways to think about a falling object.

Yeah.

We already talked about how gravity does work on a falling object.

Yeah.

And that work is what makes its kinetic energy increase, making it speed up.

Right, right.

But now we're adding this new perspective.

As the object falls, its gravitational potential energy decreases.

It's that decrease that we see as an increase in kinetic energy.

Okay.

It's not one or the other.

It's just two ways of looking at the exact same thing.

Got it.

So it's not like they're separate.

It's just two sides of the same coin.

Precisely.

But how do we actually measure this gravitational potential energy?

The chapter gives the formula u -grav equals medge.

Yeah.

Where m is the mass, g is acceleration due to gravity, and y is the vertical position.

But I noticed it specifically mentions that u increases as you go up.

That's an important detail.

It means that when u increases, when the object moves higher, the gravitational potential energy goes up too.

And when u decreases, when the object falls, the potential energy goes down.

And this change in potential energy is directly connected to the work done by gravity.

The work done by gravity, we call it w -grav, is equal to the negative of the change in gravitational potential energy.

So that's w -grav equals negative delta u -grav.

We can also write that as md1 minus md2, where y1 is the initial height and y2 is the final height.

Okay, that negative sign there, I'm guessing that's important.

Yes.

It means that when gravity does positive work, like pulling something down and making it speed up, the potential energy is decreasing.

It's like we're spending that stored energy.

So it's like we're withdrawing from the potential energy bank account.

I think the chapter used that analogy.

Exactly.

And figure 7 .2 in the chapter shows this visually.

For an object moving vertically,

it's a good way to see how the height and the potential energy are linked.

There's another important point the chapter makes that a lot of people miss.

Gravitational potential energy isn't just a property of the object itself.

Oh, it's shared.

Shared between what though?

It's shared between the object and the earth.

Gravity is an interaction between two masses.

So our formula, u -grav equals mi -gi, includes the object's mass, m, but that g is because of the earth's mass.

It's a property of the system of both the earth and the object together.

Okay, so it's not just about the one object, it's how they interact.

That's right.

That's interesting.

Now what happens when gravity is the only force acting on our system?

That's when the principle of conservation of total mechanical energy becomes really important.

If gravity is the only player, then the total mechanical energy, e, which is the kinetic energy, k, plus the gravitational potential energy, u -grav stays constant.

So the initial total mechanical energy, k one plus u -grav one, equals the final total mechanical energy, k two plus u -grav two.

That means it's a conserved quantity because it doesn't change over time in a closed system anyway.

The kingfisher.

The bird that dives into the water is the kingfisher.

That's a good example, right?

It loses height, so less gravitational potential energy, but it gains speed, so it's got more kinetic energy.

Seems like barely any energy is lost to air resistance.

It's a near perfect conversion from potential energy to kinetic energy.

So cool.

But in the real world, it's not always that simple.

There are usually other forces involved too, right?

And that's where w other comes in.

That represents the work done by any forces that are not gravity.

So now our equation becomes k one plus u -grav one plus w other equals k two plus u -grav two.

Okay, so if there are other forces doing work, they can either add energy to the system or take it away, right?

Yes.

The chapter said positive w other increases the total mechanical energy,

and negative w other decreases it.

Like that parachutist example in figure 7 .5, air resistance is doing negative work there.

Even though they're moving downwards, it's slowing them down.

Precisely.

And because air resistance opposes the motion, it reduces the total mechanical energy of the parachutist.

Makes sense.

The chapter then gave some problem solving strategies.

Strategy 7 .1, I think.

It seems like a good way to approach these energy problems.

It's a really useful framework.

It says to clearly define the beginning and the end states of the system.

Yes.

And set up a coordinate system where y is positive going upwards.

Yeah.

Then you figure out all the other forces doing work, not just gravity, and list everything you know and what you're trying to figure out.

Exactly.

And that strategy highlights when the energy approach is the best way to solve a problem.

Like when forces are changing or when the object is moving along a curved path.

In those cases, using Newton's laws directly can get pretty messy.

And you need to think about time, too.

Oh, I see.

The energy approach is just concerned with the beginning and the end, not all the details in between.

That simplifies things a lot.

Right.

Like in example 7 .1, the baseball being thrown upwards.

Oh yeah, straight up in the air.

By using the fact that mechanical energy is conserved, we know that the total energy at the start must be equal to the total energy at the highest point.

Okay.

At that point, all the kinetic energy is gone.

It's all potential.

So we can figure out the maximum height without messing around with acceleration or time.

So you're saying you can find the maximum height without having to calculate all those other things just by using conservation of energy.

Right.

And then example 7 .2 builds on that by taking into account the push you give the ball with your hand at the beginning.

Right.

That push does work.

And that work, W other, increases the total mechanical energy of the ball.

Okay.

Then as soon as it leaves your hand, it's back to just gravity doing work.

And mechanical energy is conserved again as it goes up.

Okay.

What about when things move along curved paths?

Does the work done by gravity change then?

The chapter's really clear about that.

The work done by gravity depends only on the vertical displacement, that change in y,

not the actual path the object takes.

So W grav equals negative milligram delta y, whether the object goes straight up and down or follows a more complex path.

Right, okay.

So it doesn't matter how it gets there.

It's just the change in height.

That's the beauty of the potential energy concept.

It lets us ignore the details of the path.

Conceptual example 7 .3 with the two baseballs really highlights that.

Oh, okay.

If both balls are launched with the same speed and from the same height, they'll have the same speed at any other given height as long as we can ignore air resistance.

Oh, okay.

This is because their initial total mechanical energy is the same.

And at that given height, their potential energy is the same too.

So that leaves them with the same kinetic energy and therefore the same speed.

Got it, okay.

Makes sense.

Then there's example 7 .4 with Throckmorton, the skateboarder going down a frictionless ramp.

Ugh, Throckmorton, a classic physics example.

Because there's no friction,

the only force doing work is gravity.

The normal force from the ramp doesn't do any work because it's perpendicular to the direction of motion.

Okay.

So we can use conservation of energy to find his speed at the bottom.

All that initial gravitational potential energy gets turned into kinetic energy.

But real ramps have friction, right?

Example 7 .5 shows what happens when we include friction in Throckmorton's state boarding adventure.

Right.

Now we have a non -conservative force doing work.

And that work, W other, is negative because friction always opposes motion.

So the total mechanical energy of the system decreases because of friction.

Exactly.

And example 7 .6 takes this even further.

Okay.

With a crate sliding on a ramp that has friction, it looks at what happens when it slides up the ramp and then back down.

Showing how friction does negative work the whole time, it's always removing mechanical energy from the system.

And that's why the final speed is lower than if there were no friction.

So friction's like an energy drain in these systems.

That's a good way to think about it.

At the end of that section, there's a test your understanding.

Right.

It basically says that if you have no friction and you start and end at the same heights, the final speed is the same no matter what path you take.

That's a direct consequence of the conservation of mechanical energy.

Okay, let's move on to another type of stored energy,

elastic potential energy.

All right, elastic potential energy.

It's the energy stored in something that can be deformed, like stretched or compressed, but then it returns to its original shape, like a spring or a rubber band.

Exactly.

Think about the work you do when you stretch a rubber band or compress a spring.

You're putting energy into the system and that energy gets stored as elastic potential energy.

The chapter reminds us of the work done by a spring from chapter six.

That's WLL equals 1 1⁄2 K by one squared minus 1 1⁄2 K by two squared, and then it defines elastic potential energy.

At a certain displacement, X from the equilibrium position as UL equals 1 1⁄2 K X squared, where K is the spring constant, a measure of how stiff the spring is, and X is the displacement from the equilibrium position.

Notice that X can be positive if the spring is stretched or negative if it's compressed, but because it's squared, the elastic potential energy ULL is always positive or zero.

Got it.

Figure 7 .1 free shows that parabolic relationship between displacement and potential energy.

It's a nice visual.

And it seems like there's a similar relationship here with elastic potential energy between the work done by the elastic force and the change in potential energy.

That's right.

Like with gravity, it's a negative change too, right?

So WL equals negative delta ULL, which is UL1 minus UL2.

Exactly.

When the spring does positive work, its potential energy decreases.

Like when a stretched spring pushes on something, causing it to move.

Okay.

I noticed that the chapter makes an important point about this.

With gravitational potential energy, we could kind of choose a convenient zero point for height,

but for elastic potential energy, X equals zero has to be the unstretched or uncompressed position of the spring.

That's really important to remember.

It's defined by the equilibrium state of the spring.

Okay, so if only the elastic force is doing work within a system, what happens?

Then we get conservation of total mechanical energy again, but this time it's E equals K plus ULL.

Figure 7 .3 shows a nice example of this.

With a glider attached to a spring on an air track, no friction.

Okay.

The glider oscillates back and forth.

And we see this constant conversion between the kinetic energy of the glider and the potential energy stored in the spring.

Right.

Okay, but a lot of times we'll have both gravity and springs at play, maybe even other forces too.

Of course.

And the chapter gives us the most general form of the energy equation to deal with that.

K1 plus U grab one plus UL1 plus W other equals K2 plus U grab two plus UL2.

Or we can shorten that to K1 plus U1 plus W other equals K2 plus U2, where U represents the total potential energy from both gravity and springs.

Okay.

That equation can handle pretty much anything you throw at it.

That's good.

It looks pretty powerful.

The trampoline example in figure 7 .15, that's a good visualization of how kinetic, gravitational and elastic potential energies are all changing during a jump.

It also shows that in real life, we don't have perfect conservation of mechanical energy because of things like air resistance and internal friction within the trampoline.

You're exactly right.

And this principle doesn't just apply to things we build, it shows up in nature all the time.

The chapter talks about how cheetahs use elastic potential energy stored in the flexing of their backs to run more efficiently.

It's amazing.

So cool.

Problem solving strategy 7 .2 builds on the previous one.

Now we have to consider elastic potential energy too when solving problems.

Example 7 .7 goes back to the glider on the hair track, but this time it uses conservation of mechanical energy, including the spring, to find the glider's velocity at a different location.

And example 7 .8 adds another twist.

It includes an external force pushing on the glider and the spring.

So W other isn't zero this time.

Exactly.

It increases the total mechanical energy.

Yeah.

And that changes the final velocity compared to when only the spring was doing work.

Okay.

Then example 7 .9, the falling elevator.

That stopped by a spring at the bottom with friction involved too.

That one uses the full energy equation with both gravitational and elastic potential energy.

And it includes the negative work done by friction to calculate the spring constant needed to stop the elevator safely.

That's a great example of how powerful this energy approach can be.

For dealing with complex scenarios with lots of different types of energy and forces,

the test you're understanding at the end of this section gets us thinking about the energy bar graphs for an elevator moving down while compressing a spring.

It really makes you think about the signs of K, U, grav, and UL and how they change in a dynamic situation.

Now it's time to talk about a really important distinction in the world of forces,

conservative and non -conservative forces.

This seems like a big deal.

The chapter uses the idea of reversibility to make the distinction and whether the work done by a force can be completely converted back and forth between kinetic and potential energy.

Right, a conservative force is one where the total work it does on a particle moving between two points is independent of the path taken.

So it doesn't matter how it gets there, just the start and finish points.

Exactly, and this allows for a perfect back and forth between kinetic and potential energy without losing any mechanical energy along the way.

Gravity and the spring force are prime examples.

And we'll see later that the electric force is another one.

Okay, so it's like you can store the energy and then get it all back as kinetic energy later.

The chapter lists four main characteristics of conservative forces.

It does.

First, the work done can always be written as the difference in a potential energy function between the starting and ending points.

Second, the work done is reversible.

So if you move something from A to B, the work done is the negative of the work done moving it back from B to A.

Third, like we said, the work done doesn't depend on the path taken, just the beginning and end points.

And finally, the total work done by a conservative force on a particle moving in a closed loop ending up back where it started is zero.

That's a great summary.

And remember, when only conservative forces are doing work in a system, the total mechanical energy remains constant.

It's conserved.

E equals K plus U stays the same.

Okay, got it.

But what about non -conservative forces?

How are they different?

Non -conservative forces are forces where the work done does depend on the path taken.

And it's not easily reversible.

Friction is the classic example.

Remember that crate sliding on the ramp.

The longer the path, the more work friction does.

And that energy gets dissipated usually as heat.

And air resistance or fluid resistance in general is another one.

If you throw a ball straight up in the air, it won't come back down to your hand with the same speed.

Some of that initial mechanical energy has been lost to air resistance.

So the direction of friction always opposes the motion.

So it seems like it's always removing mechanical energy from the system and turning it into heat or something.

You got it.

Forces like friction and air resistance are called dissipative forces because they lead to a loss of mechanical energy from the system.

But the chapter also points out that some non -conservative forces can actually increase the mechanical energy of a system, like the forces involved in an explosion.

Huh, okay.

The vining characteristic of a non -conservative force is that you can't express the work it does simply as the difference in a potential energy function.

So it's not just about losing energy, it's about whether we can get it back into mechanical form.

Precisely.

Now we need a more complete picture of energy conservation.

And that includes internal energy.

Internal energy.

We give it the symbol uint.

And the fundamental law of conservation of energy says that the total energy of an isolated system that includes kinetic potential and internal energy is always conserved.

The book expresses this as delta k plus delta u plus delta uint equals zero.

Where delta uint is equal to negative w other.

The work done by the non -conservative forces.

This equation emphasizes that the energy that we might say is lost from the mechanical system isn't really gone.

It's just been transformed into internal energy.

Like the friction in a car tire, it heats up the tire as you drive.

Exactly.

Mechanical energy is being converted into internal energy because of friction.

And conceptual example 7 .12 uses Throckmorton on a skateboard again with friction to show that the decrease in mechanical energy is equal to the increase in internal energy.

The skateboard and the ramp both get a little warmer because of the friction.

So energy isn't destroyed by non -conservative forces.

It just gets changed into a form that's not as readily available.

At the end of this section, it asks a question about hydroelectric power.

Reminding us that even when we're trying to convert potential energy into usable work, some energy is always lost as heat.

Because of things like friction in the water and the machinery.

It's a good real -world application of the concept.

Now let's look at the mathematical relationship between force and potential energy.

We've been talking about them qualitatively, but now let's get into the equations.

This section shows us how to find the force if we know the potential energy is a function of position.

And the book mentions this is especially helpful in electromagnetism, where it's often easier to figure out the electrical potential energy first and then figure out the force from that.

For motion in one dimension, say along the x -axis, the key equation is f sub x of x equals negative du of x dx.

That means the x component of the force at a specific position x is equal to the negative of the derivative of the potential energy function u of x with respect to x.

So the steeper the potential energy curve, the stronger the force.

And that negative sign, that's important again, right?

Absolutely.

It means the force always acts in the direction that lowers the potential energy.

Like a ball rolling down a hill.

Gravity pulls it towards lower gravitational potential energy.

Okay, I like that analogy.

The chapter then checks this equation against the potential energy functions we've already talked about.

For elastic potential energy, u of x equals one half kx squared.

If you take the derivative and apply the negative sign, you get f sub x of x equals negative kx, which is Hooke's law, the force law for an ideal spring.

And for gravitational potential energy near the earth's surface, u of y equals meningy.

The negative derivative gives us f sub y equals negative mg, which is the weight force acting downwards.

Figure 7 .22 shows these relationships with the potential energy curves and the corresponding forces.

Example 7 .13 uses an electric potential energy function.

U of x equals c over x.

By taking the derivative, we can find the corresponding electric force.

F sub x of x equals c over x squared.

Okay, what happens when we go beyond one dimension?

How does this relationship work in three dimensions?

We need to use partial derivatives.

The components of the force vector, f sub x, f sub y, and f sub z are given by the negative partial derivatives of the potential energy function u of x, y, and z with respect to each coordinate.

So f sub x equals negative partial derivative of u with respect to x and so on for y and z.

The chapter introduces this thing called the gradient operator, denoted by that upside down triangle symbol u or grad u, which is a vector containing all these partial derivatives.

The force vector f can then be written as f equals negative grad u.

So the gradient tells us how the potential energy is changing in all three spatial directions, and then the force points in the direction of the steepest decrease in potential energy because of that negative sign, right?

That's a great way to think about it.

The chapter then checks this out with a 3D gravitational potential energy, u equals melameu.

Taking the negative gradient gives us f equals negative mbj s hat, which is the familiar weight force acting downwards.

Okay.

Then there's example 7 .14, which gives a potential energy function u equals one half kr squared, where r squared equals x squared plus y squared, which represents a force directed towards the origin.

Right.

By calculating the partial derivatives for f sub x and f sub y, they show that the force vector points towards the origin and has a magnitude of kr.

Exactly.

The test you're understanding at the end of this section asks what we can say about the potential energy if the force is zero at a certain point.

The key is that a zero force means the slope of the potential energy curve in that direction is zero.

But it doesn't tell us anything about the value of the potential energy itself.

It could be zero, positive, or negative.

Okay.

Okay.

Finally, let's talk about energy diagrams.

They're a great way to visualize motion when we only have conservative forces acting.

They sound pretty useful.

They really are.

Energy diagrams are graphs that show the potential energy function u of x and the total mechanical energy E as a function of position x.

For systems with just one dimension of motion and only conservative forces doing work, figure 7 .23 uses a mass and spring system as an example.

It's easy to visualize.

Okay.

On these diagrams, the vertical distance between the horizontal line representing the total mechanical energy E and the curve representing the potential energy u of x at any given position x tells you the kinetic energy k at that point, because e equals k plus u.

And where the total energy line crosses the potential energy curve,

k equals zero.

That means the object has stopped moving monetarily.

These are called the turning points of the motion, the limits of how far it can go.

The object can only move where the total energy E is greater than or equal to the potential energy u of x, because kinetic energy can't be negative.

And we can also learn something about the force from these diagrams, right?

We learned that the force is the negative derivative of the potential energy.

Exactly.

So the slope of the u of x curve at any point tells us about the force at that position.

If the slope is zero, the net force is zero, and that means it's an equilibrium position.

You got it.

The chapter then makes a distinction between stable and unstable equilibrium based on the shape of the potential energy curve around those zero force points, a point where the potential energy is at a minimum is stable equilibrium, like a marble at the bottom of a bowl.

If you nudge it, it'll roll back to the bottom.

A point where the potential energy is at a maximum is unstable equilibrium.

That's like balancing a marble on top of an upside down bowl.

Any little nudge will make it roll away, right?

Figure 7 .24 shows these concepts with a more general potential energy function.

It has points of stable equilibrium and unstable equilibrium.

It also shows how the total energy E determines the type of motion, whether it's confined or if it has enough energy to escape.

The chapter mentions that the sign of the potential energy itself doesn't tell us the direction of the force.

It's the sign of the slope that matters.

And if you add a constant value to the potential energy, it just shifts the whole curve up or down, but it doesn't change the slope.

So the force and the physics stay the same.

That's right.

And the application section talks about acrobats balancing in unstable positions.

They had to constantly make adjustments to stay balanced and not fall to a lower potential energy.

That's a great example.

It is.

The last test you're understanding asks us to think about what the acceleration is at a point where the potential energy, U of X, is at a maximum.

The key there is that even though the slope is zero at the maximum,

the force, and therefore the acceleration, depend on the shape of the curve, if it's a sharp peak, the forces on either side will be large.

And that means a big acceleration away from that unstable equilibrium point.

So energy diagrams are a really cool way to understand motion under conservative forces.

You can see turning points, equilibrium positions, and everything without having to do all the math with Newton's equations.

That's right.

They're powerful tools.

We've covered a ton of ground today from that diver up on the platform all the way to energy diagrams.

It's been quite a deep dive into the world of energy.

It has been.

We've talked about gravitational potential energy, elastic potential energy, the conservation of mechanical energy, what non -conservative forces are, and how they relate to internal energy, the relationship between force and potential energy, and how to use energy diagrams to understand motion and equilibrium.

It's amazing how all these ideas work together to explain so much of the world around us.

From simple things like a falling ball to complex systems in biology.

It really highlights how fundamental and practical these ideas are.

And it makes me wonder,

can these concepts of potential energy and energy conservation apply to other areas beyond physics?

Like, could we talk about potential in economic systems or social dynamics?

And how changes in these potentials might cause change, even if we define energy differently in those systems.

Wow, that's a really interesting thought.

I'll have to think about that.

Hopefully this deep dive has given you a good understanding of energy and all its different forms.

And maybe it sparked some curiosity to explore these ideas even further.

Thanks for joining us today.