Work/energy problem with friction | Work and energy | Physics | Khan Academy

Summary

In this educational video, the host introduces a physics problem involving a biker and rider descending a 500-meter hill with a 5-degree incline, starting from rest. The challenge includes friction, represented by a 60-newton force, which dissipates some energy as heat. The host calculates the biker's final velocity at the hill's base, demonstrating energy conservation principles and the impact of non-conservative forces like friction. The engaging explanation also touches on where the friction might originate and the conversion of mechanical energy to heat.

Takeaways

• 🔄 The law of conservation of energy states that energy cannot be created or destroyed, only transformed.
• 🚴‍♂️ The problem introduces a 90 kg bike and rider starting from rest at the top of a 500-meter long hill with a 5-degree incline.
• 📐 The problem uses trigonometry to calculate the height of the hill, finding it to be 43.6 meters using the sine function.
• 🌐 The potential energy at the start is calculated as mass times gravity times height, resulting in 38,455 joules.
• 🛑 Friction is introduced as a nonconservative force that opposes the motion and consumes mechanical energy.
• 🔢 The average friction force is given as 60 newtons, which is used to calculate the energy lost due to friction.
• 💡 The negative work done by friction is calculated by multiplying the friction force by the distance traveled, resulting in 30,000 joules.
• 🔄 The final energy at the bottom of the hill is the initial potential energy minus the energy lost to friction, equaling 8,455 joules.
• 🚀 The final kinetic energy is calculated using the formula 1/2 mv^2, leading to a final velocity of 13.7 meters per second.
• 🔥 The energy lost to friction is converted into heat, which is a real-world consequence of nonconservative forces.

Q & A

What is the mass of the bike and rider combined?

The mass of the bike and rider combined is 90 kilograms.

How long is the hill that the bike and rider start from?

The hill is 500 meters long.

What is the incline of the hill?

The incline of the hill is 5 degrees.

What is the average friction force acting on the bike and rider?

The average friction force acting on the bike and rider is 60 newtons.

What is the initial potential energy of the bike and rider at the top of the hill?

The initial potential energy is calculated as mass times the acceleration of gravity times height, which is 90 kg * 9.8 m/s² * 43.6 m, equaling approximately 38,455 joules.

How is the height of the hill calculated?

The height of the hill is calculated using the sine function with the given angle and hypotenuse, which is 500 meters * sin(5 degrees), resulting in 43.6 meters.

What happens to the potential energy at the bottom of the hill?

At the bottom of the hill, the potential energy is mostly converted into kinetic energy, but some is lost to friction.

How is the energy lost to friction calculated?

The energy lost to friction is calculated as the negative work done by friction, which is the friction force times the distance moved against the force, or -60 N * 500 m, equaling -30,000 joules.

What is the final kinetic energy of the bike and rider at the bottom of the hill?

The final kinetic energy is the initial potential energy minus the energy lost to friction, which is 38,455 joules - 30,000 joules, equaling 8,455 joules.

What is the final velocity of the bike and rider at the bottom of the hill?

The final velocity is calculated by taking the square root of the final kinetic energy divided by the mass, which is sqrt(8,455 joules / 45 kg), resulting in approximately 13.7 meters per second.

Where does the energy lost to friction go?

The energy lost to friction is converted into heat, which could be felt as warmth in the bike's components or due to air resistance.

Outlines

00:00

🚴‍♂️ Introduction to Energy Conservation with Friction

The script begins with a discussion on energy conservation, introducing a twist involving friction. The speaker explains that while all previous problems involved conservative forces where energy was conserved, this new problem will include non-conservative forces, specifically friction. The problem is sourced from the University of Oregon's physics resources. The scenario involves a 90 kg bike and rider starting at the top of a 500-meter long hill with a 5-degree incline, facing an average friction force of 60 newtons. The goal is to find the biker's speed at the bottom of the hill. The speaker sets up the problem by calculating the potential energy at the top of the hill, using trigonometry to determine the height of the hill (43.6 meters) and then calculating the potential energy (38,455 joules). The problem is framed as one of conservation of mechanical energy, with the potential energy at the start and the kinetic energy at the end, minus the energy lost to friction.

05:01

🔥 Energy Loss Due to Friction

In the second paragraph, the script delves into the concept of nonconservative forces, specifically friction, which 'eats up' mechanical energy. The speaker explains how friction does negative work over the distance traveled by the biker, reducing the total energy of the system. The initial potential energy of the system (38,500 joules) is contrasted with the energy lost to friction, calculated as the product of the friction force (60 newtons) and the distance traveled (500 meters), resulting in 30,000 joules of energy lost. The remaining energy (8,455 joules) is then equated to the final kinetic energy of the biker at the bottom of the hill. Using the formula for kinetic energy, the speaker calculates the biker's final velocity to be 13.7 meters per second. The script concludes by noting that the energy lost to friction is not destroyed but is instead converted into heat, providing a practical example of sliding down a sandpaper slide and feeling the warmth generated by friction.

Keywords

💡Conservation of Energy

Conservation of Energy is a fundamental principle in physics stating that energy cannot be created or destroyed, only transformed from one form to another. In the video, this principle is central to solving the problem of calculating the speed of a biker at the bottom of a hill. The script uses this concept to explain that all the potential energy at the top of the hill should be converted to kinetic energy at the bottom, if not for the presence of friction.

💡Conservative Forces

Conservative forces are those forces in a system where the work done in moving a particle around any closed path is zero. These forces include gravitational force and elastic force. The script contrasts conservative forces with non-conservative forces like friction, which dissipate energy as heat rather than converting it back and forth between potential and kinetic forms.

💡Friction

Friction is a force that opposes the relative motion or tendency of such motion of two surfaces in contact. In the context of the video, friction is introduced as a non-conservative force that causes some of the energy to be lost as heat, rather than being completely converted from potential to kinetic energy. The script mentions an average friction force of 60 newtons acting against the biker's motion.

💡Potential Energy

Potential energy is the stored energy of an object due to its position relative to other objects. It has the potential to be converted into kinetic energy. In the video, the potential energy of the biker and bike at the top of the hill is calculated using the formula mass times gravity times height, which is found by using trigonometry to determine the vertical component of the hill's length.

💡Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It is calculated using the formula 0.5 * mv^2, where m is the mass and v is the velocity of the object. The video concludes by calculating the biker's velocity at the bottom of the hill by equating the remaining energy after accounting for friction to kinetic energy.

💡Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. In the video, trigonometry is used to calculate the height of the hill by relating the sine of the incline angle to the known hypotenuse (the length of the hill).

💡Incline

An incline is a sloping surface, such as a hill or ramp. The video uses a 5-degree incline to describe the hill that the biker descends. This angle is crucial for calculating the height component of the hill's length, which is necessary to determine the potential energy at the start.

💡Nonconservative Forces

Nonconservative forces are forces that do not follow the conservation of mechanical energy principle because they dissipate energy as heat or other forms of energy that cannot be completely recovered as mechanical work. Friction is an example of a nonconservative force, and the video explains how it reduces the total mechanical energy in the system.

💡Work

In physics, work is done when a force causes or tends to cause movement of an object in the direction of the force. The video discusses the negative work done by friction, which is calculated as the product of the friction force and the distance over which it acts in opposition to the motion.

💡Acceleration of Gravity

The acceleration of gravity is the rate at which objects fall towards the Earth due to gravity. It is approximately 9.8 meters per second squared. In the video, this value is used to calculate the potential energy at the top of the hill, which is then compared to the kinetic energy at the bottom after accounting for friction.

💡Velocity

Velocity is a vector quantity that describes the rate of change of an object's position with respect to time, and includes both speed (magnitude) and direction. The video concludes by finding the biker's velocity at the bottom of the hill using the remaining energy after friction has done negative work.

Highlights

Introduction to a physics problem involving energy conservation with an added element of friction.

Explanation of how friction affects energy conservation by acting as a non-conservative force.

Description of a 90 kg bike and rider starting from the top of a 500-meter long hill with a 5-degree incline.

Calculation of the hill's height using trigonometry, revealing it to be 43.6 meters.

Determination of the initial potential energy of the system as 38,455 joules.

Introduction of the concept of friction as a force that does negative work against the motion.

Calculation of the energy lost to friction, which is 30,000 joules.

Explanation of how the initial energy minus the energy lost to friction equals the final energy.

Conversion of all remaining energy into kinetic energy at the bottom of the hill.

Use of the kinetic energy formula to calculate the final velocity of the biker.

Final velocity calculation results in a speed of 13.7 meters per second.

Discussion on the practical implications of energy loss due to friction, such as heat generation.

Mention of possible sources of friction, including bike gearing, wind resistance, or skidding.

Encouragement for viewers to apply their understanding to problems involving friction.

Anticipation of future videos covering more complex physics problems.

Transcripts

Welcome back.

Welcome back. Welcome back.

I'll now do another conservation of energy

problem, and this time I'll add another twist. So far,

everything we've been doing, energy was conserved by the

law of conservation.

But that's because all of the forces that were acting in

these systems were conservative forces.

And now I'll introduce you to a problem that has a little

bit of friction, and we'll see that some of that energy gets

lost to friction.

And we can think about it a little bit.

Well where does that energy go?

And I'm getting this problem from the University of

Oregon's zebu.uoregon.edu.

And they seem to have some nice physics problems, so I'll

use theirs.

And I just want to make sure they get credit.

So let's see.

They say a 90 kilogram bike and rider.

So the bike and rider combined are 90 kilograms. So let's

just say the mass is 90 kilograms. Start at rest from

the top of a 500 meter long hill.

OK, so I think they mean that the hill is

something like this.

So if this is the hill, that the hypotenuse here is 500

hundred meters long.

So the length of that, this is 500 meters.

A 500 meter long hill with a 5 degree incline.

So this is 5 degrees.

And we can kind of just view it like a wedge, like we've

done in other problems. There you go.

That's pretty straight.

OK.

Assuming an average friction force of 60 newtons.

OK, so they're not telling us the coefficient of friction

and then we have to figure out the normal

force and all of that.

They're just telling us, what is the drag of friction?

Or how much is actually friction acting against this

rider's motion?

We could think a little bit about where that friction is

coming from.

So the force of friction is equal to 60 newtons And of

course, this is going to be going against his motion or

her motion.

And the question asks us, find the speed of the biker at the

bottom of the hill.

So the biker starts up here, stationary.

That's the biker.

My very artful rendition of the biker.

And we need to figure out the velocity at the bottom.

This to some degree is a potential energy problem.

It's definitely a conservation of mechanical energy problem.

So let's figure out what the energy of the system is when

the rider starts off.

So the rider starts off at the top of this hill.

So definitely some potential energy.

And is stationary, so there's no kinetic energy.

So all of the energy is potential, and what is the

potential energy?

Well potential energy is equal to mass times the acceleration

of gravity times height, right?

Well that's equal to, if the mass is 90, the acceleration

of gravity is 9.8 meters per second squared.

And then what's the height?

Well here we're going to have to break out a little

trigonometry.

We need to figure out this side of this triangle, if you

consider this whole thing a triangle.

Let's see.

We want to figure out the opposite.

We know the hypotenuse and we know this angle here.

So the sine of this angle is equal to opposite over

hypotenuse.

So, SOH.

Sine is opposite over hypotenuse.

So we know that the height-- so let me do a little work

here-- we know that sine of 5 degrees is equal to

the height over 500.

Or that the height is equal to 500 sine of 5 degrees.

And I calculated the sine of 5 degrees ahead of time.

Let me make sure I still have it.

That's cause I didn't have my calculator with me today.

But you could do this on your own.

So this is equal to 500, and the sine of

5 degrees is 0.087.

So when you multiply these out, what do I get?

I'm using the calculator on Google actually.

500 times sine.

You get 43.6.

So this is equal to 43.6.

So the height of the hill is 43.6 meters.

So going back to the potential energy, we have the mass times

the acceleration of gravity times the height.

Times 43.6.

And this is equal to, and then I can use just my regular

calculator since I don't have to figure out

trig functions anymore.

So 90-- so you can see the whole thing-- times 9.8 times

43.6 is equal to, let's see, roughly 38,455.

So this is equal to 38,455 joules or newton meters.

And that's a lot of potential energy.

So what happens?

At the bottom of the hill-- sorry, I have to readjust my

chair-- at the bottom of the hill, all of this gets

converted to, or maybe I should

pose that as a question.

Does all of it get converted to kinetic energy?

Almost. We have a force of friction here.

And friction, you can kind of view friction as something

that eats up mechanical energy.

These are also called nonconservative forces because

when you have these forces at play, all of the

force is not conserved.

So a way to think about it is, is that the energy, let's just

call it total energy.

So let's say total energy initial, well let me just

write initial energy is equal to the energy wasted in

friction-- I should have written just letters-- from

friction plus final energy.

So we know what the initial energy is in this system.

That's the potential energy of this bicyclist and this

roughly 38 and 1/2 kilojoules or 38,500 joules, roughly.

And now let's figure out the energy wasted from friction,

and the energy wasted from friction is the negative work

that friction does.

And what does negative work mean?

Well the bicyclist is moving 500 meters in this direction.

So distance is 500 meters.

But friction isn't acting along the same

direction as distance.

The whole time, friction is acting against the distance.

So when the force is going in the opposite direction as the

distance, your work is negative.

So another way of thinking of this problem is energy initial

is equal to, or you could say the energy initial plus the

negative work of friction, right?

If we say that this is a negative quantity, then this

is equal to the final energy.

And here, I took the friction and put it on the other side

because I said this is going to be a negative quantity in

the system.

And so you should always just make sure that if you have

friction in the system, just as a reality check, that your

final energy is less than your initial energy.

Our initial energy is, let's just say 38.5 kilojoules.

What is the negative work that friction is doing?

Well it's 500 meters.

And the entire 500 meters, it's always pushing back on

the rider with a force of 60 newtons.

So force times distance.

So it's minus 60 newtons, cause it's going in the

opposite direction of the motion, times 500.

And this is going to equal the ending, oh, no.

This is going to equal the final energy, right?

And what is this?

60 times 500, that's 3,000.

No, 30,000, right.

So let's subtract 30,000 from 38,500.

So let's see.

Minus 30.

I didn't have to do that.

I could have done that in my head.

So that gives us 8,455 joules is equal to the final energy.

And what is all the final energy?

Well by this time, the rider's gotten back to, I guess we

could call the sea level.

So all of the energy is now going to be

kinetic energy, right?

What's the formula for kinetic energy?

It's 1/2 mv squared.

And we know what m is.

The mass of the rider is 90.

So we have this is 90.

So if we divide both sides.

So the 1/2 times 90.

That's 45.

So 8,455 divided by 45.

So we get v squared is equal to 187.9.

And let's take the square root of that and we get the

velocity, 13.7.

So if we take the square root of both sides of this, so the

final velocity is 13.7.

I know you can't read that.

13.7 meters per second.

And this was a slightly more interesting problem because

here we had the energy wasn't completely conserved.

Some of the energy, you could say, was eaten by friction.

And actually that energy just didn't

disappear into a vacuum.

It was actually generated into heat.

And it makes sense.

If you slid down a slide of sandpaper, your pants would

feel very warm by the time you got to the bottom of that.

But the friction of this, they weren't specific on where the

friction came from, but it could have come from the

gearing within the bike.

It could have come from the wind.

Maybe the bike actually skidded a little

bit on the way down.

I don't know.

But hopefully you found that a little bit interesting.

And now you can not only work with conservation of

mechanical energy, but you can work problems where there's a

little bit of friction involved as well.

Anyway, I'll see you in the next video.