### Video Transcript

In this video, we’re going to learn about conservative and nonconservative forces. We’ll see what these two force types are and how they’re different from one another.

To start out, imagine that you are the captain of a gigantic, inflatable blimp. After a long day of riding, you’re running low on fuel and you want to make it back to the landing pad using as little fuel as possible. Of all the many pads they can get you to the landing pad, which one will be the most fuel efficient?

To better understand this question, we want to learn something about conservative and nonconservative forces. At the heart of this question of conservative and nonconservative forces is the work done by these forces. Say that we have a spring and it’s unstressed and extended to its natural length. If we then compress that spring using a force, then the energy used in doing that work is not lost energy rather it’s stored as potential energy in the spring. In this way, we can say that the spring force conserves energy. It is a conservative force.

Or think of another example, say we take a mass 𝑚 and we move it on top of a table of height ℎ. The work it takes to do this is equal to the force of gravity 𝑚 times 𝑔 multiplied by the height against that force we moved this mass. Considering the mass on the top of the table, we know that that energy isn’t lost. It’s stored as gravitational potential energy. So gravity like the spring force conserves the energy used to do work. We can say that a force is a conservative force if the work it does results in potential energy. In other words, the energy that goes into doing the work is conserved rather than dissipated or lost.

There’s something else that’s true though about conservative forces. And we can use our mass in this example of gravity to show it. One question we might have when we see the mass in its starting position and then in its ending position is “how did it move between those two points?” For example, maybe we picked up the mass and then walked it over to the table top and placed it down or maybe we slid it along the ground with our foot then picked it up when it got to the edge of the table and placed it on top or perhaps we picked the box up as we’re moving towards the table in one smooth motion.

Considering these different possible pathways for getting the box from its start to its end point, we realize that from the perspective of gravitational potential energy the path makes no difference. Given the mass of the box, all that matters is the height difference between where it started and where it ended. This leads us to a second way of describing a conservative force. A force is conservative if the work it does does not depend on the path taken — that is, its path independent.

We’ve seen that two examples of conservative forces are gravity and the spring force. Now, what about nonconservative forces? We can say that a force is nonconservative if the work it does does depend on the path taken.

Imagine we had a large heavy crate sitting on a rough concrete floor and say we wanted to slide the crate along the floor to the opposite corner in the room. We can see that the particular path we choose affects how much work it takes to move the box from its start to its end point. Not only that, but for all the energy we input to do that work to move the box, when the box reaches its final location, we don’t have any potential energy to show for it. That energy has been dissipated as heat energy through friction. This is why another name for a nonconservative force is a dissipative force. It’s not easy to recover the energy that goes into the work done by a nonconservative force.

We’ve seen that one example of a nonconservative force is friction. Another example is air resistance, which at the molecular level is also due to friction. Let’s practice using these ideas of conservative and nonconservative forces through a couple of examples.

You are in a room in a basement with a smooth concrete floor and a nice rug. The rug is three metres wide and four metres long. You have to push a very heavy box from one corner of the rug to its opposite corner. The magnitude of friction between the box and the rug is 55 newtons. But the magnitude of friction between the box and the concrete floor is only 40 newtons. Will you do more work against friction going around the floor or across the rug? How much extra work would it take?

This is an exercise involving nonconservative forces. And to start out, let’s draw a diagram. In this situation, we have a very heavy crate on the corner of a three-by-four-metre rug. We want to move the crate to the opposite corner. And we’re told that if we move the crate straight across the rug, the force of friction is 55 newtons, while if we choose instead to move the crate on the smooth concrete floor around the rug, the overall force of friction is 40 newtons. We want to figure out which of these two pads will take more work.

Recalling that work is equal to force times distance, we can write that the work required to move the crate across the rug equals 𝐹 sub 𝑟, given as 55 newtons, times the distance across the rug the crate moves. Since the rug is a rectangle, that distance 𝑑 sub 𝑟 is equal to the square root of three metres squared plus four metres squared or five metres. Plugging in these values and calculating 𝑊 sub 𝑟, it’s equal to 275 joules.

Next, we want to calculate the work done if we move the crate across the floor instead of the rug. This is equal to 𝐹 sub 𝑓, which is 40 newtons, times 𝑑 sub 𝑓, which is the distance the crate would move. This distance skirting the edge of the rug is equal to four plus three or seven metres. Plugging in for these two values, when we calculate 𝑊 sub 𝑓, we find it’s 280 joules. So comparing the work done on the rug to the work done on the floor, we see that more work is required to move the crate across the floor by five joules.

Now, let’s look at an example involving a conservative force.

A particle with a mass 𝑚 is suspended from a string of negligible mass and a length of 1.0 metres, as shown in the diagram. The particle is displaced to a position where the taut string is at an angle of 30 degrees from the vertical. And the particle is released from rest at that position. The particle moves through an arc, where the lowest point of the arc is the point 𝑃. What is the instantaneous speed of the particle at point 𝑃? What is the vertically upward displacement of the particle from point 𝑃 when its instantaneous speed is 0.81 metres per second?

We can call this instantaneous particle speed at point 𝑃 𝑣 and the vertically upward displacement of the particle from 𝑃 we’ll name 𝑑. We’re told that point 𝑃 on our diagram is the location of where the mass would be when the string is hanging straight down. We start off solving for the speed of the mass when it’s at point 𝑃.

Because gravity is a conservative force, we know that the potential energy of the mass 𝑚 when it’s in its original position can be converted into kinetic energy of the mass when it’s at point 𝑃. We can write our energy balance equation to say that the initial kinetic plus potential energy is equal to the final kinetic plus potential energy of the mass. At the outsaid, the speed of the mass is zero. So its initial kinetic energy is zero. And if we set the altitude of the mass at point 𝑃 to be zero, then its final potential energy will be zero as well. So we can say the initial potential energy of the mass is equal to its final kinetic energy.

Recalling that gravitational potential energy equals mass times the acceleration due to gravity times height and that kinetic energy equals one-half an object’s mass times its speed squared, we can write that 𝑚 times 𝑔 times ℎ is equal to one-half 𝑚𝑣 squared. We see that the mass of this object cancels out. And rearranging to solve for the speed 𝑣, we find it’s equal to the square root of two times 𝑔 times ℎ.

𝑔, the acceleration due to gravity, we treat as exactly 9.8 metres per second squared. And we see on our diagram that ℎ is the height difference between the point 𝑃 and the original location of the mass. That height difference is equal to 1.0 metres, the length of the string, minus the length of the string times the cosine of 30 degrees. Plugging this value for ℎ and the known value for 𝑔 into our expression for 𝑣, to two significant figures 𝑣 is 1.6 metres per second. That’s how fast the mass is moving when it’s at its lowest point 𝑃.

In part two, we imagine that the speed of our mass — we’ll call it 𝑣 sub 𝑑 — is 0.81 metres per second. We want to solve for the vertical distance 𝑑 of our mass at that speed above point 𝑃. In this case, when we write out our energy balance equation, we’re only able to eliminate the initial potential energy of our mass because we imagine its initial point is at point 𝑃. At point 𝑃, the mass has kinetic energy. And at the point where its speed is 0.81 metres per second, it will have both kinetic and potential energy.

Using the variables we’ve decided on, we can write that one-half 𝑚𝑣 squared, where 𝑣 is the speed we solved for in part one, is equal to one-half 𝑚𝑣 sub 𝑑 squared plus 𝑚 times 𝑔 times 𝑑 — the distance we want to solve for. Again, the object’s mass cancels out and we can rearrange to solve for 𝑑. It’s 𝑣 squared minus 𝑣 sub 𝑑 squared all over two times 𝑔. When we plug in for these values and calculate 𝑑, to two significant figures we find it’s 3.3 centimetres. That’s the vertical displacement of the mass above point 𝑃.

Let’s summarize what we’ve learnt so far about conservative and nonconservative forces. We’ve seen that a force is conservative if the work it does does not depend on the path taken; that is, its path independent. And a force is nonconservative if it does. Another way to say this is that a force is conservative if the work it does results in potential energy and nonconservative if the work it does dissipates energy. And finally, by way of example of these forces, we’ve seen that gravity and the spring force are examples of conservative forces, while friction and air resistance are examples of nonconservative forces.