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UExcel Physics: Study Guide & Test Prep18 chapters | 201 lessons | 13 flashcard sets

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Lesson Transcript

Instructor:
*Damien Howard*

Damien has a master's degree in physics and has taught physics lab to college students.

Learn how to tell if a force is conservative and what exactly is being conserved. Then look at a couple of specific examples of forces to see how they are conservative.

Every once in a while you'll hear the topic of conservation come up. You might be watching a movie where an injured person is told to conserve their energy, or you might hear on the news that people need to conserve water due to a drought. No matter how it comes up, the basic concept of conservation is that you are maintaining a supply of something instead of using it all up.

Conservation is also a concept that comes up in physics, such as with **conservative forces**. Despite the name, what's being conserved with conservative forces is energy. Here, force and energy are related through work. Work is a force multiplied by a displacement, and the result comes in the form of energy. Often, conserving this energy is accomplished by transferring potential energy to kinetic energy and vice versa within a system instead of giving off the energy in some other form.

We can tell whether or not a force is conservative by looking at something called path independence. A force is **path independent** and, therefore, conservative when the work done by the force does not depend on the route the object travels. Say we have a particle moving from point A to point B. In this image, we see a few examples of different paths a particle could take to move between the points. However, no matter which path is chosen, the work done by the force moving the particle will be the same as long as it is conservative. To help further understand path independence, we'll look at a couple examples of forces and show that they are path independent and therefore conservative.

**Force due to gravity** is the attractive force between all objects with mass. On Earth, we feel this force as weight. This is one of the most common conservative forces we can look at. For this example, the work being done on an object is the force due to gravity multiplied by height. The work done turns out to be the change in potential energy, which can be written as follows.

Delta *PE* = *m* * *g* * (*h*{*f*} - *h*{*i*})

*m* = mass

*g* = acceleration due to gravity

*h*{*f*} = final height

*h*{*i*} = initial height

If we think of someone dropping a ball while standing still, this equation makes sense. The ball will drop straight down, so the change in height, (*h*{*f*} - *h*{*i*}), will be the exact distance it travels. Now, imagine a ball being dropped out of a moving car instead. It doesn't drop straight down. It now travels horizontally in the direction the car was moving while falling to the ground. The change in height is no longer necessarily the exact distance the ball travels since it's moving in two dimensions.

However, the equation for change in potential energy doesn't adjust to show the horizontal travel. The only distances given are still the final and initial heights. So for the work done by the force due to gravity, it doesn't matter what path the ball takes to travel from the initial height to the final height. This lets us see that force due to gravity is path independent and a conservative force.

Another common conservative force is the **elastic spring force**, which is the force a spring enacts on an object attached to it when stretching or compressing. The set up for this turns out to be quite similar to the force due to gravity, as the work done by the elastic spring force is also the change in potential energy. The equation for change in potential energy in this case is written as follows.

Delta *PE* = (1/2) * *k* * (*s*{2}^2 - *s*{1}^2)

*k* = spring constant

*s*{2} = the final displacement of the spring (stretching or compression)

*s*{1} = the initial displacement of the spring (compression or stretching)

Note that if the initial displacement is compression, the final displacement will be stretching and vice versa.

To see how this is a conservative force, let's imagine a spring pendulum. Here, the spring is the arm of the pendulum with a weight attached to one end. The spring oscillates up and down, stretching and compressing, while also swinging back and forth. In this example, the elastic spring force is doing work on the weight attached to the spring.

Looking at the equation for change in potential energy, we can see that just like the force due to gravity, there are only two distances. How much the spring is compressed and how much it is stretched out. Only those two points matter, and how the weight is swinging on the pendulum doesn't come into play. This shows that the work done on the weight isn't dependent on the path the weight is travelling, and the elastic spring force is a conservative force.

In a **conservative force**, energy is being conserved by keeping it within the system instead of giving it off. We can tell that a force is conservative when the work done by it is **path independent**. This means that the work being done does not depend on the route the object travels. Two examples of conservative forces are the **force due to gravity** and the **elastic spring force**. In both cases, the work being done by those forces is the change in potential energy. In their equations, we can see that the change in potential energy is path independent as they both only depend on two distances, and not how the object travels between those two distances.

After this lesson is finished you should be able to:

- Define conservative force and explain how it is related to energy
- Explain path independence
- Discuss two examples of conservative force

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UExcel Physics: Study Guide & Test Prep18 chapters | 201 lessons | 13 flashcard sets

- Go to Vectors

- Go to Kinematics

- What is Energy? - Definition and Significance in Nature 9:40
- Work: Definition, Characteristics, and Examples 4:38
- Work Done by a Variable Force 7:10
- Work-Energy Theorem: Definition and Application 4:29
- Kinetic Energy to Potential Energy: Relationship in Different Energy Types 5:59
- Gravitational Potential Energy: Definition, Formula & Examples 4:41
- Elastic Potential Energy: Definition, Formula & Examples 5:16
- Conservative Forces: Examples & Effects 5:17
- Conservation of Mechanical Energy 6:39
- Power: Definition and Mathematics 5:24
- Go to Work and Energy in Physics

- Go to Relativity

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