Physics
Conservative Force
A conservative force is a type of force that does not depend on the path taken by an object. The work done by a conservative force is independent of the path taken by the object and only depends on the initial and final positions of the object. Examples of conservative forces include gravity and electrostatic forces.
Written by Perlego with AI-assistance
Related key terms
1 of 5
12 Key excerpts on "Conservative Force"
eBook - PDF- Raymond Serway, Chris Vuille(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
This observation motivates the following definition of a Conservative Force: A force is conservative if the work it does moving an object between two points is the same no matter what path is taken. NonConservative Forces, as we’ve seen, don’t have this property. The work–energy theorem, Equation 5.7, can be rewritten in terms of the work done by Conservative Forces W c c and the work done by nonConservative Forces c and the work done by nonConservative Forces c W nc nc because the net work is because the net work is just the sum of these two: W nc nc 1 W c c 5 D KE [5.8] It turns out that Conservative Forces have another useful property: The work they do can be recast as something called potential energy, a quantity that depends only on the beginning and end points of a curve, not the path taken. 5.3 Gravitational Potential Energy An object with kinetic energy (energy of motion) can do work on another object, just like a moving hammer can drive a nail into a wall. A brick on a high shelf can also do work: it can fall off the shelf, accelerate downward, and hit a nail squarely, driving it into the floorboards. The brick is said to have potential energy associated y associated y with it, because from its location on the shelf it can potentially do work. Potential energy is a property of a system, rather than of a single object, because it’s due to the relative positions of interacting objects in the system, such as the position of the diver in Figure 5.11 relative to the Earth. In this topic we define a system as a collection of objects interacting via forces or other processes that are internal to the system. It turns out that potential energy is another way of looking at the work done by Conservative Forces. b Conservative Force Figure 5.11 Because the gravity field is conservative, the diver regains as kinetic energy the work she did against gravity in climbing the ladder. Taking the frictionless slide gives the same result.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
Version 2 A force is conservative when it does no net work on an object moving around a closed path, starting and finishing at the same point. Figure 6.13 helps us to illustrate version 2 of the definition of a Conservative Force. The picture shows a roller coaster car racing through dips and double dips, ultimately returning to its starting point. This kind of path, which begins and ends at the same place, is called a closed path. Gravity provides the only force that does work on the car, assuming that there is no friction or air resistance. Of course, the track exerts a normal force, but this force is always directed perpendicular to the motion and does no work. On the downward parts of the trip, the gravitational force does positive work, increasing the car’s kinetic energy. Conversely, on the upward parts of the motion, the gravitational force does negative work, decreasing the car’s kinetic energy. Over the entire trip, the gravitational force does as much positive work as negative work, so the net work is zero, and the car returns to its starting point with the same kinetic energy it had at the start. Therefore, consistent with version 2 of the definition of a Conservative Force, W gravity 5 0 J for a closed path. The gravitational force is our first example of a Conservative Force. Later, we will encounter others, such as the elastic force of a spring and the electrical force of elec- trically charged particles. With each Conservative Force we will associate a potential energy, as we have already done in the gravitational case (see Equation 6.5). For other Conservative Forces, however, the algebraic form of the potential energy will differ from that in Equation 6.5. Not all forces are conservative. A force is nonconservative if the work it does on an object moving between two points depends on the path of the motion between the points. The kinetic frictional force is one example of a nonConservative Force.
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
Conservative Forces, NonConservative Forces, and Potential Energy | 159 A closed path is any set of displacements that start and end at the same point. In Figure 6.4.1, the starting and finishing point for the diver is the bottom of the ladder. Practice Problem 6.4.1 explores definition 1 of a Conservative Force. It follows from definition 1 that the work done by a Conservative Force is independent of the path along which it acts: Conservative Force (Definition 2) A force is conservative if the work it does on an object as it moves from place to place is always independent of the details of the path it takes. Definitions 1 and 2 of a Conservative Force are abstract, but they are fundamentally impor- tant in physics. We can define potential energy only for Conservative Forces. Going Deeper Proof of Definition 2 The following figure shows two points, A and B, and three paths connecting those two points. B 1 2 3 A Suppose that a specific Conservative Force acts on an object that moves along these paths. From definition 1 of a Conservative Force, the net work is zero if the object moves from A to B along path 1, then returns to A along path 2: W 1 + W 2 = 0 or W 2 = −W 1 Similarly, if the object moves from A to B along path 1 and returns to A along path 3, then W 1 + W 3 = 0 or W 3 = −W 1 It follows that W 2 = W 3 . This analysis applies to any return path from B to A, and thus proves definition 2. NonConservative Forces In addition to gravity, the force exerted by an ideal spring (i.e., one that obeys Hooke’s law) is also a Conservative Force. For now, these are the only Conservative Forces we have dealt with. Most forces are nonconservative. Consider sliding a box around a room by pushing it along with a horizontal force. The work done by the pushing force is positive because the force is in the same direction as the displacement.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Therefore, consistent with version 2 of the definition of a Conservative Force, W gravity = 0 J for a closed path. The gravitational force is our first example of a Conservative Force. Later, we will encounter others, such as the elastic force of a spring and the electrical force of electrically charged par- ticles. With each Conservative Force we will associate a potential energy, as we have already done in the gravitational case (see Equation 6.5). For other Conservative Forces, however, the algebraic form of the potential energy will differ from that in Equation 6.5. Not all forces are conservative. A force is nonconservative if the work it does on an object moving between two points depends on the path of the motion between the points. The kinetic frictional force is one example of a nonConservative Force. When an object slides over a surface and kinetic friction is present, the frictional force points opposite to the sliding motion and does negative work. Between any two points, greater amounts of work are done over longer paths between the points, so that the work depends on the choice of path. Thus, the kinetic frictional force is nonconservative. Air resistance is another nonConservative Force. The concept of poten- tial energy is not defined for a nonConservative Force. For a closed path, the total work done by a nonConservative Force is not zero as it is for a Conservative Force. In Figure 6.14, for instance, a frictional force would oppose the motion and slow down the car. Unlike gravity, friction would do negative work on the car throughout the entire trip, on both the up and down parts of the motion. Assuming that the car makes it back to the starting point, the car would have less kinetic energy than it had originally. Table 6.2 gives some examples of conservative and nonConservative Forces. Start FIGURE 6.14 A roller coaster track is an example of a closed path.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Later, we will encounter others, such as the elastic force of a spring and the electrical force of electri- cally charged particles. With each Conservative Force we will associate a potential energy, as we have already done in the gravitational case (see Equation 6.5). For other conser- vative forces, however, the algebraic form of the potential energy will differ from that in Equation 6.5. Not all forces are conservative. A force is nonconservative if the work it does on an object moving between two points depends on the path of the motion between the points. The kinetic frictional force is one example of a nonConservative Force. When an object slides over a surface and kinetic friction is present, the frictional force points opposite to the sliding motion and does negative work. Between any two points, greater amounts of work are done over longer paths between the points, so that the work depends on the choice of path. Thus, the kinetic frictional force is nonconservative. Air resistance is another nonConservative Force. The concept of potential energy is not defined for a nonConservative Force. For a closed path, the total work done by a nonConservative Force is not zero as it is for a Conservative Force. In Figure6.14, for instance, a frictional force would oppose the motion and slow down the car. Unlike gravity, friction would do negative work on the car throughout the entire trip, on both the up and down parts of the motion. Assuming that the car makes it back to the starting point, the car would have less kinetic energy than it had originally. Table6.2 gives some examples of conservative and nonConservative Forces. In normal situations, Conservative Forces (such as gravity) and nonConservative Forces (such as friction and air resistance) act simultaneously on an object.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
175 C H A P T E R 8 After reading this module, you should be able to . . . 8.1.1 Distinguish a Conservative Force from a nonConservative Force. 8.1.2 For a particle moving between two points, identify that the work done by a Conservative Force does not depend on which path the particle takes. 8.1.3 Calculate the gravitational potential energy of a particle (or, more properly, a particle–Earth system). 8.1.4 Calculate the elastic potential energy of a block–spring system. 8.1 POTENTIAL ENERGY KEY IDEAS 1. A force is a Conservative Force if the net work it does on a particle moving around any closed path, from an initial point and then back to that point, is zero. Equivalently, a force is conservative if the net work it does on a particle moving between two points does not depend on the path taken by the par- ticle. The gravitational force and the spring force are Conservative Forces; the kinetic frictional force is a nonConservative Force. 2. Potential energy is energy that is associated with the configuration of a system in which a Conservative Force acts. When the Conservative Force does work W on a particle within the system, the change ∆U in the potential energy of the system is ∆U = −W. If the particle moves from point x i to point x f , the change in the potential energy of the system is ΔU = − x t x f F(x) dx . 3. The potential energy associated with a system consisting of Earth and a nearby particle is gravitational potential energy. If the particle moves from height y i to height y f , the change in the gravitational potential energy of the particle–Earth system is ∆U = mg( y f − y i ) = mg ∆y. 4. If the reference point of the particle is set as y i = 0 and the corresponding gravitational potential energy of the system is set as U i = 0, then the gravi- tational potential energy U when the particle is at any height y is U( y) = mgy. 5. Elastic potential energy is the energy associated with the state of compression or extension of an elastic object.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Therefore, consistent with version 2 of the definition of a Conservative Force, W gravity = 0 J for a closed path. FIGURE 6.14 A roller coaster track is an example of a closed path. Start CHAPTER 6 Work and energy 155 The gravitational force is our first example of a Conservative Force. Later, we will encounter others, such as the elastic force of a spring and the electrical force of electrically charged particles. With each Conservative Force we will associate a potential energy, as we have already done in the gravitational case (see equation 6.5). For other Conservative Forces, however, the algebraic form of the potential energy will differ from that in equation 6.5. Not all forces are conservative. A force is nonconservative if the work it does on an object moving between two points depends on the path of the motion between the points. The kinetic frictional force is one example of a nonConservative Force. When an object slides over a surface and kinetic friction is present, the frictional force points opposite to the sliding motion and does negative work. Between any two points, greater amounts of work are done over longer paths between the points, so that the work depends on the choice of path. Thus, the kinetic frictional force is nonconservative. Air resistance is another nonConservative Force. The concept of potential energy is not defined for a nonConservative Force. TABLE 6.2 Some conservative and nonConservative Forces Conservative Forces Gravitational force (ch. 4) Elastic spring force (ch. 10) Electric force (ch. 18, 19) NonConservative Forces Static and kinetic frictional forces Air resistance Tension Normal force Propulsion force of a rocket For a closed path, the total work done by a nonConservative Force is not zero as it is for a Conservative Force. In figure 6.14, for instance, a frictional force would oppose the motion and slow down the car.
eBook - PDF- Paul Peter Urone, Roger Hinrichs(Authors)
- 2012(Publication Date)
- Openstax(Publisher)
Plot velocity squared versus the distance traveled by the marble. What is the shape of each plot? If the shape is a straight line, the plot shows that the marble’s kinetic energy at the bottom is proportional to its potential energy at the release point. Figure 7.9 A marble rolls down a ruler, and its speed on the level surface is measured. 7.4 Conservative Forces and Potential Energy Potential Energy and Conservative Forces Work is done by a force, and some forces, such as weight, have special characteristics. A Conservative Force is one, like the gravitational force, for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. We can define a potential energy (PE) for any Conservative Force, just as we did for the gravitational force. For example, when you wind up a toy, an egg timer, or an old-fashioned watch, you do work against its spring and store energy in it. (We treat these springs as ideal, in that we assume there is no friction and no production of thermal energy.) This stored energy is recoverable as work, and it is useful to think of it as potential energy contained in the spring. Indeed, the reason that the spring has this characteristic is that its force is conservative. That is, a Conservative Force results in stored or potential energy. Gravitational potential energy is one example, as is the energy stored in a spring. We will also see how Conservative Forces are related to the conservation of energy. Potential Energy and Conservative Forces Potential energy is the energy a system has due to position, shape, or configuration. It is stored energy that is completely recoverable. A Conservative Force is one for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. We can define a potential energy (PE) for any Conservative Force.
eBook - PDF- Robert Resnick, David Halliday, Kenneth S. Krane(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
If this total work is zero, we call the force a Conservative Force. If the total work for the round trip is not zero, we call the force a nonconser- vative force. The elastic restoring force (spring force) and gravity are two examples of Conservative Forces. Friction is an example of a nonConservative Force. A second way to identify a force as conservative or non- conservative is based on a comparison of the work done when the object on which the force acts moves from one lo- cation to another by several different paths. For example, suppose you are moving packages of mass m from the base- ment to the first floor in a building that has several floors, each of height h. If you move a package directly from the basement to the first floor, the (conservative) gravitational force acting on the package does work W g mgh. If in- stead you first move it to the fifth floor (W g 5mgh) and then return it to the first floor (W g 4mgh), the total work done by gravity for the entire process is W g mgh, exactly the same as if you had carried the package directly. No matter how many intermediate stopping points or how many times you go back and forth over the same path, when you finally deliver the package to the first floor, the total work done by gravity between the original location of the package (the basement) and its final location (the first floor) will be mgh. On the other hand, consider the behavior of the noncon- servative frictional force for the system illustrated in Fig. 12-3 as the disk moves along two different paths from posi- 258 Chapter 12 / Energy 2: Potential Energy + kd 2 1 2 – kd 2 1 2 + kd 2 1 2 – kd 2 W= W= W= W= 1 2 (a) (b) (c) (d) (e) x x = 0 x = + d x = – d k m FIGURE 12-1. A block moves under the action of a spring force from (a) x d to (b) x 0, moving left, to (c) x d, to (d ) x 0, moving right, and (e) back to x d.
eBook - PDFWorkshop Physics Activity Guide Module 2
Mechanics II
- Priscilla W. Laws, David P. Jackson, Brett J. Pearson(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
In other words, the system’s mechanical energy is conserved. Defining a potential energy function is the first step toward developing the law of conservation of energy. Like the law of conservation of momentum, this principle is extremely powerful and allows us to analyze certain types of prob- lems with minimal effort. 11.2 CONSERVATIVE AND NON-Conservative ForceS In the example above, the change in the ball’s kinetic energy is a direct result of the gravitational force between the ball and Earth. Any force that allows the kinetic energy to be temporarily “stored” and then returned in its entirety is called a Conservative Force. Gravity is an example of a Conservative Force. Some forces act on objects in such a way that the kinetic energy is not stored, and so mechanical energy is “lost” from the system; such forces are called non-conservative. As the following activity demonstrates, friction is an example of a non-Conservative Force. You may find it helpful to have a wooden block available while doing the activity. 11.2.1. Activity: Kinetic Energy of a Sliding Block a. Consider placing an object such as a wooden block (m = 0.35 kg) on a tabletop in which the coefficient of sliding friction is k = 0.4. Imagine giving the block a quick push so that it has an initial speed of v = 2.8 m/s. It then slides to a stop (before falling off the end of the table!). We are interested in the motion of the block after you release it. Take the system to be composed of only the block and treat it as a point particle. Draw a free-body diagram showing all the forces acting on the block and explain whether the work done by each force is positive, negative, or zero. b. Use the work-energy principle for a point particle (or, equivalently, the center-of-mass equation) to determine the distance the block slides before coming to rest. Hint: You may wish to review Section 10.7.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
This kind of path, which begins and ends at the same place, is called a closed path. Gravity provides the only force that does work on the car, assuming that there is no friction or air resistance. Of course, the track exerts a normal force, but this force is always directed perpendicular to the motion and does no work. On the downward parts of the trip, the gravitational force does positive work, increasing the car’s kinetic energy. Start Figure 6.14 A roller coaster track is an example of a closed path. 6.4 | Conservative Versus NonConservative Forces 155 Conversely, on the upward parts of the motion, the gravitational force does negative work, decreasing the car’s kinetic energy. Over the entire trip, the gravitational force does as much positive work as negative work, so the net work is zero, and the car returns to its starting point with the same kinetic energy it had at the start. Therefore, consistent with version 2 of the definition of a Conservative Force, W gravity 5 0 J for a closed path. The gravitational force is our first example of a Conservative Force. Later, we will encounter others, such as the elastic force of a spring and the electrical force of elec- trically charged particles. With each Conservative Force we will associate a potential energy, as we have already done in the gravitational case (see Equation 6.5). For other Conservative Forces, however, the algebraic form of the potential energy will differ from that in Equation 6.5. Not all forces are conservative. A force is nonconservative if the work it does on an object moving between two points depends on the path of the motion between the points. The kinetic frictional force is one example of a nonConservative Force. When an object slides over a surface and kinetic friction is present, the frictional force points opposite to the sliding motion and does negative work.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
Potential energy for an ideal spring U(x) = 1 2 kx 2 + const. Work done by Conservative Force over a closed path W closed path = ∮ E → cons · d r → = 0 Condition for Conservative Force in two dimensions ⎛ ⎝ dF x dy ⎞ ⎠ = ⎛ ⎝ dF y dx ⎞ ⎠ Conservative Force is the negative derivative of potential energy F l = − dU dl Conservation of energy with no non-Conservative Forces 0 = W nc, AB = Δ(K + U) AB = ΔE AB . SUMMARY 8.1 Potential Energy of a System • For a single-particle system, the difference of potential energy is the opposite of the work done by the forces acting on the particle as it moves from one position to another. • Since only differences of potential energy are physically meaningful, the zero of the potential energy function can be chosen at a convenient location. 386 Chapter 8 | Potential Energy and Conservation of Energy This OpenStax book is available for free at http://cnx.org/content/col12031/1.5 • The potential energies for Earth’s constant gravity, near its surface, and for a Hooke’s law force are linear and quadratic functions of position, respectively. 8.2 Conservative and Non-Conservative Forces • A Conservative Force is one for which the work done is independent of path. Equivalently, a force is conservative if the work done over any closed path is zero. • A non-Conservative Force is one for which the work done depends on the path. • For a Conservative Force, the infinitesimal work is an exact differential. This implies conditions on the derivatives of the force’s components. • The component of a Conservative Force, in a particular direction, equals the negative of the derivative of the potential energy for that force, with respect to a displacement in that direction. 8.3 Conservation of Energy • A conserved quantity is a physical property that stays constant regardless of the path taken. • A form of the work-energy theorem says that the change in the mechanical energy of a particle equals the work done on it by non-Conservative Forces.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
