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Law of Conservation of Energy

The law of conservation of energy is a physical law that states that the total energy of an isolated system is a constant , although energy can change forms . In other words, energy is conserved over time. The law of conservation of energy is the first law of thermodynamics . French mathematician and philosopher Émilie du Châtelet first proposed and tested the law in the 18th century.

Formulas for the Law of Conservation of Energy

There are a few different ways of writing the formula for the law of conservation of energy. One of the most common formulas describes the relationship between kinetic energy (K) and potential energy (U):

K 1 + U 1 = K 2 + U 2

In this case, the total energy of the system is a constant, but energy converts between potential and kinetic energy.

For calculations involving frictionless carts, swings, pendulums, throwing a ball, etc., there is another useful form of the equation for the conservation of energy, which uses the following formulas for potential and kinetic energy:

U = mgh ; where U is potential energy, m is mass, g is acceleration due to gravity, and h is height K = ½mv 2 ; where m is mass and v is velocity

Total energy is the sum of potential and kinetic energy:

E total = mgh + ½mv 2

This formula works well for physics problems where there is no friction. More complex equation cover the situation where some energy gets converted into heat via friction.

Conservation of Energy Example Problem

See the formula for conservation of energy in action with this common physics problem involving a cart traveling on a frictionless track.

Another form of the laws of conservation of energy states that the internal energy (∆E) of a system is the sum of the heat flow (Q) across its boundaries (q) and the work done on the system (W).

Examples of the Law of Conservation of Energy

There are many examples of the law of conservation of energy in everyday life:

• The energy of a child on a swing changes between potential and kinetic energy. At the top of the swing, all of the energy is potential. At the bottom of the swing, it’s all kinetic. The energy is a mixture of kinetic and potential energy between these two points. In a frictionless system, the potential energy at the top equals the kinetic energy at the bottom, which equals the sum of the kinetic and potential energy at the other points.
• A swinging pendulum also illustrates a conversion between kinetic and potential energy, exactly like a swing. Of course, in both the swing and pendulum examples, friction plays a role. The conserved energy really is a mixture of kinetic energy, potential energy, and thermal energy or heat.
• A car converts chemical energy (gasoline) into kinetic energy. Here again, so energy is lost as heat, but the sum of the forms of energy remains constant.
• As an apple falls from a tree, it starts out with potential energy. As it falls, it has a mixture of kinetic and potential energy. In the instant it strikes the ground, all of its energy is kinetic. The sum of its potential and kinetic energy is a constant value.
• A flashlight converts chemical energy from its battery into electrical energy, which is then converted into light and heat.
• A speaker converts electrical energy into sound, which is another form of energy.
• Generators convert mechanical energy into electrical energy.
• Your body converts chemical energy from food into mechanical energy (moving muscles), different chemical energy molecules, and heat.
• An exploding firework converts chemical potential energy into kinetic energy, light, heat, and sound.

Classical Mechanics vs General Relativity

In classical mechanics, the law of conservation of energy and the law of conservation of mass are two separate laws. However, they combine in relativity in Einstein’s famous equation:

This equation shows mass can convert into energy, and vice versa. The law of conservation of energy still holds true, as long as the reference from of the observer remains unchanged.

Perpetual Motion Machines

One consequence of the law of conservation of energy is that is means perpetual motion machines of the first kind are impossible. These are machines that do work forever without any additional energy input. While perpetual motion that does work might look good on paper, it doesn’t work in the real world because some energy in a machine changes form into heat. Usually, this is from friction. So, to keep a machine running actually requires a continuous input of energy.

Remember, the law of conservation of energy applies to a closed system. Sometimes it isn’t easy or even possible to define or isolate a system. This comes into play in general relativity, where systems don’t always have time translation symmetry. For example, conservation of energy isn’t necessarily defined for curved spacetime or time crystals.

• Feynman, Richard (1970). The Feynman Lectures on Physics Vol I . Addison Wesley. ISBN 978-0-201-02115-8.
• Gibney, Elizabeth (2017). “The quest to crystallize time”. Nature . 543 (7644): 164–166. doi: 10.1038/543164a
• Hagengruber, Ruth (ed.) (2011). Émilie du Chatelet: Between Leibniz and Newton . Springer. ISBN 978-94-007-2074-9.
• Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. ISBN 978-0-7167-1088-2.
• Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks/Cole. ISBN 978-0-534-40842-8.

Related Posts

8.3 Conservation of Energy

Learning objectives.

By the end of this section, you will be able to:

• Formulate the principle of conservation of mechanical energy, with or without the presence of non-conservative forces
• Use the conservation of mechanical energy to calculate various properties of simple systems

In this section, we elaborate and extend the result we derived in Potential Energy of a System , where we re-wrote the work-energy theorem in terms of the change in the kinetic and potential energies of a particle. This will lead us to a discussion of the important principle of the conservation of mechanical energy. As you continue to examine other topics in physics, in later chapters of this book, you will see how this conservation law is generalized to encompass other types of energy and energy transfers. The last section of this chapter provides a preview.

The terms ‘conserved quantity’ and ‘conservation law’ have specific, scientific meanings in physics, which are different from the everyday meanings associated with the use of these words. (The same comment is also true about the scientific and everyday uses of the word ‘work.’) In everyday usage, you could conserve water by not using it, or by using less of it, or by re-using it. Water is composed of molecules consisting of two atoms of hydrogen and one of oxygen. Bring these atoms together to form a molecule and you create water; dissociate the atoms in such a molecule and you destroy water. However, in scientific usage, a conserved quantity for a system stays constant, changes by a definite amount that is transferred to other systems, and/or is converted into other forms of that quantity. A conserved quantity, in the scientific sense, can be transformed, but not strictly created or destroyed. Thus, there is no physical law of conservation of water.

Systems with a Single Particle or Object

We first consider a system with a single particle or object. Returning to our development of Equation 8.2 , recall that we first separated all the forces acting on a particle into conservative and non-conservative types, and wrote the work done by each type of force as a separate term in the work-energy theorem. We then replaced the work done by the conservative forces by the change in the potential energy of the particle, combining it with the change in the particle’s kinetic energy to get Equation 8.2 . Now, we write this equation without the middle step and define the sum of the kinetic and potential energies, K + U = E ; K + U = E ; to be the mechanical energy of the particle.

Conservation of Energy

The mechanical energy E of a particle stays constant unless forces outside the system or non-conservative forces do work on it, in which case, the change in the mechanical energy is equal to the work done by the non-conservative forces:

This statement expresses the concept of energy conservation for a classical particle as long as there is no non-conservative work. Recall that a classical particle is just a point mass, is nonrelativistic, and obeys Newton’s laws of motion. In Relativity , we will see that conservation of energy still applies to a non-classical particle, but for that to happen, we have to make a slight adjustment to the definition of energy.

It is sometimes convenient to separate the case where the work done by non-conservative forces is zero, either because no such forces are assumed present, or, like the normal force, they do zero work when the motion is parallel to the surface. Then

In this case, the conservation of mechanical energy can be expressed as follows: The mechanical energy of a particle does not change if all the non-conservative forces that may act on it do no work. Understanding the concept of energy conservation is the important thing, not the particular equation you use to express it.

Problem-Solving Strategy

• Identify the body or bodies to be studied (the system). Often, in applications of the principle of mechanical energy conservation, we study more than one body at the same time.
• Identify all forces acting on the body or bodies.
• Determine whether each force that does work is conservative. If a non-conservative force (e.g., friction) is doing work, then mechanical energy is not conserved. The system must then be analyzed with non-conservative work, Equation 8.12 .
• For every conservative force that does work, choose a reference point and determine the potential energy function for the force. The reference points for the various potential energies do not have to be at the same location.
• If no non-conservative work is done, apply the principle of mechanical energy conservation by setting the sum of the kinetic energies and potential energies equal at every point of interest.

Example 8.7

Simple pendulum.

Since the particle is released from rest, the initial kinetic energy is zero. At the lowest point, we define the gravitational potential energy to be zero. Therefore our conservation of energy formula reduces to

The vertical height of the particle is not given directly in the problem. This can be solved for by using trigonometry and two givens: the length of the pendulum and the angle through which the particle is vertically pulled up. Looking at the diagram, the vertical dashed line is the length of the pendulum string. The vertical height is labeled h . The other partial length of the vertical string can be calculated with trigonometry. That piece is solved for by

Therefore, by looking at the two parts of the string, we can solve for the height h ,

We substitute this height into the previous expression solved for speed to calculate our result:

Significance

Check your understanding 8.7.

How high above the bottom of its arc is the particle in the simple pendulum above, when its speed is 0.81 m / s ? 0.81 m / s ?

Example 8.8

Air resistance on a falling object.

Step 2: Gravitational force is acting on the panel, as well as air resistance, which is stated in the problem.

Step 3: Gravitational force is conservative; however, the non-conservative force of air resistance does negative work on the falling panel, so we can use the conservation of mechanical energy, in the form expressed by Equation 8.12 , to find the energy dissipated. This energy is the magnitude of the work:

Step 4: The initial kinetic energy, at y i = 1 km , y i = 1 km , is zero. We set the gravitational potential energy to zero at ground level out of convenience.

Step 5: The non-conservative work is set equal to the energies to solve for the work dissipated by air resistance.

Check Your Understanding 8.8

You probably recall that, neglecting air resistance, if you throw a projectile straight up, the time it takes to reach its maximum height equals the time it takes to fall from the maximum height back to the starting height. Suppose you cannot neglect air resistance, as in Example 8.8 . Is the time the projectile takes to go up (a) greater than, (b) less than, or (c) equal to the time it takes to come back down? Explain.

In these examples, we were able to use conservation of energy to calculate the speed of a particle just at particular points in its motion. But the method of analyzing particle motion, starting from energy conservation, is more powerful than that. More advanced treatments of the theory of mechanics allow you to calculate the full time dependence of a particle’s motion, for a given potential energy. In fact, it is often the case that a better model for particle motion is provided by the form of its kinetic and potential energies, rather than an equation for force acting on it. (This is especially true for the quantum mechanical description of particles like electrons or atoms.)

We can illustrate some of the simplest features of this energy-based approach by considering a particle in one-dimensional motion, with potential energy U ( x ) and no non-conservative interactions present. Equation 8.12 and the definition of velocity require

Separate the variables x and t and integrate, from an initial time t = 0 t = 0 to an arbitrary time, to get

If you can do the integral in Equation 8.14 , then you can solve for x as a function of t .

Example 8.9

Constant acceleration.

Solving for the position, we obtain x ( t ) = x 0 − 1 2 ( E / m x 0 ) t 2 x ( t ) = x 0 − 1 2 ( E / m x 0 ) t 2 .

Check Your Understanding 8.9

What potential energy U ( x ) U ( x ) can you substitute in Equation 8.13 that will result in motion with constant velocity of 2 m/s for a particle of mass 1 kg and mechanical energy 1 J?

We will look at another more physically appropriate example of the use of Equation 8.13 after we have explored some further implications that can be drawn from the functional form of a particle’s potential energy.

Systems with Several Particles or Objects

Systems generally consist of more than one particle or object. However, the conservation of mechanical energy, in one of the forms in Equation 8.12 or Equation 8.13 , is a fundamental law of physics and applies to any system. You just have to include the kinetic and potential energies of all the particles, and the work done by all the non-conservative forces acting on them. Until you learn more about the dynamics of systems composed of many particles, in Linear Momentum and Collisions , Fixed-Axis Rotation , and Angular Momentum , it is better to postpone discussing the application of energy conservation to then.

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Conservation of Energy

10th - 12th grade.

25 questions

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No student devices needed.   Know more

What kind(s) of energy is(are) in a person climbing a ladder?

Need the mass of the person to determine.

As a pendulum swings from its highest to lowest position, what happens to its kinetic and potential energy?

The PE and KE increase and decrease together

The PE decreases while the KE increases

Impossible to know without knowing the mass

The PE and the KE decrease and increase together.

According to the Law of Conservation of energy, the amount of energy before and after a reaction must be the same. Which statement below is true?

The total amount of energy is conserved

The total amount of energy is less after a reaction

The total amount of energy is more after a reaction

Need the mass of the object to determine if energy is conserved.

When velocity triples, kinetic energy increases by ______ times. KE =1/2mv 2

What is the Total Mechanical Energy at point A?

What is the Kinetic Energy at point B?

What is the Kinetic Energy at point C?

What is the Kinetic Energy at point D?

What is the GPE at point E?

What is the GPE at point F?

What is the KE at point F?

When something is dropped, what type of energy transfer takes place?

Kinetic -> Gravitational Potential

Gravitational Potential -> Kinetic -> Sound

Electrical -> Chemical

Heat -> Nuclear

If two marbles are rolled down a ramp from some height towards a container the _______ marble will move the container farther because it possess more _________ energy as it approaches the end of the ramp.

heavier, kinetic

heavier, potential

lighter, kinetic

lighter, potential

In diagram 2, if the mass of the car is 600 kg, what is the change in potential energy from point A to point B?

If a 0.5 kg ball is thrown up with 250 J of kinetic energy, how high will it go?

(HINT: At the ball's highest point it has stopped.)

KE = 1/2mv 2 , PE = mgh

g = 9.8 m/s 2

A cliff diver with a mass of 125 kg jumps off a 50m high cliff, how fast will he be traveling just before he hits the water? (His height at this point is 0m)

(HINT: Find PE first. then KE = 1/2mv 2

A person with a mass of 60 kg jumps in the air. If the gravitational potential energy is 1,470 J at the highest point, how did they jump?

The rollercoaster has both PE and KE, but where will it have more PE than KE?

Where will the roller coaster have the most total mechanical energy? Check all that apply. Neglect friction and air resistance.

A book has 200J of potential energy and then it falls onto the floor. After it hits the floor and is sitting still, what happened to the energy?

It has disappeared.

It goes into the floor and book as thermal energy and sound.

It changes to chemical potential energy.

It turns into gravity.

The skateboarder shown above has a potential energy of 2,800 J and kinetic energy of 0J at a height of 4 meters. How much total mechanical energy will she have after 2 seconds? Neglect friction and air resistance.

This graph depicts what type of relationship between kinetic and potential energy?

Poly-atomic

A 5.0 kg object is pushed across the floor and the attached force vs. displacement graph describes the forces and displacement. What is the change in KE of the object? (Hint: Net work = change in kinetic energy)

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