Conservation of Energy

FIELDS OF STUDY: Classical Mechanics; Electromagnetism; Nuclear Physics

ABSTRACT: The motion of an object can be described by considering the various forms of energy the object has. In the absence of dissipative forces, the energy of a system can be neither created nor destroyed. This principle can be used to find the varying amount of kinetic energy of an object, which directly relates to the speed of that object.

Principal Terms

• kinetic energy: energy due to any kind of motion, be it rotation, vibration, or translation.
• potential energy: energy that is stored in objects and has the potential to become other forms of energy, such as kinetic energy.
• total mechanical energy: the sum of all the kinetic and potential energies of an object in a closed system.

Conservation of Energy and Mass

To study the motion, or kinematics, of an object, one can apply Isaac Newton’s (1642–1727) laws of motion and obtain results that match what happens in the real world. However, there is one small issue with this approach: it gets a lot more complicated when dealing with variable accelerations. When a car stops at a red light, it is not accelerating at that moment. When the light turns green, the driver applies a variable amount of pressure to the gas pedal. That variable amount of pressure produces a variable acceleration of the car. In these cases, applying Newton’s laws of motion produce different results depending on the acceleration value used. Calculating energy is a simple approach to solving these kinds of problems. This is due to the fact that the total amount of energy an object has never changes. Physicists call this principle the conservation of energy.

Energy is conserved by all objects in the absence of friction or any other dissipative force. If a person rubs a finger on a table for a long period of time, his or her finger will get hotter and hotter. If the person hits the table with an open palm, his or her palm will also be warmer than before. This is because some of that person’s energy was turned into heat, which is a form of energy. Heat then moves away from the person’s hand in the form of infrared radiation and goes back into the environment. While it may seem like the energy was not conserved, it is still part of a larger overall system and did not disappear completely.

The law of conservation of energy states that the total energy in the universe is always the same, and therefore energy can be neither created nor destroyed. In other words, it is not possible to add energy to the universe or take some energy out of the universe. This law can be applied to any system in which one can assume no dissipating forces exist. In 1905, Albert Einstein (1879–1955) recognized in his theory of relativity that mass is itself a form of energy. Thus the law of conservation of energy also addresses the conservation of mass, in that the total amount of mass and energy in the universe is constant.

Inside the sun, mass is constantly being turned into energy. This is the energy that warms Earth. In a way, fossil fuels are a form of energy from the sun, if one considers the law of conservation of mass and energy. Eons ago, the sun converted mass into energy in the form of light and heat. When that light and heat reached Earth, plants, bacteria, and algae turned the sun’s energy into mass in the form of food, which provided them with energy to live and grow. That energy was converted back into mass in the form of the newly grown plants, bacteria, and algae. Animals then ate those plants, bacteria, and algae as food, which they converted to energy. Over time, the remains of flora and fauna fossilized, becoming coal and oil. Humans use coal and oil as fuel to provide energy. Conservation of energy is all around, and it affects everyone in more ways than one might think.

Different Forms of Energy

In order to understand conservation of energy, one must understand the different forms and properties of energy. Everything that is in motion has a form of energy called kinetic energy. In fact, the temperature of a room is defined as the average kinetic energy of the particles in the room. The air molecules in a room are in a constant state of motion. Not only are they moving around, they are also vibrating. If on average they are moving faster, then they have more kinetic energy, which makes the whole room warmer. If on average they move slower, then less kinetic energy leads to lower temperatures. The kinetic energy (K) of an object, in joules (J), is mathematically defined by the object’s mass (m) and its velocity (v), as in the following equation:

But that is not the only form of energy objects can have. Objects tend to fall to Earth’s surface due to the planet’s gravitational pull. Another simple way to say this is that the object has extra amounts of energy when it is above the surface of Earth. This energy is known as the gravitational potential energy (U_g), and it is determined by the object’s height above the surface (h), its mass (m), and the strength of Earth’s gravitational field (g). When a pencil is held, it has potential energy. As the pencil falls to the ground, it starts to lose some of that potential energy, which becomes kinetic energy. The pencil’s velocity increases as it continues to fall. When it hits the ground, all of its potential energy has been turned into kinetic energy, and it lands at its highest possible speed. At that point, dissipative forces cause the energy to be lost to the surrounding environment. Mathematically speaking, the gravitational potential energy, in joules, is expressed as

U_g = mgh

where g equals 9.8 meters per second per second, or meters per second squared (m/s²), near the surface of Earth. There are many other forms of stored or potential energy. There is stored energy in chemicals that are about to react in a chemical reaction. Some of this energy is released in the reaction as heat.

Another form of potential energy is found by the stretching and compression of a spring. If there is a mass (m) attached to a spring that has been fixed to a wall, and someone pulls on the mass without letting go, the mass now has potential energy. If the mass is released, it will begin to oscillate, gaining kinetic energy. At one point during this oscillation, the mass will compress the spring to the maximum possible amount and stop moving for a fraction of a second. When this happens, all the kinetic energy gained has been transformed back into potential energy. Then the spring will push back on the mass, allowing the stored energy to be transformed into kinetic energy. The entire process repeats itself for as long as the mass is allowed to oscillate. The potential energy stored in a spring is a function of the distance the spring is stretched or compressed (x) and the properties of the particular spring used, summarized as its unique spring constant (k). The spring constant is a measurement of how rigid the spring is and how it reacts to being stretched or compressed. In the International System of Units, it is measured in newtons per meter (N/m). Mathematically, the potential energy of a spring in joules, known as the elastic potential energy (U_e), is found using the following equation:

Elastic potential energy only exists if a spring is part of the system in question. An object can have multiple forms of energy at once. A pendulum that is oscillating is moving, therefore it has kinetic energy, and is at a distance from the surface, so it has gravitational potential energy. When the pendulum is at its highest point, all its energy is in the form of gravitational potential energy. When it is at its lowest point, it has zero potential energy and the highest amount of kinetic energy it can have. In between, it has different amounts of potential and kinetic energies. As described above, energy in a closed system is conserved. That means that all of the energy in the pendulum is always the same. Physicists have defined the total mechanical energy (E) as the sum of all the kinetic and potential energies of an object in a closed system:

E = K + U_g + U_e

When physicists say that the energy is conserved, they mean that mechanical energy is conserved. This means that there is no change in the total mechanical energy, or that the initial mechanical energy (E_i) equals the final mechanical energy (E_f):

E_i = E_f

Substituting the definitions of the different forms of energy into the mechanical energy conservation equation, the equation becomes

K_i + U_g,i+ U_e,i = K_f + U_g,f + U_e,f

Sample Problem

A single-car roller coaster with a mass of 500 kilograms (kg) sits at the top of the track, waiting to begin its motion. After its initial descent, it travels up a smaller hill with an altitude of 10 meters (m). When at the top of this hill, it has a speed of 15 m/s. What is the kinetic energy of the car when it reaches the ground at the bottom of the smaller hill?

Answer:

In order to calculate the final kinetic energy, the initial kinetic and potential energies must be found. There are no springs in this system, so there is no elastic potential energy. First, use the information about the speed on the smaller hill to calculate the initial kinetic energy:

Then use the same information to calculate the initial gravitational potential energy:

U_g,i= mgh_i

U_g,i = (500 kg) (9.8 m/s²) (10 m)

U_g,i = 49,000 J

Do the same thing for the final potential energy:

U_g,f = mgh_f

U_g,f = (500 kg) (9.8 m/s²) (0 m)

U_g,f = 0 J

Note that when the car is at the bottom of the hill, its height above the ground is 0 meters, so it has no final gravitational potential energy. Using the values for initial kinetic energy and initial and final gravitational potential energy, use the equation for the conservation of total mechanical energy to find the final kinetic energy:

K_i + U_g,i = K_f + U_g,f

56,250 J + 49,000 J = K_f + 0 J

K_f = 105,250 J

The final kinetic energy is 105,250 joules.

Energy Production

By using energy to solve problems about motion, one can arrive at the same result without having to deal with variable forces and accelerations. This has wide implications and applications for the real world. When hydroelectric power plants produce energy, they do so by converting potential and kinetic energy into electrical energy. As water falls down from the top of a lake behind a dam through pipes called penstocks, it loses potential energy and gains kinetic energy. This allows the water to move faster and faster as it falls. It then hits the blades of a turbine and transfers its energy into the turbine, causing it to spin. The turbine turns a generator, which produces electrical energy.

Conservation of energy can also be seen in the production of energy by other means. Most of the electricity produced in the United States comes from the burning of coal. When coal is used to heat water to produce electricity, the power plant cannot produce more energy than is stored in the coal as chemical potential energy. The same can be said about gasoline cars. The kinetic energy obtained by the explosive reaction of gasoline in an engine cannot be greater than the chemical energy stored in that gasoline before the reaction. This means that no one will ever be able to attain infinite speeds by means of propulsion, as an ever-increasing need for kinetic energy comes from an ever-increasing amount of stored potential energy, and there is a limited, and not infinite, amount of energy in the universe. Nothing in the universe that has mass can propel itself at or faster than the speed of light.

Bibliography

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