Kinetic & Potential Energy

FIELDS OF STUDY: Classical Mechanics; Relativity

ABSTRACT: This article describes different types of kinetic and potential energy and how they are related to the total energy of a system. Kinetic energy is expressed in terms of classical mechanics and relativistic mechanics in order to investigate a particle as it approaches the speed of the light.

Principal Terms

• center of mass: the point of a system that moves as though all of the mass of the system were located or concentrated there and all of the forces were applied there.
• center of momentum: the inertial frame of a system where the vector sum of the moments of all of the particles in that system is zero.
• frame of reference: a set of coordinate axes that serve to describe position or movement of an object with reference to that coordinate system.
• special relativity: the theory that states that for all inertial nonaccelerating reference frames, the laws of motion remain the same, and that speed of light in a vacuum is the same for all observers, regardless of the observer’s movement relative to the source of light or the movement of the source itself.

Kinetic and Potential Energy Explained

Energy comes in many different forms. Some common forms of energy are thermal or heat energy, chemical energy, gravitational energy, and electrical energy. Kinetic energy is the energy an object possesses because of its motion. Any kind of motion produces kinetic energy, including translation (movement along a straight line), rotation, and vibration. Objects at rest have zero kinetic energy. The faster an object moves, the greater its kinetic energy.

Work is used to describe the energy transferred to or from an object by a force acting on that object. When energy is transferred to an object, the work is said to be positive. When energy is transferred away from the object, the work is negative. "Work" is defined as the displacement of the object being acted on multiplied by the force acting on it. The change in the kinetic energy of an object is equal to the net work done on that object. Also, the kinetic energy after the work is done is equal to the sum of the initial kinetic energy and the net work done. This is known as the work–kinetic energy theorem. It is true for both negative and positive work.

Potential energy is the energy a system possesses due to the configuration or positioning of objects within the system that exert forces on each other. The potential energy of a system can change if the configuration of the system changes. Potential energy can be thought of as a stored energy of an object, or as the difference between the energy of an object at a certain position and its energy at a reference position. There are different types of potential energy that relate to various types of forces, including gravitational, elastic, and electric potential energy.

A conservative force is a force for which the work done to move an object from one location to another does not depend on the path taken to get there. The work done by a conservative force is equal to the negative change in potential energy.

Measuring Kinetic Energy

To correctly measure kinetic energy, one must keep in mind the frame of reference in which it is being measured. If a bird were to fly over the head of a stationary observer, that bird would have velocity and, thus, kinetic energy. However, if another observer were moving at the same speed as the bird, then the bird would be stationary relative to that observer, with no velocity and therefore no kinetic energy. Two observers who are moving in two different reference frames measure two different values. Even so, the total energy of an isolated system never changes within the measured reference frame.

A system’s total kinetic energy is the sum of the total kinetic energy in its center-of-momentum frame and the kinetic energy the total mass would have if it were entirely concentrated in its center of mass. The center-of-momentum frame, or center of momentum for short, is the reference frame in which the sum of the momentum of a particular system is equal to zero. This frame gives the minimum values of a system’s energy. The kinetic energy of a system in its center of momentum is a quantity that is conserved, meaning that its value never changes, and is the same to all observers. In other reference frames, one must take into account additional kinetic energy resulting from the total mass moving at the speed of the center of mass.

The two main forms of kinetic energy are translational kinetic energy, produced by an object moving in a straight line, and rotational kinetic energy, produced by an object rotating about its center of mass. The kinetic energy (K) of an object moving in a straight line at nonrelativistic speeds (that is, much slower than the speed of light) is described by the equation

where m is the mass of the object and v is its velocity. The equation for rotational kinetic energy is very similar, only with moment of inertia (I) instead of mass and angular velocity (ω) instead of linear velocity.

Types of Potential Energy

The energy stored in elastic materials such as bungee cords, rubber bands, and springs due to their compressing or stretching is called elastic potential energy. The force required to compress or stretch a spring is equal to the force that will restore the spring back to its initial relaxed state. Hooke’s law states that this force, called the spring force (F), is proportional to the distance from equilibrium that the spring is extended or compressed (x), as shown in the following equation:

F = −kx

In this equation, k the spring constant, a measure of spring stiffness that is specific to the particular spring. The stiffer the spring, the greater the value of k. In the International System of Units (SI), the value of k is given in newtons per meter (N/m). The negative sign indicates that the spring force always acts in the direction opposite the displacement of the spring.

The same variables are used to calculate the elastic potential energy (U_e) for a spring, as shown in the following equation:

When a spring is not being compressed or stretched, it is in its equilibrium position and has no elastic potential energy stored in it.

If an object moves along the y axis relative to the ground, the gravitational force does work on the object. Gravity exerts a downward force on the center of mass of an object near Earth’s surface. Gravitational potential energy (U_g) is the stored energy associated with an object due to its vertical position y, or the height of the object relative to its reference position. The gravitational potential energy of an object depends on the object’s mass (m), the height (h) to which it is raised, and the rate of acceleration due to gravity (g), which on Earth is approximately 9.8 meters per second per second, or meters per second squared (m/s²). This relationship is shown in the following equation:

U_g = mgh

It is important to establish a reference point or zero-height position. Depending on the specific situation, the ground or a tabletop can be considered the zero position. The potential energy of an object then depends on the height relative to the ground or tabletop.

Relativistic Mechanics

Special relativity applies the principle of relativity to inertial reference frames. Inertial reference frames are frames of reference that move with a constant velocity relative to each other and in which the laws of physics hold. Newton’s laws of motion for mechanical systems do not work for systems that accelerate relative to an inertial frame. When an object’s speed approaches the speed of light, one must use relativistic mechanics to calculate kinetic energy. Albert Einstein’s theory of relativity defines relativistic kinetic energy (K) as

where c is the speed of light in a vacuum, equal to 299,792,458 meters per second. As the velocity of a particle approaches the speed of light, the energy of the particleapproaches infinity. In the center of momentum, the system’s total energy is the rest energy (E), which can be related to mass and the speed of light by Einstein’s famous mass-energy equation:

E = mc²

The relativistic total energy is the sum of the kinetic energy and the rest energy. When a particle is at rest relative to an observer, K is equal to zero, and the relativistic total energy equation is equal to the mass-energy equation.

Conservation of Mechanical Energy

Energy can be converted from kinetic energy to potential energy and back again. The SI unit of kinetic energy is the joule (J), equal to 1 newton-meter (N·m), or 1 kg·m²/s². The total mechanical energy (E) of a system is the sum of its potential energy (U) and its kinetic energy (K):

E = U + K

Because the total energy of a system is constant, any decrease in the system’s potential energy must result in a corresponding increase in its kinetic energy, and vice versa. Thus, this equation can also be written as

E_i = E_f

where E_i is initial energy and E_fis final energy. When written in terms of potential and kinetic energy, the equation becomes

U_i + K_i = U_f + K_f

where U_i and K_i are the initial potential and kinetic energies, respectively, and U_f and K_f are the final potential and kinetic energies.

A pendulum consists of a string connected to a pivot point at one end and a mass at the other end. The pendulum moves back and forth periodically. As the pendulum mass swings from its maximum height to its minimum height, its speed increases. Every time the mass falls, it loses potential energy, but it gains speed and thus kinetic energy. The mechanical energy of this system is thereby conserved.

Sample Problem

A 0.50 kg crate resting on a horizontal, frictionless surface is pushed into a spring and then released. The crate compresses the spring a distance of 11.0 cm. The spring has a force constant of k = 250 N/m. What is the velocity and kinetic energy of the crate just after it leaves the spring?

Answer

Recall that the initial total energy (E_i) must be equal to the final total energy (E_f), and that the total energy of a system is equal to the sum of its potential energy (U) and its kinetic energy (K):

E_i = E_f

E = U + K

U_i + K_i = U_f + K_f

First, convert the compression distance from centimeters to meters. Then use the equation for elastic potential energy to calculate the initial potential energy (U_i) of the compressed spring. The initial kinetic energy (K_i) is 0 J because the crate is at rest. To keep units consistent throughout, recall that 1 J = 1 N·m = 1 kg·m²/s².

Use these values to calculate the final kinetic energy (K_f). This can be done because the final total energy consists solely of the kinetic energy of the released crate, just like the initial total energy consisted solely of the potential energy of the compressed spring. The final potential energy (U_f) is 0 J because the spring is no longer compressed.

U_i + K_i = U_f + K_f

(1.5125 J) + (0 J) = (0 J) + K_f

K_f = 1.5125 J

Now use the equation for kinetic energy to calculate the crate’s velocity after it leaves the spring:

The crate’s velocity is 2.46 m/s, and its final kinetic energy is 1.5125 J.

Kinetic and Potential Energy

The total amount of energy in the universe is always constant. It can change from kinetic to potential and back again, as seen when a high jumper leaps into the air and then comes back down, picking up speed along the way. Relativistic mechanics are used for objects approaching the speed of light and for transforming measurements between reference frames that move relative to each other. Einstein’s special theory of relativity is necessary for modern long-range navigation, in which the precise location and speed of a moving craft are constantly monitored and updated.

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