Table of Contents

## Kinetic Energy

**E ⇒ energy (J or N m)**

Energy is one of the most fundamental parts of the universe. Energy supports life itself. It allows objects to move despite other forces, the pressure to be exerted, substances to be heated, and electric current to flow. The 2 most essential forms of energy can be described as follows:

Energy |
Equation |
Description |

Potential energy E_{pot} |
E_{pot} = F_{G }h |
Potential energy is the energy stored by an object of a certain mass as a result of its position (for example, when it is held in an elevated position).
h = height of the object (m) |

Kinetic energy E_{kin } |
E_{kin }= ½ m v^{2} |
Kinetic energy is the energy of motion or acceleration or when a resting body is set into motion.
m = mass (kg) v = velocity (m/s) |

## Gravitational Potential Energy

Gravitational potential energy is the energy that an object contains due to its position in a gravitational field. It is most commonly used when an object is present near the earth’s surface, where acceleration is downward towards the earth (gravity). Gravity is a constant force of 9.8 m/sec^{2} (for earth). Thus, an item released from the top of a building will fall towards the earth to the base of the building at an acceleration rate of 9.8 m/sec^{2}. The acceleration is denoted as negative due to the downward motion.

### Potential energy vs. kinetic energy

These 2 energy states transfer energy back and forth to each other. Kinetic energy is the energy of motion. As an object moves, it is gaining kinetic energy. Potential energy is stored energy, which is the energy that can be used ‘potentially’ for motion.

Let us look at an example of a person sitting down and holding an apple in their hand. As they raise their hand, the apple is gaining potential energy. When they release the apple, potential energy gets converted into kinetic energy as it falls. Consider another example of a battery available at a store. The battery, when not in use, contains potential energy. When the battery is placed into a toy that uses it to move, the toy will now use the energy stored in the battery. This process is also converting potential energy into kinetic (usable) energy.

The gravitational potential energy can be calculated based on the equation U_{g }= m g h where U_{g} is gravitational potential energy, m = mass of the object, g = acceleration due to gravity, and h = the height that the object has reached. If an apple is on the ground, it has no gravitational potential energy. As the apple is tossed upwards, it gains gravitational potential energy up to a peak value occurring at the maximum height of the apple’s travel.

### Total energy

The total energy in a system is equal to the sum of the kinetic energy and the gravitational potential energy. When the apple is on the ground, all the energy is kinetic. As it ascends, the kinetic energy is converted to potential energy. At the peak, all the energy is potential energy. As it descends, the reverse occurs: the energy gets converted from potential energy to kinetic energy until it is back on the ground with no potential energy and all kinetic energy. The interesting concept that results is associated with total energy. Total energy at every single point along the path of the apple’s travel is always the same. The total energy simply gets divided into the changing kinetic and potential energies.

## Spring Potential Energy

Spring potential energy is the potential energy that is stored due to the deformation of the spring, which is an elastic object. The potential energy that is stored is equal to the work done to stretch the spring. It is dependent on the spring constant k, and the distance the spring is stretched. The spring constant is a measure of the stiffness of the spring. Various factors can affect the stiffness of the spring, including the material of the spring, the diameter of the spring, the diameter of the wire of the spring, and the length of the spring.

### Energy of the spring

The energy that is contained in the spring is referred to as spring potential energy. This potential energy is denoted as U_{s}. The spring is initially at its ‘happy’ spot, referred to as its equilibrium position. This position is where the spring is not expanded or compressed.

So, where does the energy exist in the spring? On starting to pull the spring, the spring will expand easily without much force. The more one pulls the spring and the further it stretches, the harder it becomes to pull it even a little more. The same thing occurs in compressing the spring. The more one compresses the spring, the harder it becomes. So, any expansion or compression of the spring will cause the spring to tend to go towards the equilibrium point. This tendency is the conversion of potential energy by compression or expansion to kinetic energy as it moves towards the equilibrium point.

The other important aspect of a string is the presence of elastic recoil: When a spring is stretched, it exerts a restoring force which tends to bring it back to its original length. This restoring force (transfer of potential energy to kinetic energy), is proportional to the amount of stretch as described by Hooke’s law.

Hooke’s law was named after the 17th-century British physicist Robert Hooke. The basis of his law is that for relatively small deformation of spring, the displacement or size of the deformation is directly proportional to the deforming force or load. In equation form, it is

**F = – k x**

x = amount of displacement

k = proportionality constant specific for each spring

F = resultant force

### Potential energy equation

In order to determine the potential energy present from the expansion or compression of a spring, it is necessary to know the amount of displacement of the spring from its equilibrium point. Thus, the equation form is as follows:

**U _{s} = ½ k (x – x_{o})^{2}**

x_{o} = equilibrium point

x = amount of displacement

k = proportionality constant specific for each spring (kg/s^{2})

This equation is simplified since x_{o} = 0. Thus, U_{s} = ½ k x^{2}.

## Energy: Important Facts

A summary of the interrelated concepts is important as it gives a clear understanding of the information regarding energy. Comprehension of the fundamental types of energy is essential before memorizing the equations to solve problems.

### Types of energy

There are 3 types of energy: kinetic, potential, and total energy.

Kinetic energy is the energy of motion.

**K = ½ m v ^{2}**

Potential energy is the stored energy that can be used for motion.

Gravitational potential energy:

**U _{g} = m g h**

Spring potential energy:

**U _{s} = ½ k (x – x_{o})^{2}**

Total energy:

**E = K + U**

The total energy is the combined energy that stays constant when an object is moving and at rest or when spring is in equilibrium and expanding/compressing. The changes that take place occur in the kinetic and potential energies.

Variable |
Meaning |
Units |

K, U, E | Kinetic, potential, and total energy | Joules (J) |

m | Mass | kg |

v | Velocity | m/s |

h | ‘Height’ (vertical distance) from chosen origin | m |

k | Spring constant (spring “stiffness”) | kg/s^{2} |

### Potential energy in relation to ‘ground level’

The question that comes up in dealing with problems is whether the potential energy changes if the ‘ground level’ is changed. The potential energy U is dependent on h. h is defined as the vertical distance from the chosen origin. So, it is quantifying not how far an object is from the ground, but the change in motion from a relative point. Thus, U is independent of the coordinate system. The important aspect is the change that takes place in U, as well as the change that takes place in h for gravitational potential energy calculations or the change that takes place in x^{2} for spring potential energy calculations. Thus, potential energy is relative to the change taking place and is not absolute. It is independent of the coordinate system.

### Energy in context

The table below shows the different properties relating to energy. They are all different in what they are calculating but provide information about related concepts of energy. Again, it is crucial to understand the concepts first before trying to use the equations. It will make more sense in that manner, and retention of the material will be better in the long run.