Forces of Nature Weblecture

Notes on Faraday's Lectures are currently under revision

- Faraday,
*On the Various Forces of Nature*: Lecture II - Breithaupt,
*Understanding Physics*: Chapters 4 and 5

The concept of energy is an abstraction: we do not observe "energy" except as work (a change in potential energy) or movement (kinetic energy). Notice the difference in how we measure the two kinds of energy! I can calculate kinetic energy by measuring mass and velocity: the kinetic energy or energy of movement of an object is (by convention) half of the product of its motion and the square of its velocity: KE = ½mv^{2}. This kinetic energy is similar to momentum "p" where p = mv. Notice that kinetic energy is ½ mv ^{2}= ½(mv)v = ½ pv.

Potential energy is always determined as a change of state: ΔPE = PE_{final} - PE_{initial}. It doesn't really matter what PE_{initial} actually is, so we usually pick some convenient starting point as the "zero" point.

Consider what happens when I pick a book off the floor and put it on the book shelf. I have lifted it up against the gravitational pull of the Earth on the book, so I have done work that changes its potential energy state from the "lower" energy state (on the floor) to the "higher" energy state (on the shelf). In one sense, I have put work into the book by changing its position in the Earth's force field and moving it against the direction of that force. ΔPE = Work done = Force * distance = mg * h where h is the height change for the book.

The conservation of energy law says that if we convert energy from one form to another, the energy must all be accounted for: it doesn't disappear, although it may be "lost" and become a form that can no longer be used to do work. If I pull the book off the shelf and drop it from the shelf height, the potential energy or work we used to pick the book up will be converted back to kinetic energy as it falls: ΔPE = mg * h = KE = ½mv^{2}. The velocity of the book as it hits the floor will be v = √(2gh).

All forms of energy are either kinetic energy of motion, or potential energy changes due to change of position in a force field. Nuclear energy is energy released by a disintegrating atomic nucleus, a change in particle positions moving within the strong and weak nuclear forces of the atom. Chemical energy is energy released by changing positions of atoms affected by each others electrical force fields from positively charged protons and negatively charged electrons. Thermal energy or heat energy is sum of the kinetic energy of motion of the particles within a substance.

We've seen how work is a form of energy, often determined from the distance d we move an object against a force F: energy = work = force * distance. Power is the transfer or use of energy over time: power = work done/time. In an ideal machine, the amount of power going into the machine and the amount of power coming out of the machine will be the same. But no machine is ideal: some of the work done by the machine is dissipated as heat in friction. We can measure how efficient a machine is by comparing the power output to the power input. Much of modern industrial research is concerned with improving the efficiency of machines.

Students often confuse or conflate the concepts of energy, momentum, and force, but in physics, these concepts are distinct, even though they are closely related.

We've already seen how force, as Newton defined it, is the product of mass times acceleration, where acceleration is a change in velocity with respect to time:

F = ma = mΔv/Δt

as we shall see later on, Einstein had to modify this definition slightly to account for changes in mass at near-light speeds:

F = ma =Δm * Δ v/Δt = Δ(mv)/Δt

The quantity "mv" was already recognized indigenous time as related to changes in the state of motion, the *vis* of the motion. when kinetic energy was eventually defined as KE = ½mv!2^{2}, this vis gained its own term: momentum p = mv. over time, physicists realized that both momentum and kinetic energy were conserved in collision situations, providing kinetic energy was not converted to potential energy through changes in height or the action of gravitational forces. Given the equations above, we can explain force not only as mass times acceleration, but also as a change in momentum with respect to time.

If you have studied calculus, you will be familiar with the derivative of the polynomial. If f(x) = cx^{r}, then f'(x) = d f(x)/dx = c * rx^{r-1}. If we take the derivative of kinetic energy with respect to velocity, we get momentum: f(v) = ½mv^{2}; f'(v) = d f(v)/dv = 2 * ½ mv^{2-1} = mv. In other words, momentum is a rate at which kinetic energy changes with respect to its velocity — which is what we would expect if momentum is also the application of force creating that change over time!

Another way to look at the relationship of these concepts is to look at the units used to measure the quantities involved in the formulas. If we look the standard international units for kinetic energy, we have

KE = ½mv^{2} = kilograms * (meters/second)^{2}

If we look at force acting through a distance, we discover the same units are involved:

Force * distance = ma * d =

kilograms * ((meters/second)/second) * meters =

kilograms * meters^{2} / seconds^{2} =

kilograms * (meters/second)^{2}

Physicists use dimensional analysis to explore the relationships between different physical quantities. These relationships can reveal tightly bound concepts are dependent on one another and allow us to understand how momentum, energy, and force are related in collision situations.

Thermal physics covers the phenomena of heat. Before we can look at how heat changes form or moves from one object to another, we need to understand what heat is, and how we measure it. We start by making a distinction between **heat**, which is a form of energy, and **temperature**, which is a measure of the average kinetic energy in an object.

Consider the heat and temperature of water in two containers: the water in a pool, heated to a comfortable swimming temperature by the light of the sun on a warm day, and water in a teakettle heated to the boiling point on the stove. The amount of water in the pool is much greater than the amount of water in the teakettle, but the temperatures much lower. Which has the greater amount of heat?

If we represent heat energy as money, we can visualize the low amount of heat that each unit of water (say, one cubic centimeter or one milliliter) has as one cent ($0.01). In the teakettle, the same volume of water has a value of one quarter ($0.25) because this volume has more energy. Despite the fact that the teakettle is filled with the more valuable quarters, the swimming pool has so many pennies that the total amount of heat value is greater than the heat value of the teakettle. The amount of matter in this case determines the total amount of heat energy available in each object.

If heat is energy, then temperature — for the physicist — becomes a measure of the average kinetic energy or heat energy of the particles in a substance. Heat energy is an *extensive* property: the more matter there is in an object, the more heat energy that object can hold at the same temperature. Temperature is an *intensive* property: regardless of how much matter we have in the object, temperature doesn't change unless heat flows into or out of the substance. So heat depends on an amount of matter, but temperature is independent of the amount of matter in the object.

Physicists explain heat energy by using the kinetic-molecular theory of matter. In this concept of matter, individual atoms and molecules have kinetic energy of motion that depends on the mass of the atom or molecule, and the velocity at which the object is traveling. This "traveling" could simply be vibration in place as well as translational motion through space. When heat energy is added to a substance, the velocity of the individual particles, or the rate of vibration, increases.

The genetic-molecular theory of matter allows us to explain heat transfer at the microscopic level. In *convection*, Clumps of matter whose molecules how high velocity and therefore high heat energy move from one place to another, carrying that energy with them. Examples of heat transfer by convection include boiling water, when the hot water near the heated bottom of the pan moves upward, and atmospheric thermals, where warm volumes of air rise because their increased heat energy causes them to expand and "float" above the surrounding cooler air. Convection currents in the atmosphere and the ocean are responsible for the Earth's weather patterns and climate change, transporting solar energy from where it strikes the Earth's to other areas at different rates.

In *conduction*, two objects touch one another. If the temperatures are different, collisions between the atoms and molecules in the hotter substance transfer energy to individual atoms and molecules in the cooler substance. The average kinetic energy of molecules in the hotter substance drops, and the hotter substance temperature decreases as it cools. The average kinetic energy of molecules in the cooler substance increases, and the cooler substance temperature increases as it heats up.

The third form of heat transfer, radiation, is a form of electromagnetic energy, so we will discuss how it works when we talk about light.

One of the consequences of the kinetic-molecular theory of matter and its explanation of heat energy is the realization that temperature should never be negative. Kinetic energy is proportional to mass and velocity: KE = ½mv^{2}. Mass is an extensive property of matter: for a material object, it must be greater than zero; it can never be zero or less (or we don't have matter). Velocity could be negative (depending on direction), but the square will always be positive. Given these conditions, the average kinetic energy of a system is always positive, so the temperature that measures this quantity should always be positive.

The commonly used scale for temperature is the centigrade (Celsius) scale, with the zero point set at the freezing point of water. This introduces problems in calculations where values are determined by dividing by temperature. A more appropriate scale for these situtions would place the zero point at "absolute zero" — the lowest possible temperature for matter to achieve.

We can determine this zero point from the behavior of gases at different temperatures. Suppose that we measure the pressure of a gas at 0 °C and 100 °C:

Temperature °C | Pressure (kPa) |

0 | 250 |

100 | 183 |

The ratio of pressure change to temperature change is is (250 - 183) / (100 - 0) = 67/100.

Now let us consider what happens when pressure drops to 0 (as it should if nothing is moving), using the same ratio of pressure change to temperature change is

(183 - 0) / (0 - T_{abs}) = 67/100.

We can solve for T_{abs}:

T_{abs} = -(183 - 0) * 100/67

T_{abs} = -18300/67 = -273.15 °C.

Many repetitions of this experiment have confirmed that absolute zero in the Celsius scale is -273.15 °C, so this becomes the 0 point of a new scale, the Kelvin scale. For simplity in converting, the size of a Kelvin step is the same size as a Celsius degree; the only difference in the two scales is their zero point.

The ability of a substance to absorb or transfer heat energy depends on the composition and density of the substance, and to some extent, its current temperature. The *specific heat capacity* of a substance such as water is determined as the amount of energy required to raise the temperature of the substance by 1 °C. The heat capacity of water is very high compared to similar substances because polar water molecules (with a negative oxygen side and a positive hydrogen side) are attracted to each other, and resist being pulled apart. Any increase in their velocity relative to each other must overcome this attractive force, so more heat energy is required to increase water temperature by one degree than is required by other non-polar liquids, such as gasoline (octane).

Adding heat energy will increase the temperature of a solid substance until it reaches its melting point. At this temperature, heat energy is spent breaking the bonds which hold molecules together in a solid lattice. The temperature doesn't rise again until these bonds are broken and the individual molecules are free to move in a liquid. The energy required for the melting phase change is called the *latent heat of fusion*. It can be calculated as the amount of heat applied during the melting period for some amount of substance. If we continue to apply heat after melting, the temperature of the liquid will rise until the substance reaches its boiling point. The substance will remain at its boiling point temperature until all of the substance has evaporated, then the temperature of the vapor will rise with additional heat. The heat required for the evaporation phase is the *latent heat of evaporation.*.

We can graph the temperature change for any substance. The graph below assumes that we apply heat steadily over time, so we plot temperature change against the total amount of heat applied over time:

- Consider how engineers try to make cars and planes, which use gasoline, more efficient. What is the original form of the energy in gasoline? What are common ways to measure energy output and efficiency for cars?
- Consider energy (work), power, force, and momentum. How can dimensional analysis help us to understand how these characteristics and quantities are related?
- The temperature of the swimming pool, filled to the brim with sun-warmed water, is about 75°F. The temperature of the water in the tea kettle is just below 212 °F. Which has more total kinetic energy? Which has individual molecules with a higher average kinetic energy? Which quantity is intrinsic? Which is extrinsic?

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