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0 Introduction To Study Modules For Calculus-Based General Physics

Calculus-Based General Physics

These modules were prepared by fifteen college physics professors for use in self-paced, mastery-oriented, student-tutored. calculus-based general physics courses. This style of teaching offers student's a personalized system of instruction (PSI), in which they increase their knowledge of physics and experience a positive learning environment. We hope our efforts in preparing these modules will enable you to try and enjoy teaching physics using PSI.

This is the second printing of the CBP modules. We have tried to remove all of the errors from this material. No doubt, we missed a few, please write to us and tell us any that …

Ampère's Law

Calculus-Based General Physics

Everyone has seen a bar magnet in the form of a compass or a door catch. Anyone who has ever casually played with magnets or magnetic toys knows that magnets interact with other magnets; i.e., a magnet experiences a force caused by thp presence of an external magnetic field produced by the other magnet. A wire carrying a current experiences a force caused by the presence of a nearby magnet (as you saw in the module Magnetic Forces). We then expect the converse to also hold true, i.e., that the bar magnet will also experience a force from the presence …

Diffraction

Calculus-Based General Physics

Have you ever wondered why you can hear around corners, but cannot see around them? You know that light and sound are waves, and should therefore share the same basic properties. Why then do they seem so different in the property of their "shadows"?

In this module you will learn that light does exhibit all the bending properties of sound and water waves. The effect, however, depends on the size of the obstacle compared to the wavelength. It is only the largeness of everyday obstacles compared to the very small wavelength of light that deemphasizes the bending, or diffraction, of …

Electric Potential

Calculus-Based General Physics

You have no doubt noticed that TV sets, light bulbs, and other electric appliances operate on 115 V, but electric ovens and clothes dryers usually need 220 V. Batteries may be rated at a harmless 1.5, 6, 9, or 12 V, but a high-tension electric transmission line may provide electric power at 400,000 V. Now just what physical quantity is measured by all these volts? How do volts relate to force, energy, and power, about which you have learned in earlier modules? The answer is that volts measure electric potential difference (sometimes called "voltage"), which is derived from the potential …

Faraday's Law

Calculus-Based General Physics

Consider the electric light you may be using to read this module by and the influence on your life style of the vast amounts of electrical energy produced in the United States. This module treats the fundamental principle that allows for the transformation of mechanical energy into electrical energy. The physical law that governs the production of electric current is named after its discoverer, Michael Faraday.

Fluid Mechanics

Calculus-Based General Physics

An invigorating shower in the morning is usually a pleasant experience except for the pesky shower curtain slapping your legs and allowing water to run on the floor. You would think that the downward stream of water would be enough to keep the curtain back even without water striking the curtain. But not so: fast-moving fluids (water spray causing a downdraft of air) contain a low-pressure region. Thus the pressure outside the shower is greater than the pressure inside - with the result that the curtain is blown in and flops against your legs.

More technical applications of fluid mechanics …

Kinetic Theory Of Gases

Calculus-Based General Physics

As you read this sentence you will experimentally demonstrate the general gas law at least once by breathing in and out. As you expand the volume in your lungs, the pressure drops and air comes in; as you decrease the lung volume the pressure rises and air goes out. Pressure (p) and volume (V) are related; at constant temperature, pV = const.

This relation, called Boyle's law, was well established before the atomic theory of matter was accepted. In this module you will learn how to apply much of your knowledge of Newton's laws, kinetic energy, momentum, and elastic collisions …

Lenses And Mirrors

Calculus-Based General Physics

If you have ever worn glasses, used a magnifying glass, looked through a telescope, or looked in a mirror, you have some idea of the effect that transmitting and reflecting materials have on light. When light passes from one material to another it is refracted. It is this property of light that is used in making eye glasses and magnifying glasses. The laws of reflection and refraction have immediate application in the construction of optical instruments. Two main objectives of most optical devices are to increase the light-gathering area and to provide a magnified image. Magnification is not usually the …

Introduction To Quantum Physics

Calculus-Based General Physics

You have probably encountered a system known as an "electric eye," which senses light from an artificial source or the sun. This information is used to open doors, count pedestrian or auto traffic, turn on lights at sunset, read holes in punched card, and for a host of other applications. Most of these devices are based on the photoelectric effect, which is the light-induced emission of electrons from atoms.

The photoelectric effect completely baffled physicists at the time of its discovery. Einstein's explanation of this process, which won the Nobel Prize in 1921, was a major part of the twentieth-century …

Maxwell's Predictions

Calculus-Based General Physics

With this module, you will reach a milestone in your study of electromagnetic phenomena. From past modules, you now have (at your fingertips, hopefully!) the same basic laws of electromagnetism that Maxwell collected together in the nineteenth century. However, as powerful as these laws were, Maxwell found that there was a basic flaw -- a logical inconsistency -- in the one known as Ampere's law. He was able to deduce (in advance of any direct experimental test) precisely the correction that was needed. With this correction, the addition of what is called the "displacement-current" term to Ampere's law, it follows …

Optical Instruments

Calculus-Based General Physics

You are now familiar with some of the properties of idealized single lenses and simple spherical and plane mirrors. Almost all optical instruments are made up of a combination of lenses, some close together and others far apart. Real lenses and mirrors have many undesirable properties intimately interconnected with their desirable properties. By making careful and clever combinations of lenses one can enhance the desirable and minimize the undesirable characteristics. In this module you will begin the study of some simple combinations of mirrors and simple lenses; it will give you some insight into the complications and fascinating possibilities of …

Rectilinear Motion

Calculus-Based General Physics

How long does it take you to go home? This depends on how far you are from home (displacement), how fast (velocity) you can travel, and how often you must start and stop (acceleration).

This module treats kinematics, which is the part of physics concerned with the description of the motion of a body. The body may be an automobile, a baseball, a raindrop, a flower in the wind, or a running horse. The change in position of a body can be described in terms of the vector quantities: displacement, velocity, and acceleration. Calculus can be used to define the …

Trigonometry

Calculus-Based General Physics

Many of the applications of physics will require you to have a thorough knowledge of the basic properties of right triangles, i.e., triangles that have one angle equal to 90°.

The trigonometric functions are defined with respect to a right triangle as follows:

sin θ = y/r

cos θ = x/r

tan θ = y/x

The values of the trigonometric sine, cosine, and tangent functions for a given θ can be determined from a table such as in the appendix to your text or the last page of this module. You can also get the values by use of most …

Simple Harmonic Motion

Calculus-Based General Physics

Have you ever felt you were the slave of a clock? Clocks are mechanisims that include a pendulum or balance wheel whose repeated patterns of movement define equal time intervals, one after another. Such repeated movements are called periodic motion. Periodic motion may occur when a particle or body is confined to a limited region of space by the forces acting on it and does not have sufficient energy to escape.

In this module you will study the special kind of periodic motion that results when the net force acting on a particle, often called the restoring force, is directly …

Capacitors

Calculus-Based General Physics

Capacitors are important components of electronic circuits and of electrical machinery and power grids. You can find large oil-insulated capacitors on powerline poles or small ceramic-insulated capacitors in a radio. In each application the capacitor is used to store electrical charge and electrical energy - for example, sometimes for a short time in an alternating-current cycle, sometimes for a long time until the energy is needed, as in a strobe light for a camera. Your body can be a capacitor, storing up enough charge and energy to cause a painful spark when the capacitor discharqes.

Practical capacitors are basically two …

Coulomb's Law And The Electric Field

Calculus-Based General Physics

This module beqins the study of electricity. Not only is it true that we see nature's gigantic electrical show in thunderstorm displays with lightning, but the very functioning of our smallest cells depends on the balance of electrically charged ions, and their movement through cell membranes. On a larger scale than cell membranes, water-purification studies with large membranes show promise of "electrically" removing undesired ions or debris from water. The electronic air cleaner is yet another direct application of the material to come: a 7000-V potential difference between a thin wire and flat collecting plates ionizes the air, and the …

Conservation Of Energy

Calculus-Based General Physics

Imagine a bicycle rider coasting without pedaling along a road that is very smooth but has a lot of small hills. As he coasts up a hill, the force of gravity will, of course, slow him down; but it speeds him up again as he goes down the other side. We say that gravity is a conservative force because it gives back as much kinetic energy (KE) to the cyclist when he returns to a lower level, as it took away when he ascended to the top. We therefore assign a gravitational potential energy (PE) U_g to the cyclist, …

Alternating-Current Circuits

Calculus-Based General Physics

The electric clock on the wall, radio and television, the incredibly rapid handling
of information by computers, and the transmission of signals by our own nerves
are among countless devices and effects that depend on circuits in which currents
or voltages vary with time. Alternatinq-current (ac) circuits, in which charges
oscillate back and forth in a wire in such a way that the average current is
zero,are among the simpler time-varying circuits. In this module you will study
the behavior of simple ac circuits containing resistors, inductors, and capacitors.

Gravitation

Calculus-Based General Physics

The members of the solar system -- the Sun, the Moon, and the planets -- have held a strong fascination for mankind since prehistoric times. The motions of these heavenly bodies were thought to have important specific influences on persons' lives -- a belief that is reflected even today in horoscopes and astrological publications. A revolution in man's thinking that occurred about four hundred years ago established the concept of a solar system with planets orbiting about the Sun and moons orbiting about some of the planets. Copernicus, Kepler, Galileo, and Newton were the four scientific leaders chiefly responsible for …

Interference

Calculus-Based General Physics

You may have observed the sound from your radio fade in and out as you listened to some distant station. Or perhaps you have sat in a "dead" seat in a poorly designed concert hall where, despite the fact that no physical object is between you and the performer, the sound is distorted and weak. Perhaps you have held two fingers close together and looked with one eye through the narrow slit separating the fingers and observed those mysterious black lines in and parallel to the slit. These and many other similar phenomena result from the interference of two or …

Flux And Gauss' Law

Calculus-Based General Physics

Charles Augustine de Coulomb (1736-1806) designed his famous experiment to measure the force relationships between charged bodies: Coulomb's law is the resulting empirical statement. Gauss' law (Karl Friedrich Gauss, 1777-1855), which you will learn in this module, has a more obscure origin. It was originally a mathematical theorem. Scientists in Gauss' nineteenth century were much more inclined than we are today to equate mathematical correctness with physical correctness. When it was realized that Gauss' (mathematical) theorem could be applied to the electric-field concepts of Faraday to produce Gauss' (physical) law, this extension was eagerly accepted. The origins of the law, …

Planar Motion

Calculus-Based General Physics

Enough of this physics where things move along straight lines only! We know that most interesting real-life motions involve curves of many and varied shapes. This module extends your understanding of kinematics from one dimension to two dimensions. To accomplish this, you will combine your knowledge of calculus and vectors with concepts like position, displacement, velocity, speed, and acceleration.

Two important applications that will be utilized many times in later modules are covered here. First is the motion of a particle experiencing constant acceleration, e.g., a baseball in flight. Second is the motion of a particle in a circular path …

Rotational Motion

Calculus-Based General Physics

There is a motion of a system of masses that is as simple as the motion of a point mass on a straight line. It is the rotation of a rigid body about a fixed axis. For example, we live on a rotating earth, use rotating devices such as a potter's wheel or a phonograph turntable, and test our luck with a spinning roulette wheel. All of these are objects whose motion is described by the time dependence of a single variable, the angle of rotation. We shall study the angular equivalent of uniformly accelerated motion for some rotating objects. …

Partial Derivatives

Calculus-Based General Physics

If we have a function of more than one independent variable, then we can define a partial derivative with respect to one of the variables, which is simply the derivative of the function with all the other variables fixed, The notation using ∂, which we will use below, tells you it is a partial derivative, For example, suppose we have the function y that depends on the independent variables x and t:

y = A sin(kx – ωt),

where A, k, and ω are constants, Then, "the partial derivative of y with respect to x" …

Calculus

Calculus-Based General Physics

Differentiation of a function, say f(x), is a mathematical operation which yields a second function called the derivative of f [symbolized by f '(x) or dy/dx]. This procedure is represented in the diagram, which shows

Input Differentiation Operation Output

f(x) → f ' (x) = df/dx

the function f(x) being input to an "analytical machine" that manufactures as output the derivative of f. A detailed mathematical prescription for the differentiation operation is nontrivial. It usually involves a quarter or semester course, which requires time, attention, and effort on the student's part.

Traveling Waves

Calculus-Based General Physics

"For many people -- perhaps for most -- the word "wave" conjures up a picture of an ocean, with the rollers sweeping onto the beach from the open sea. If you have stood and watched this phenomenon, you may have felt that for all its grandeur it contains an element of anticlimax. You see the crests racing in, you get a sense of the massive assault by the water on the land -- and indeed the waves can do great damage, which means that they are carriers of energy -- but yet when it is all over, when the wave …

Magnetic Forces

Calculus-Based General Physics

It may surprise you to learn that the conversion of electrical energy to mechanical work in electric motors or stereo loud speakers is seldom done by electrostatic forces (Coulomb's law). Magnetic forces associated with moving charges (currents) are the basis of most electromechanical devices. In analogy with the electrostatic case, we introduce an intermediary called the magnetic field. This module considers the forces on currents or moving charges in a magnetic field; the module Ampere's Law will show how magnetic fields are generated by currents.

Ohm's Law

Calculus-Based General Physics

Recently you have heard many ways of reducing energy consumption in the home. One of the suggested ways is to use 60-W liqht bulbs rather than 100-W bulbs; another is to cut back on the use of electrical appliances. You readily identify these suggestions with decreasing the amount of "electricity" beinq transported through the wires coming into your home.

You were warned quite early in life not to stick a metal knife into the toaster to force out burning bread; you either unplug the toaster or use a utensil with a wooden handle. Why? Because you were warned of the …

Newton's Laws

Calculus-Based General Physics

When a body is at rest, we know from experience that it will remain at rest unless something is done to change that state. A heavy box on the floor will stay in place unless it is pushed or pulled. We walk without fear beside a massive rock on level ground because we know it won't suddenly move and crush us.

Undoubtedly you have leaned against a chair only to have it move and send you scurrying for your balance. Did you then question the relationship of the interaction between you and the chair to the ensuing motion of the …

Reflection And Refraction

Calculus-Based General Physics

Sight is certainly one of our most important senses and depends on the interaction of electromagnetic waves in the visible portion of the spectrum with the eye. The use of materials that reflect light and that refract or "bend" light extends throughout our industrialized society. In this module we deal with light traveling in two dimensions and encounterinq the boundaries between media under those conditions in which the wavelength is small compared with the size of the obstacles or apertures. Under such conditions, since diffraction and interference effects are negligible, the principal phenomena occurring at the interfaces, reflection and refraction, …