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College Physics 2e | Frontiers of Physics | Cosmology and Particle Physics | m42686 | |||
College Physics 2e | Frontiers of Physics | General Relativity and Quantum Gravity | m42689 | General Relativity | Einstein first considered the case of no observer acceleration when he developed the revolutionary special theory of relativity, publishing his first work on it in 1905. By 1916, he had laid the foundation of general relativity, again almost on his own. Much of what Einstein did to develop his ideas was to mentally analyze certain carefully and clearly defined situations—doing this is to perform a thought experiment. [link] illustrates a thought experiment like the ones that convinced Einstein that light must fall in a gravitational field. Think about what a person feels in an elevator that is accelerated upward. It is identical to being in a stationary elevator in a gravitational field. The feet of a person are pressed against the floor, and objects released from hand fall with identical accelerations. In fact, it is not possible, without looking outside, to know what is happening—acceleration upward or gravity. This led Einstein to correctly postulate that acceleration and gravity will produce identical effects in all situations. So, if acceleration affects light, then gravity will, too. [link] shows the effect of acceleration on a beam of light shone horizontally at one wall. Since the accelerated elevator moves up during the time light travels across the elevator, the beam of light strikes low, seeming to the person to bend down. (Normally a tiny effect, since the speed of light is so great.) The same effect must occur due to gravity, Einstein reasoned, since there is no way to tell the effects of gravity acting downward from acceleration of the elevator upward. Thus gravity affects the path of light, even though we think of gravity as acting between masses and photons are massless. |
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College Physics 2e | Frontiers of Physics | General Relativity and Quantum Gravity | m42689 | Quantum Gravity | **Black holes radiate**Quantum gravity is important in those situations where gravity is so extremely strong that it has effects on the quantum scale, where the other forces are ordinarily much stronger. The early universe was such a place, but black holes are another. The first significant connection between gravity and quantum effects was made by the Russian physicist Yakov Zel’dovich in 1971, and other significant advances followed from the British physicist Stephen Hawking. (See [link].) These two showed that black holes could radiate away energy by quantum effects just outside the event horizon (nothing can escape from inside the event horizon). Black holes are, thus, expected to radiate energy and shrink to nothing, although extremely slowly for most black holes. The mechanism is the creation of a particle-antiparticle pair from energy in the extremely strong gravitational field near the event horizon. One member of the pair falls into the hole and the other escapes, conserving momentum. (See [link].) When a black hole loses energy and, hence, rest mass, its event horizon shrinks, creating an even greater gravitational field. This increases the rate of pair production so that the process grows exponentially until the black hole is nuclear in size. A final burst of particles and $\gamma$ rays ensues. This is an extremely slow process for black holes about the mass of the Sun (produced by supernovas) or larger ones (like those thought to be at galactic centers), taking on the order of ${\text{10}}^{\text{67}}$ years or longer! Smaller black holes would evaporate faster, but they are only speculated to exist as remnants of the Big Bang. Searches for characteristic $\gamma$-ray bursts have produced events attributable to more mundane objects like neutron stars accreting matter. |
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College Physics 2e | Frontiers of Physics | Superstrings | m42691 | |||
College Physics 2e | Frontiers of Physics | Dark Matter and Closure | m42692 | Evidence | The first clues that there is more matter than meets the eye came from the Swiss-born American astronomer Fritz Zwicky in the 1930s; major work was also done by the American astronomer Vera Rubin. Zwicky measured the velocities of stars orbiting the galaxy, using the relativistic Doppler shift of their spectra (see [link](a)). He found that velocity varied with distance from the center of the galaxy, as graphed in [link](b). If the mass of the galaxy was concentrated in its center, as are its luminous stars, the velocities should decrease as the square root of the distance from the center. Instead, the velocity curve is almost flat, implying that there is a tremendous amount of matter in the galactic halo. Using instruments and methods that offered a greater degree of precision, Rubin investigated the movement of spiral galaxies and observed that their outermost reaches were rotating as quickly as their centers. She also calculated that the rotational velocity of galaxies should have been enough to cause them to fly apart, unless there was a significant discrepancy between their observable matter and their actual matter. This became known as the galaxy rotation problem, which can be "solved" by the presence of unobserved or dark matter. Although not immediately recognized for its significance, such measurements have now been made for many galaxies, with similar results. Further, studies of galactic clusters have also indicated that galaxies have a mass distribution greater than that obtained from their brightness (proportional to the number of stars), which also extends into large halos surrounding the luminous parts of galaxies. Observations of other EM wavelengths, such as radio waves and X rays, have similarly confirmed the existence of dark matter. Take, for example, X rays in the relatively dark space between galaxies, which indicates the presence of previously unobserved hot, ionized gas (see [link](c)). |
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College Physics 2e | Frontiers of Physics | Dark Matter and Closure | m42692 | Theoretical Yearnings for Closure | Is the universe open or closed? That is, will the universe expand forever or will it stop, perhaps to contract? This, until recently, was a question of whether there is enough gravitation to stop the expansion of the universe. In the past few years, it has become a question of the combination of gravitation and what is called the cosmological constant. The cosmological constant was invented by Einstein to prohibit the expansion or contraction of the universe. At the time he developed general relativity, Einstein considered that an illogical possibility. The cosmological constant was discarded after Hubble discovered the expansion, but has been re-invoked in recent years. |
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College Physics 2e | Frontiers of Physics | Dark Matter and Closure | m42692 | What Is the Dark Matter We See Indirectly? | There is no doubt that dark matter exists, but its form and the amount in existence are two facts that are still being studied vigorously. As always, we seek to explain new observations in terms of known principles. However, as more discoveries are made, it is becoming more and more difficult to explain dark matter as a known type of matter. |
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College Physics 2e | Frontiers of Physics | Complexity and Chaos | m42694 | |||
College Physics 2e | Frontiers of Physics | High-temperature Superconductors | m42696 | |||
College Physics 2e | Frontiers of Physics | Some Questions We Know to Ask | m42704 | On the Largest Scale | null |
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College Physics 2e | Frontiers of Physics | Some Questions We Know to Ask | m42704 | On the Intermediate Scale | null |
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College Physics 2e | Frontiers of Physics | Some Questions We Know to Ask | m42704 | On the Smallest Scale | null |
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College Physics for AP® Courses 2e | Introduction: The Nature of Science and Physics | Connection for AP® Courses | m54763 | |||
College Physics for AP® Courses 2e | Introduction: The Nature of Science and Physics | Physics: An Introduction | m42092 | Science and the Realm of Physics | Science consists of the theories and laws that are the general truths of nature as well as the body of knowledge they encompass. Scientists are continually trying to expand this body of knowledge and to perfect the expression of the laws that describe it. Physics is concerned with describing the interactions of energy, matter, space, and time, and it is especially interested in what fundamental mechanisms underlie every phenomenon. The concern for describing the basic phenomena in nature essentially defines the *realm of physics*. |
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College Physics for AP® Courses 2e | Introduction: The Nature of Science and Physics | Physics: An Introduction | m42092 | Applications of Physics | You need not be a scientist to use physics. On the contrary, knowledge of physics is useful in everyday situations as well as in nonscientific professions. It can help you understand how microwave ovens work, why metals should not be put into them, and why they might affect pacemakers. (See [link] and [link].) Physics allows you to understand the hazards of radiation and rationally evaluate these hazards more easily. Physics also explains the reason why a black car radiator helps remove heat in a car engine, and it explains why a white roof helps keep the inside of a house cool. Similarly, the operation of a car’s ignition system as well as the transmission of electrical signals through our body’s nervous system are much easier to understand when you think about them in terms of basic physics. |
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College Physics for AP® Courses 2e | Introduction: The Nature of Science and Physics | Physics: An Introduction | m42092 | Models, Theories, and Laws; The Role of Experimentation | The laws of nature are concise descriptions of the universe around us; they are human statements of the underlying laws or rules that all natural processes follow. Such laws are intrinsic to the universe; humans did not create them and so cannot change them. We can only discover and understand them. Their discovery is a very human endeavor, with all the elements of mystery, imagination, struggle, triumph, and disappointment inherent in any creative effort. (See [link] and [link].) The cornerstone of discovering natural laws is observation; science must describe the universe as it is, not as we may imagine it to be. |
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College Physics for AP® Courses 2e | Introduction: The Nature of Science and Physics | Physics: An Introduction | m42092 | The Evolution of Natural Philosophy into Modern Physics | Physics was not always a separate and distinct discipline. It remains connected to other sciences to this day. The word *physics* comes from Greek, meaning nature. The study of nature came to be called “natural philosophy.” From ancient times through the Renaissance, natural philosophy encompassed many fields, including astronomy, biology, chemistry, physics, mathematics, and medicine. Over the last few centuries, the growth of knowledge has resulted in ever-increasing specialization and branching of natural philosophy into separate fields, with physics retaining the most basic facets. (See [link], [link], and [link].) Physics as it developed from the Renaissance to the end of the 19th century is called classical physics. It was transformed into modern physics by revolutionary discoveries made starting at the beginning of the 20th century. |
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College Physics for AP® Courses 2e | Introduction: The Nature of Science and Physics | Physical Quantities and Units | m42091 | SI Units: Fundamental and Derived Units | [link] gives the fundamental SI units that are used throughout this textbook. This text uses non-SI units in a few applications where they are in very common use, such as the measurement of blood pressure in millimeters of mercury (mm Hg). Whenever non-SI units are discussed, they will be tied to SI units through conversions. |
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College Physics for AP® Courses 2e | Introduction: The Nature of Science and Physics | Physical Quantities and Units | m42091 | Units of Time, Length, and Mass: The Second, Meter, and Kilogram | null |
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College Physics for AP® Courses 2e | Introduction: The Nature of Science and Physics | Physical Quantities and Units | m42091 | Metric Prefixes | SI units are part of the metric system. The metric system is convenient for scientific and engineering calculations because the units are categorized by factors of 10. [link] gives metric prefixes and symbols used to denote various factors of 10. |
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College Physics for AP® Courses 2e | Introduction: The Nature of Science and Physics | Physical Quantities and Units | m42091 | Known Ranges of Length, Mass, and Time | The vastness of the universe and the breadth over which physics applies are illustrated by the wide range of examples of known lengths, masses, and times in [link]. Examination of this table will give you some feeling for the range of possible topics and numerical values. (See [link] and [link].) |
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College Physics for AP® Courses 2e | Introduction: The Nature of Science and Physics | Physical Quantities and Units | m42091 | Unit Conversion and Dimensional Analysis | It is often necessary to convert from one type of unit to another. For example, if you are reading a European cookbook, some quantities may be expressed in units of liters and you need to convert them to cups. Or, perhaps you are reading walking directions from one location to another and you are interested in how many miles you will be walking. In this case, you will need to convert units of feet to miles. |
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College Physics for AP® Courses 2e | Introduction: The Nature of Science and Physics | Accuracy, Precision, and Significant Figures | m42120 | Accuracy and Precision of a Measurement | Science is based on observation and experiment—that is, on measurements. Accuracy is how close a measurement is to the correct value for that measurement. For example, let us say that you are measuring the length of standard computer paper. The packaging in which you purchased the paper states that it is 11.0 inches long. You measure the length of the paper three times and obtain the following measurements: 11.1 in., 11.2 in., and 10.9 in. These measurements are quite accurate because they are very close to the correct value of 11.0 inches. In contrast, if you had obtained a measurement of 12 inches, your measurement would not be very accurate. |
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College Physics for AP® Courses 2e | Introduction: The Nature of Science and Physics | Accuracy, Precision, and Significant Figures | m42120 | Accuracy, Precision, and Uncertainty | The degree of accuracy and precision of a measuring system are related to the uncertainty in the measurements. Uncertainty is a quantitative measure of how much your measured values deviate from a standard or expected value. If your measurements are not very accurate or precise, then the uncertainty of your values will be very high. In more general terms, uncertainty can be thought of as a disclaimer for your measured values. For example, if someone asked you to provide the mileage on your car, you might say that it is 45,000 miles, plus or minus 500 miles. The plus or minus amount is the uncertainty in your value. That is, you are indicating that the actual mileage of your car might be as low as 44,500 miles or as high as 45,500 miles, or anywhere in between. All measurements contain some amount of uncertainty. In our example of measuring the length of the paper, we might say that the length of the paper is 11 in., plus or minus 0.2 in. The uncertainty in a measurement, $A$, is often denoted as *$\mathrm{\delta A}$* (“delta $A$”), so the measurement result would be recorded as
$A±\mathrm{\delta A}$. In our paper example, the length of the paper could be expressed as |
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College Physics for AP® Courses 2e | Introduction: The Nature of Science and Physics | Accuracy, Precision, and Significant Figures | m42120 | Precision of Measuring Tools and Significant Figures | An important factor in the accuracy and precision of measurements involves the precision of the measuring tool. In general, a precise measuring tool is one that can measure values in very small increments. For example, a standard ruler can measure length to the nearest millimeter, while a caliper can measure length to the nearest 0.01 millimeter. The caliper is a more precise measuring tool because it can measure extremely small differences in length. The more precise the measuring tool, the more precise and accurate the measurements can be. |
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College Physics for AP® Courses 2e | Introduction: The Nature of Science and Physics | Approximation | m42121 | |||
College Physics for AP® Courses 2e | Kinematics | Connection for AP® Courses | m54768 | |||
College Physics for AP® Courses 2e | Kinematics | Displacement | m42033 | Position | In order to describe the motion of an object, you must first be able to describe its position—where it is at any particular time. More precisely, you need to specify its position relative to a convenient reference frame. Earth is often used as a reference frame, and we often describe the position of an object as it relates to stationary objects in that reference frame. For example, a rocket launch would be described in terms of the position of the rocket with respect to the Earth as a whole, while a professor’s position could be described in terms of where she is in relation to the nearby white board. (See [link].) In other cases, we use reference frames that are not stationary but are in motion relative to the Earth. To describe the position of a person in an airplane, for example, we use the airplane, not the Earth, as the reference frame. (See [link].) |
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College Physics for AP® Courses 2e | Kinematics | Displacement | m42033 | Displacement | If an object moves relative to a reference frame (for example, if a professor moves to the right relative to a white board or a passenger moves toward the rear of an airplane), then the object’s position changes. This change in position is known as displacement. The word “displacement” implies that an object has moved, or has been displaced. |
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College Physics for AP® Courses 2e | Kinematics | Displacement | m42033 | Distance | Although displacement is described in terms of direction, distance is not. Distance is defined to be *the magnitude or size of displacement between two positions*. Note that the distance between two positions is not the same as the distance traveled between them. Distance traveled is *the total length of the path traveled between two positions*. Distance has no direction and, thus, no sign. For example, the distance the professor walks is 2.0 m. The distance the airplane passenger walks is 4.0 m. |
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College Physics for AP® Courses 2e | Kinematics | Vectors, Scalars, and Coordinate Systems | m42124 | Coordinate Systems for One-Dimensional Motion | In order to describe the direction of a vector quantity, you must designate a coordinate system within the reference frame. For one-dimensional motion, this is a simple coordinate system consisting of a one-dimensional coordinate line. In general, when describing horizontal motion, motion to the right is usually considered positive, and motion to the left is considered negative. With vertical motion, motion up is usually positive and motion down is negative. In some cases, however, as with the jet in [link], it can be more convenient to switch the positive and negative directions. For example, if you are analyzing the motion of falling objects, it can be useful to define downwards as the positive direction. If people in a race are running to the left, it is useful to define left as the positive direction. It does not matter as long as the system is clear and consistent. Once you assign a positive direction and start solving a problem, you cannot change it. |
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College Physics for AP® Courses 2e | Kinematics | Time, Velocity, and Speed | m42096 | Time | As discussed in Physical Quantities and Units, the most fundamental physical quantities are defined by how they are measured. This is the case with time. Every measurement of time involves measuring a change in some physical quantity. It may be a number on a digital clock, a heartbeat, or the position of the Sun in the sky. In physics, the definition of time is simple—time is *change*, or the interval over which change occurs. It is impossible to know that time has passed unless something changes. |
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College Physics for AP® Courses 2e | Kinematics | Time, Velocity, and Speed | m42096 | Velocity | Your notion of velocity is probably the same as its scientific definition. You know that if you have a large displacement in a small amount of time you have a large velocity, and that velocity has units of distance divided by time, such as miles per hour or kilometers per hour. |
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College Physics for AP® Courses 2e | Kinematics | Time, Velocity, and Speed | m42096 | Speed | In everyday language, most people use the terms “speed” and “velocity” interchangeably. In physics, however, they do not have the same meaning and they are distinct concepts. One major difference is that speed has no direction. Thus *speed is a scalar*. Just as we need to distinguish between instantaneous velocity and average velocity, we also need to distinguish between instantaneous speed and average speed. |
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College Physics for AP® Courses 2e | Kinematics | Acceleration | m42100 | Instantaneous Acceleration | Instantaneous acceleration $a$, or the *acceleration at a specific instant in time*, is obtained by the same process as discussed for instantaneous velocity in Time, Velocity, and Speed—that is, by considering an infinitesimally small interval of time. How do we find instantaneous acceleration using only algebra? The answer is that we choose an average acceleration that is representative of the motion. [link] shows graphs of instantaneous acceleration versus time for two very different motions. In [link](a), the acceleration varies slightly and the average over the entire interval is nearly the same as the instantaneous acceleration at any time. In this case, we should treat this motion as if it had a constant acceleration equal to the average (in this case about $1\text{.}8 m{\text{/s}}^{2}$). In [link](b), the acceleration varies drastically over time. In such situations it is best to consider smaller time intervals and choose an average acceleration for each. For example, we could consider motion over the time intervals from 0 to 1.0 s and from 1.0 to 3.0 s as separate motions with accelerations of $+3\text{.}0 m{\text{/s}}^{2}$ and $\text{–2}\text{.}0 m{\text{/s}}^{2}$, respectively. |
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College Physics for AP® Courses 2e | Kinematics | Acceleration | m42100 | Sign and Direction | Perhaps the most important thing to note about these examples is the signs of the answers. In our chosen coordinate system, plus means the quantity is to the right and minus means it is to the left. This is easy to imagine for displacement and velocity. But it is a little less obvious for acceleration. Most people interpret negative acceleration as the slowing of an object. This was not the case in [link], where a positive acceleration slowed a negative velocity. The crucial distinction was that the acceleration was in the opposite direction from the velocity. In fact, a negative acceleration will *increase* a negative velocity. For example, the train moving to the left in [link] is sped up by an acceleration to the left. In that case, both $v$ and $a$ are negative. The plus and minus signs give the directions of the accelerations. If acceleration has the same sign as the velocity, the object is speeding up. If acceleration has the opposite sign as the velocity, the object is slowing down. |
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College Physics for AP® Courses 2e | Kinematics | Motion Equations for Constant Acceleration in One Dimension | m42099 | Notation: *t*, *x*, *v*, *a* | First, let us make some simplifications in notation. Taking the initial time to be zero, as if time is measured with a stopwatch, is a great simplification. Since elapsed time is $\Delta t={t}_{f}-{t}_{0}$, taking ${t}_{0}=0$ means that $\Delta t={t}_{f}$, the final time on the stopwatch. When initial time is taken to be zero, we use the subscript 0 to denote initial values of position and velocity. That is, ${x}_{0}$ *is the initial position* and ${v}_{0}$ *is the initial velocity*. We put no subscripts on the final values. That is, $t$ *is the final time*, $x$ *is the final position*, and $v$ *is the final velocity*. This gives a simpler expression for elapsed time—now, $\Delta t=t$. It also simplifies the expression for displacement, which is now $\Delta x=x-{x}_{0}$. Also, it simplifies the expression for change in velocity, which is now $\Delta v=v-{v}_{0}$. To summarize, using the simplified notation, with the initial time taken to be zero, |
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College Physics for AP® Courses 2e | Kinematics | Motion Equations for Constant Acceleration in One Dimension | m42099 | Putting Equations Together | In the following examples, we further explore one-dimensional motion, but in situations requiring slightly more algebraic manipulation. The examples also give insight into problem-solving techniques. The box below provides easy reference to the equations needed. |
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College Physics for AP® Courses 2e | Kinematics | Problem-Solving Basics for One-Dimensional Kinematics | m42125 | Problem-Solving Steps | While there is no simple step-by-step method that works for every problem, the following general procedures facilitate problem solving and make it more meaningful. A certain amount of creativity and insight is required as well. |
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College Physics for AP® Courses 2e | Kinematics | Problem-Solving Basics for One-Dimensional Kinematics | m42125 | Unreasonable Results | Physics must describe nature accurately. Some problems have results that are unreasonable because one premise is unreasonable or because certain premises are inconsistent with one another. The physical principle applied correctly then produces an unreasonable result. For example, if a person starting a foot race accelerates at $0\text{.}{\text{40 m/s}}^{2}$ for 100 s, his final speed will be 40 m/s (about 150 km/h)—clearly unreasonable because the time of 100 s is an unreasonable premise. The physics is correct in a sense, but there is more to describing nature than just manipulating equations correctly. Checking the result of a problem to see if it is reasonable does more than help uncover errors in problem solving—it also builds intuition in judging whether nature is being accurately described. |
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College Physics for AP® Courses 2e | Kinematics | Falling Objects | m42102 | Gravity | The most remarkable and unexpected fact about falling objects is that, if air resistance and friction are negligible, then in a given location all objects fall toward the center of Earth with the *same constant acceleration*, *independent of their mass*. This experimentally determined fact is unexpected, because we are so accustomed to the effects of air resistance and friction that we expect light objects to fall slower than heavy ones. |
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College Physics for AP® Courses 2e | Kinematics | Falling Objects | m42102 | One-Dimensional Motion Involving Gravity | The best way to see the basic features of motion involving gravity is to start with the simplest situations and then progress toward more complex ones. So we start by considering straight up and down motion with no air resistance or friction. These assumptions mean that the velocity (if there is any) is vertical. If the object is dropped, we know the initial velocity is zero. Once the object has left contact with whatever held or threw it, the object is in free-fall. Under these circumstances, the motion is one-dimensional and has constant acceleration of magnitude $g$. We will also represent vertical displacement with the symbol $y$ and use $x$ for horizontal displacement. |
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College Physics for AP® Courses 2e | Kinematics | Graphical Analysis of One-Dimensional Motion | m42103 | Slopes and General Relationships | First note that graphs in this text have perpendicular axes, one horizontal and the other vertical. When two physical quantities are plotted against one another in such a graph, the horizontal axis is usually considered to be an independent variable and the vertical axis a dependent variable. If we call the horizontal axis the $x$-axis and the vertical axis the $y$-axis, as in [link], a straight-line graph has the general form |
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College Physics for AP® Courses 2e | Kinematics | Graphical Analysis of One-Dimensional Motion | m42103 | Graph of Position vs. Time (*a* = 0, so *v* is constant) | Time is usually an independent variable that other quantities, such as position, depend upon. A graph of position versus time would, thus, have $x$ on the vertical axis and $t$ on the horizontal axis. [link] is just such a straight-line graph. It shows a graph of position versus time for a jet-powered car on a very flat dry lake bed in Nevada. |
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College Physics for AP® Courses 2e | Kinematics | Graphical Analysis of One-Dimensional Motion | m42103 | Graphs of Motion when $a$ is constant but $a\ne 0$ | The graphs in [link] below represent the motion of the jet-powered car as it accelerates toward its top speed, but only during the time when its acceleration is constant. Time starts at zero for this motion (as if measured with a stopwatch), and the position and velocity are initially 200 m and 15 m/s, respectively. |
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College Physics for AP® Courses 2e | Kinematics | Graphical Analysis of One-Dimensional Motion | m42103 | Graphs of Motion Where Acceleration is Not Constant | Now consider the motion of the jet car as it goes from 165 m/s to its top velocity of 250 m/s, graphed in [link]. Time again starts at zero, and the initial velocity is 165 m/s. (This was the final velocity of the car in the motion graphed in [link].) Acceleration gradually decreases from $5\text{.}{\text{0 m/s}}^{2}$ to zero when the car hits 250 m/s. The velocity increases until 55 s and then becomes constant, since acceleration decreases to zero at 55 s and remains zero afterward. |
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College Physics for AP® Courses 2e | Two-Dimensional Kinematics | Connection for AP® Courses | m54777 | |||
College Physics for AP® Courses 2e | Two-Dimensional Kinematics | Kinematics in Two Dimensions: An Introduction | m42104 | Two-Dimensional Motion: Walking in a City | Suppose you want to walk from one point to another in a city with uniform square blocks, as pictured in [link]. |
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College Physics for AP® Courses 2e | Two-Dimensional Kinematics | Kinematics in Two Dimensions: An Introduction | m42104 | The Independence of Perpendicular Motions | The person taking the path shown in [link] walks east and then north (two perpendicular directions). How far they walk east is only affected by their motion eastward. Similarly, how far they walk north is only affected by their motion northward. |
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College Physics for AP® Courses 2e | Two-Dimensional Kinematics | Vector Addition and Subtraction: Graphical Methods | m42127 | Vectors in Two Dimensions | A vector is a quantity that has magnitude and direction. Displacement, velocity, acceleration, and force, for example, are all vectors. In one-dimensional, or straight-line, motion, the direction of a vector can be given simply by a plus or minus sign. In two dimensions (2-d), however, we specify the direction of a vector relative to some reference frame (i.e., coordinate system), using an arrow having length proportional to the vector’s magnitude and pointing in the direction of the vector. |
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College Physics for AP® Courses 2e | Two-Dimensional Kinematics | Vector Addition and Subtraction: Graphical Methods | m42127 | Vector Addition: Head-to-Tail Method | The head-to-tail method is a graphical way to add vectors, described in [link] below and in the steps following. The tail of the vector is the starting point of the vector, and the head (or tip) of a vector is the final, pointed end of the arrow. |
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College Physics for AP® Courses 2e | Two-Dimensional Kinematics | Vector Addition and Subtraction: Graphical Methods | m42127 | Vector Subtraction | Vector subtraction is a straightforward extension of vector addition. To define subtraction (say we want to subtract $\mathbf{\text{B}}$ from
$\mathbf{\text{A}}$
, written $\mathbf{\text{A}}–\mathbf{\text{B}}$
, we must first define what we mean by subtraction. The *negative* of a vector $\mathbf{\text{B}}$
is defined to be $\mathbf{\text{–B}}$; that is, graphically *the negative of any vector has the same magnitude but the opposite direction*, as shown in [link]. In other words, $\mathbf{\text{B}}$ has the same length as $\mathbf{\text{–B}}$, but points in the opposite direction. Essentially, we just flip the vector so it points in the opposite direction. |
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College Physics for AP® Courses 2e | Two-Dimensional Kinematics | Vector Addition and Subtraction: Graphical Methods | m42127 | Multiplication of Vectors and Scalars | If we decided to walk three times as far on the first leg of the trip considered in the preceding example, then we would walk , or 82.5 m, in a direction $\text{66}\text{.}0\text{º}$ north of east. This is an example of multiplying a vector by a positive scalar. Notice that the magnitude changes, but the direction stays the same. |
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College Physics for AP® Courses 2e | Two-Dimensional Kinematics | Vector Addition and Subtraction: Graphical Methods | m42127 | Resolving a Vector into Components | In the examples above, we have been adding vectors to determine the resultant vector. In many cases, however, we will need to do the opposite. We will need to take a single vector and find what other vectors added together produce it. In most cases, this involves determining the perpendicular components of a single vector, for example the *x*- *and* *y*-components, or the north-south and east-west components. |
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College Physics for AP® Courses 2e | Two-Dimensional Kinematics | Vector Addition and Subtraction: Analytical Methods | m42128 | Resolving a Vector into Perpendicular Components | Analytical techniques and right triangles go hand-in-hand in physics because (among other things) motions along perpendicular directions are independent. We very often need to separate a vector into perpendicular components. For example, given a vector like $\mathbf{A}$ in [link], we may wish to find which two perpendicular vectors, ${\mathbf{A}}_{x}$ and ${\mathbf{A}}_{y}$, add to produce it. |
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College Physics for AP® Courses 2e | Two-Dimensional Kinematics | Vector Addition and Subtraction: Analytical Methods | m42128 | Calculating a Resultant Vector | If the perpendicular components ${\mathbf{A}}_{x}$ and ${\mathbf{A}}_{y}$ of a vector $\mathbf{A}$ are known, then $\mathbf{A}$ can also be found analytically. To find the magnitude $A$ and direction $\theta$ of a vector from its perpendicular components ${\mathbf{A}}_{x}$ and ${\mathbf{A}}_{y}$, relative to the *x*-axis, we use the following relationships: |
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College Physics for AP® Courses 2e | Two-Dimensional Kinematics | Vector Addition and Subtraction: Analytical Methods | m42128 | Adding Vectors Using Analytical Methods | To see how to add vectors using perpendicular components, consider [link], in which the vectors $\mathbf{A}$ and $\mathbf{B}$ are added to produce the resultant $\mathbf{R}$. |
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College Physics for AP® Courses 2e | Two-Dimensional Kinematics | Projectile Motion | m42042 | |||
College Physics for AP® Courses 2e | Two-Dimensional Kinematics | Addition of Velocities | m42045 | Relative Velocity | If a person rows a boat across a rapidly flowing river and tries to head directly for the other shore, the boat instead moves *diagonally* relative to the shore, as in [link]. The boat does not move in the direction in which it is pointed. The reason, of course, is that the river carries the boat downstream. Similarly, if a small airplane flies overhead in a strong crosswind, you can sometimes see that the plane is not moving in the direction in which it is pointed, as illustrated in [link]. The plane is moving straight ahead relative to the air, but the movement of the air mass relative to the ground carries it sideways. |
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College Physics for AP® Courses 2e | Two-Dimensional Kinematics | Addition of Velocities | m42045 | Relative Velocities and Classical Relativity | When adding velocities, we have been careful to specify that the *velocity is relative to some reference frame*. These velocities are called relative velocities. For example, the velocity of an airplane relative to an air mass is different from its velocity relative to the ground. Both are quite different from the velocity of an airplane relative to its passengers (which should be close to zero). Relative velocities are one aspect of relativity, which is defined to be the study of how different observers moving relative to each other measure the same phenomenon. |
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College Physics for AP® Courses 2e | Dynamics: Force and Newton's Laws of Motion | Connection for AP® Courses | m54802 | |||
College Physics for AP® Courses 2e | Dynamics: Force and Newton's Laws of Motion | Development of Force Concept | m42069 | |||
College Physics for AP® Courses 2e | Dynamics: Force and Newton's Laws of Motion | Newton’s First Law of Motion: Inertia | m42130 | Mass | The property of a body to remain at rest or to remain in motion with constant velocity is called inertia. Newton’s first law is often called the law of inertia. As we know from experience, some objects have more inertia than others. It is obviously more difficult to change the motion of a large boulder than that of a basketball, for example. The inertia of an object is measured by its mass. Roughly speaking, mass is a measure of the amount of “stuff” (or matter) in something. The quantity or amount of matter in an object is determined by the numbers of atoms and molecules of various types it contains. Unlike weight, mass does not vary with location. The mass of an object is the same on Earth, in orbit, or on the surface of the Moon. In practice, it is very difficult to count and identify all of the atoms and molecules in an object, so masses are not often determined in this manner. Operationally, the masses of objects are determined by comparison with the standard kilogram. |
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College Physics for AP® Courses 2e | Dynamics: Force and Newton's Laws of Motion | Newton’s Second Law of Motion: Concept of a System | m42073 | Units of Force | ${\mathbf{\text{F}}}_{\text{net}}=m\mathbf{\text{a}}$ is used to define the units of force in terms of the three basic units for mass, length, and time. The SI unit of force is called the newton (abbreviated N) and is the force needed to accelerate a 1-kg system at the rate of $1{\text{m/s}}^{2}$. That is, since ${\mathbf{\text{F}}}_{\text{net}}=m\mathbf{\text{a}}$, |
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College Physics for AP® Courses 2e | Dynamics: Force and Newton's Laws of Motion | Newton’s Second Law of Motion: Concept of a System | m42073 | Weight and the Gravitational Force | When an object is dropped, it accelerates toward the center of Earth. Newton’s second law states that a net force on an object is responsible for its acceleration. If air resistance is negligible, the net force on a falling object is the gravitational force, commonly called its weight $\mathbf{\text{w}}$. Weight can be denoted as a vector $\mathbf{\text{w}}$ because it has a direction; *down* is, by definition, the direction of gravity, and hence weight is a downward force. The magnitude of weight is denoted as $w$*.* Galileo was instrumental in showing that, in the absence of air resistance, all objects fall with the same acceleration $g$. Using Galileo’s result and Newton’s second law, we can derive an equation for weight. |
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College Physics for AP® Courses 2e | Dynamics: Force and Newton's Laws of Motion | Newton’s Third Law of Motion: Symmetry in Forces | m42074 | |||
College Physics for AP® Courses 2e | Dynamics: Force and Newton's Laws of Motion | Normal, Tension, and Other Examples of Forces | m42075 | Normal Force | Weight (also called force of gravity) is a pervasive force that acts at all times and must be counteracted to keep an object from falling. You definitely notice that you must support the weight of a heavy object by pushing up on it when you hold it stationary, as illustrated in [link](a). But how do inanimate objects like a table support the weight of a mass placed on them, such as shown in [link](b)? When the bag of dog food is placed on the table, the table actually sags slightly under the load. This would be noticeable if the load were placed on a card table, but even rigid objects deform when a force is applied to them. Unless the object is deformed beyond its limit, it will exert a restoring force much like a deformed spring (or trampoline or diving board). The greater the deformation, the greater the restoring force. So when the load is placed on the table, the table sags until the restoring force becomes as large as the weight of the load. At this point the net external force on the load is zero. That is the situation when the load is stationary on the table. The table sags quickly, and the sag is slight so we do not notice it. But it is similar to the sagging of a trampoline when you climb onto it. |
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College Physics for AP® Courses 2e | Dynamics: Force and Newton's Laws of Motion | Normal, Tension, and Other Examples of Forces | m42075 | Tension | A tension is a force along the length of a medium, especially a force carried by a flexible medium, such as a rope or cable. The word “tension*”* comes from a Latin word meaning “to stretch.” Not coincidentally, the flexible cords that carry muscle forces to other parts of the body are called *tendons*. Any flexible connector, such as a string, rope, chain, wire, or cable, can exert pulls only parallel to its length; thus, a force carried by a flexible connector is a tension with direction parallel to the connector. It is important to understand that tension is a pull in a connector. In contrast, consider the phrase: “You can’t push a rope.” The tension force pulls outward along the two ends of a rope. |
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College Physics for AP® Courses 2e | Dynamics: Force and Newton's Laws of Motion | Normal, Tension, and Other Examples of Forces | m42075 | Extended Topic: Real Forces and Inertial Frames | There is another distinction among forces in addition to the types already mentioned. Some forces are real, whereas others are not. *Real forces* are those that have some physical origin, such as the gravitational pull. Contrastingly, *fictitious forces* are those that arise simply because an observer is in an accelerating frame of reference, such as one that rotates (like a merry-go-round) or undergoes linear acceleration (like a car slowing down). For example, if a satellite is heading due north above Earth’s northern hemisphere, then to an observer on Earth it will appear to experience a force to the west that has no physical origin. Of course, what is happening here is that Earth is rotating toward the east and moves east under the satellite. In Earth’s frame this looks like a westward force on the satellite, or it can be interpreted as a violation of Newton’s first law (the law of inertia). An inertial frame of reference is one in which all forces are real and, equivalently, one in which Newton’s laws have the simple forms given in this chapter. |
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College Physics for AP® Courses 2e | Dynamics: Force and Newton's Laws of Motion | Problem-Solving Strategies | m42076 | Problem-Solving Strategy for Newton’s Laws of Motion | Step 1. As usual, it is first necessary to identify the physical principles involved. *Once it is determined that Newton’s laws of motion are involved (if the problem involves forces), it is particularly important to draw a careful sketch of the situation*. Such a sketch is shown in [link](a). Then, as in [link](b), use arrows to represent all forces, label them carefully, and make their lengths and directions correspond to the forces they represent (whenever sufficient information exists).
(a) A sketch of Tarzan hanging from a vine. (b) Arrows are used to represent all forces. $\mathbf{\text{T}}$ is the tension in the vine above Tarzan, ${\mathbf{\text{F}}}_{\text{T}}$ is the force he exerts on the vine, and $\mathbf{\text{w}}$ is his weight. All other forces, such as the nudge of a breeze, are assumed negligible. (c) Suppose we are given the ape man’s mass and asked to find the tension in the vine. We then define the system of interest as shown and draw a free-body diagram. ${\mathbf{\text{F}}}_{\text{T}}$ is no longer shown, because it is not a force acting on the system of interest; rather, ${\mathbf{\text{F}}}_{\text{T}}$ acts on the outside world. (d) Showing only the arrows, the head-to-tail method of addition is used. It is apparent that $\mathbf{\text{T}}=-\mathbf{w}$, if Tarzan is stationary.
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College Physics for AP® Courses 2e | Dynamics: Force and Newton's Laws of Motion | Further Applications of Newton’s Laws of Motion | m42132 | Integrating Concepts: Newton’s Laws of Motion and Kinematics | Physics is most interesting and most powerful when applied to general situations that involve more than a narrow set of physical principles. Newton’s laws of motion can also be integrated with other concepts that have been discussed previously in this text to solve problems of motion. For example, forces produce accelerations, a topic of kinematics, and hence the relevance of earlier chapters. When approaching problems that involve various types of forces, acceleration, velocity, and/or position, use the following steps to approach the problem: |
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College Physics for AP® Courses 2e | Dynamics: Force and Newton's Laws of Motion | Extended Topic: The Four Basic Forces—An Introduction | m42137 | Action at a Distance: Concept of a Field | All forces act at a distance. This is obvious for the gravitational force. Earth and the Moon, for example, interact without coming into contact. It is also true for all other forces. Friction, for example, is an electromagnetic force between atoms that may not actually touch. What is it that carries forces between objects? One way to answer this question is to imagine that a force field surrounds whatever object creates the force. A second object (often called a *test object*) placed in this field will experience a force that is a function of location and other variables. The field itself is the “thing” that carries the force from one object to another. The field is defined so as to be a characteristic of the object creating it; the field does not depend on the test object placed in it. Earth’s gravitational field, for example, is a function of the mass of Earth and the distance from its center, independent of the presence of other masses. The concept of a field is useful because equations can be written for force fields surrounding objects (for gravity, this yields $w=\text{mg}$ at Earth’s surface), and motions can be calculated from these equations. (See [link].) |
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College Physics for AP® Courses 2e | Further Applications of Newton's Laws: Friction, Drag, and Elasticity | Connection for AP® Courses | m54888 | |||
College Physics for AP® Courses 2e | Further Applications of Newton's Laws: Friction, Drag, and Elasticity | Friction | m42139 | |||
College Physics for AP® Courses 2e | Further Applications of Newton's Laws: Friction, Drag, and Elasticity | Drag Forces | m42080 | |||
College Physics for AP® Courses 2e | Further Applications of Newton's Laws: Friction, Drag, and Elasticity | Elasticity: Stress and Strain | m42081 | Changes in Length—Tension and Compression: Elastic Modulus | A change in length $\Delta L$ is produced when a force is applied to a wire or rod parallel to its length ${L}_{0}$, either stretching it (a tension) or compressing it. (See [link].) |
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College Physics for AP® Courses 2e | Further Applications of Newton's Laws: Friction, Drag, and Elasticity | Elasticity: Stress and Strain | m42081 | Sideways Stress: Shear Modulus | [link] illustrates what is meant by a sideways stress or a *shearing force*. Here the deformation is called $\Delta x$ and it is perpendicular to ${L}_{0}$, rather than parallel as with tension and compression. Shear deformation behaves similarly to tension and compression and can be described with similar equations. The expression for shear deformation is |
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College Physics for AP® Courses 2e | Further Applications of Newton's Laws: Friction, Drag, and Elasticity | Elasticity: Stress and Strain | m42081 | Changes in Volume: Bulk Modulus | An object will be compressed in all directions if inward forces are applied evenly on all its surfaces as in [link]. It is relatively easy to compress gases and extremely difficult to compress liquids and solids. For example, air in a wine bottle is compressed when it is corked. But if you try corking a brim-full bottle, you cannot compress the wine—some must be removed if the cork is to be inserted. The reason for these different compressibilities is that atoms and molecules are separated by large empty spaces in gases but packed close together in liquids and solids. To compress a gas, you must force its atoms and molecules closer together. To compress liquids and solids, you must actually compress their atoms and molecules, and very strong electromagnetic forces in them oppose this compression. |
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College Physics for AP® Courses 2e | Uniform Circular Motion and Gravitation | Connection for AP® Courses | m54986 | |||
College Physics for AP® Courses 2e | Uniform Circular Motion and Gravitation | Rotation Angle and Angular Velocity | m42083 | Rotation Angle | When objects rotate about some axis—for example, when the CD (compact disc) in [link] rotates about its center—each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each pit used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the rotation angle $\text{Δ}\theta$ to be the ratio of the arc length to the radius of curvature: |
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College Physics for AP® Courses 2e | Uniform Circular Motion and Gravitation | Rotation Angle and Angular Velocity | m42083 | Angular Velocity | How fast is an object rotating? We define angular velocity $\omega$ as the rate of change of an angle. In symbols, this is |
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College Physics for AP® Courses 2e | Uniform Circular Motion and Gravitation | Centripetal Acceleration | m42084 | |||
College Physics for AP® Courses 2e | Uniform Circular Motion and Gravitation | Centripetal Force | m42086 | |||
College Physics for AP® Courses 2e | Uniform Circular Motion and Gravitation | Fictitious Forces and Non-inertial Frames: The Coriolis Force | m42142 | |||
College Physics for AP® Courses 2e | Uniform Circular Motion and Gravitation | Newton’s Universal Law of Gravitation | m42143 | Tides | Ocean tides are one very observable result of the Moon’s gravity acting on Earth. [link] is a simplified drawing of the Moon’s position relative to the tides. Because water easily flows on Earth’s surface, a high tide is created on the side of Earth nearest to the Moon, where the Moon’s gravitational pull is strongest. Why is there also a high tide on the opposite side of Earth? The answer is that Earth is pulled toward the Moon more than the water on the far side, because Earth is closer to the Moon. So the water on the side of Earth closest to the Moon is pulled away from Earth, and Earth is pulled away from water on the far side. As Earth rotates, the tidal bulge (an effect of the tidal forces between an orbiting natural satellite and the primary planet that it orbits) keeps its orientation with the Moon. Thus there are two tides per day (the actual tidal period is about 12 hours and 25.2 minutes), because the Moon moves in its orbit each day as well). |
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College Physics for AP® Courses 2e | Uniform Circular Motion and Gravitation | Newton’s Universal Law of Gravitation | m42143 | ”Weightlessness” and Microgravity | In contrast to the tremendous gravitational force near black holes is the apparent gravitational field experienced by astronauts orbiting Earth. What is the effect of “weightlessness” upon an astronaut who is in orbit for months? Or what about the effect of weightlessness upon plant growth? Weightlessness doesn’t mean that an astronaut is not being acted upon by the gravitational force. There is no “zero gravity” in an astronaut’s orbit. The term just means that the astronaut is in free-fall, accelerating with the acceleration due to gravity. If an elevator cable breaks, the passengers inside will be in free fall and will experience weightlessness. You can experience short periods of weightlessness in some rides in amusement parks. |
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College Physics for AP® Courses 2e | Uniform Circular Motion and Gravitation | Newton’s Universal Law of Gravitation | m42143 | The Cavendish Experiment: Then and Now | As previously noted, the universal gravitational constant $G$ is determined experimentally. This definition was first done accurately by Henry Cavendish (1731–1810), an English scientist, in 1798, more than 100 years after Newton published his universal law of gravitation. The measurement of $G$ is very basic and important because it determines the strength of one of the four forces in nature. Cavendish’s experiment was very difficult because he measured the tiny gravitational attraction between two ordinary-sized masses (tens of kilograms at most), using apparatus like that in [link]. Remarkably, his value for $G$ differs by less than 1% from the best modern value. |
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College Physics for AP® Courses 2e | Uniform Circular Motion and Gravitation | Satellites and Kepler’s Laws: An Argument for Simplicity | m42144 | Kepler’s Laws of Planetary Motion | **Kepler’s First Law** |
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College Physics for AP® Courses 2e | Uniform Circular Motion and Gravitation | Satellites and Kepler’s Laws: An Argument for Simplicity | m42144 | Derivation of Kepler’s Third Law for Circular Orbits | We shall derive Kepler’s third law, starting with Newton’s laws of motion and his universal law of gravitation. The point is to demonstrate that the force of gravity is the cause for Kepler’s laws (although we will only derive the third one). |
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College Physics for AP® Courses 2e | Uniform Circular Motion and Gravitation | Satellites and Kepler’s Laws: An Argument for Simplicity | m42144 | The Case for Simplicity | The development of the universal law of gravitation by Newton played a pivotal role in the history of ideas. While it is beyond the scope of this text to cover that history in any detail, we note some important points. The definition of planet set in 2006 by the International Astronomical Union (IAU) states that in the solar system, a planet is a celestial body that: |
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College Physics for AP® Courses 2e | Work, Energy, and Energy Resources | Connection for AP® Courses | m55016 | |||
College Physics for AP® Courses 2e | Work, Energy, and Energy Resources | Work: The Scientific Definition | m42146 | What It Means to Do Work | The scientific definition of work differs in some ways from its everyday meaning. Certain things we think of as hard work, such as writing an exam or carrying a heavy load on level ground, are not work as defined by a scientist. The scientific definition of work reveals its relationship to energy—whenever work is done, energy is transferred. |
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College Physics for AP® Courses 2e | Work, Energy, and Energy Resources | Work: The Scientific Definition | m42146 | Calculating Work | Work and energy have the same units. From the definition of work, we see that those units are force times distance. Thus, in SI units, work and energy are measured in newton-meters. A newton-meter is given the special name joule (J), and $1{\rule{0.25em}{0ex}}\text{J}=1{\rule{0.25em}{0ex}}\text{N}\cdot \text{m}=1{\rule{0.25em}{0ex}}\text{kg}\cdot {\text{m}}^{2}{\text{/s}}^{2}$. One joule is not a large amount of energy; it would lift a small 100-gram apple a distance of about 1 meter. |
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College Physics for AP® Courses 2e | Work, Energy, and Energy Resources | Kinetic Energy and the Work-Energy Theorem | m42147 | Work Transfers Energy | What happens to the work done on a system? Energy is transferred into the system, but in what form? Does it remain in the system or move on? The answers depend on the situation. For example, if the lawn mower in [link](a) is pushed just hard enough to keep it going at a constant speed, then energy put into the mower by the person is removed continuously by friction, and eventually leaves the system in the form of heat transfer. In contrast, work done on the briefcase by the person carrying it up stairs in [link](d) is stored in the briefcase-Earth system and can be recovered at any time, as shown in [link](e). In fact, the building of the pyramids in ancient Egypt is an example of storing energy in a system by doing work on the system. Some of the energy imparted to the stone blocks in lifting them during construction of the pyramids remains in the stone-Earth system and has the potential to do work. |
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College Physics for AP® Courses 2e | Work, Energy, and Energy Resources | Kinetic Energy and the Work-Energy Theorem | m42147 | Net Work and the Work-Energy Theorem | We know from the study of Newton’s laws in Dynamics: Force and Newton's Laws of Motion that net force causes acceleration. We will see in this section that work done by the net force gives a system energy of motion, and in the process we will also find an expression for the energy of motion. |
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College Physics for AP® Courses 2e | Work, Energy, and Energy Resources | Gravitational Potential Energy | m42148 | Work Done Against Gravity | Climbing stairs and lifting objects is work in both the scientific and everyday sense—it is work done against the gravitational force. When there is work, there is a transformation of energy. The work done against the gravitational force goes into an important form of stored energy that we will explore in this section. |
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College Physics for AP® Courses 2e | Work, Energy, and Energy Resources | Gravitational Potential Energy | m42148 | Converting Between Potential Energy and Kinetic Energy | Gravitational potential energy may be converted to other forms of energy, such as kinetic energy. If we release the mass, gravitational force will do an amount of work equal to $\text{mgh}$ on it, thereby increasing its kinetic energy by that same amount (by the work-energy theorem). We will find it more useful to consider just the conversion of ${\text{PE}}_{\text{g}}$ to $\text{KE}$ without explicitly considering the intermediate step of work. (See [link].) This shortcut makes it is easier to solve problems using energy (if possible) rather than explicitly using forces. |
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College Physics for AP® Courses 2e | Work, Energy, and Energy Resources | Gravitational Potential Energy | m42148 | Using Potential Energy to Simplify Calculations | The equation $\text{Δ}{\text{PE}}_{\text{g}}=\text{mgh}$ applies for any path that has a change in height of $h$, not just when the mass is lifted straight up, as long as $h$ is small compared to the radius of Earth. Note that, as we learned in Uniform Circular Motion and Gravitation, the force of Earth’s gravity does decrease with distance from Earth. The change is negligible for small changes in distance, but if you want to use potential energy in problems involving travel to the moon or even further, Newton’s universal law of gravity must be taken into account. This more complete treatment is beyond the scope of this text and is not necessary for the problems we consider here. (See [link].) It is much easier to calculate $\text{mgh}$ (a simple multiplication) than it is to calculate the work done along a complicated path. The idea of gravitational potential energy has the double advantage that it is very broadly applicable and it makes calculations easier. From now on, we will consider that any change in vertical position $h$ of a mass $m$ is accompanied by a change in gravitational potential energy $\text{mgh}$, and we will avoid the equivalent but more difficult task of calculating work done by or against the gravitational force. |
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College Physics for AP® Courses 2e | Work, Energy, and Energy Resources | Conservative Forces and Potential Energy | m42149 | Potential Energy and Conservative Forces | Work is done by a force, and some forces, such as weight, have special characteristics. A conservative force is one, like the gravitational force, for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. We can define a potential energy $\left(\text{PE}\right)$ for any conservative force, just as we did for the gravitational force. For example, when you wind up a toy, an egg timer, or an old-fashioned watch, you do work against its spring and store energy in it. (We treat these springs as ideal, in that we assume there is no friction and no production of thermal energy.) This stored energy is recoverable as work, and it is useful to think of it as potential energy contained in the spring. Indeed, the reason that the spring has this characteristic is that its force is *conservative*. That is, a conservative force results in stored or potential energy. Gravitational potential energy is one example, as is the energy stored in a spring. We will also see how conservative forces are related to the conservation of energy. |
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College Physics for AP® Courses 2e | Work, Energy, and Energy Resources | Conservative Forces and Potential Energy | m42149 | Potential Energy of a Spring | First, let us obtain an expression for the potential energy stored in a spring (${\text{PE}}_{s}$). We calculate the work done to stretch or compress a spring that obeys Hooke’s law. (Hooke’s law was examined in Elasticity: Stress and Strain, and states that the magnitude of force $F$ on the spring and the resulting deformation $\Delta L$ are proportional, $F=k\Delta L$.) (See [link].) For our spring, we will replace $\Delta L$ (the amount of deformation produced by a force $F$) by the distance $x$ that the spring is stretched or compressed along its length. So the force needed to stretch the spring has magnitude $\text{F = kx}$, where $k$ is the spring’s force constant. The force increases linearly from 0 at the start to $\text{kx}$ in the fully stretched position. The average force is $\text{kx}/2$. Thus the work done in stretching or compressing the spring is ${W}_{s}=\text{Fd}=\left(\frac{\text{kx}}{2}\right)x=\frac{1}{2}{\text{kx}}^{2}$. Alternatively, we noted in Kinetic Energy and the Work-Energy Theorem that the area under a graph of $F$ vs. $x$ is the work done by the force. In [link](c) we see that this area is also $\frac{1}{2}{\text{kx}}^{2}$. We therefore define the potential energy of a spring, ${\text{PE}}_{s}$, to be |
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