Units
When asked about height, responses can vary in units, such as 5 feet 8 inches, 68 inches, or 1.73 meters, all representing the same measurement This variation highlights the importance of clearly defining measurement units in scientific contexts, particularly in physics, where precision is crucial for experiments and predictions Misunderstandings regarding units can lead to significant practical issues, as evidenced by a past scientific experiment that failed due to different manufacturers using incompatible unit systems while assuming uniformity among them.
Nature can be fundamentally described using four essential quantities: length, mass, time, and electric charge All other physical quantities can be derived from these four core dimensions, highlighting their significance in understanding the natural world.
Length can be measured using the English system (inches, feet, miles) or the metric system (millimeters, centimeters, meters, kilometers), with the metric system being preferred by most scientists Smaller or larger lengths are often expressed as multiples of powers of 10 For reference, 1 inch is equivalent to 2.54 cm, and 1 foot is approximately 30 cm, while a meter is slightly more than a yard.
In this article, we will focus on the metric system for measuring mass, specifically using grams (g) and kilograms (kg) The prefix "kilo" indicates a factor of 1,000, meaning that one kilogram equals 1,000 grams, similar to how one kilometer equals 1,000 meters It's important to note that a kilogram weighs approximately 2.2 pounds; however, we must distinguish between mass and weight, as a kilogram measures mass while weight is a force, a concept we will explore further in Chapter 4.
Time, of course, is measured in seconds (s or sec), minutes (min), or hours (h) Most of the time we will use seconds.
Electric charge is quantified in coulombs, a unit named after Charles Coulomb, who formulated the fundamental force law governing interactions between electric charges Notably, one coulomb represents a significant quantity of charge, especially when compared to the charge of an electron or proton, which is approximately 1.6 × 10^−19 coulombs.
In this book, we focus on the MKS (meter-kilogram-second) system of units, which is the standard used by most physicists All other physical quantities can be expressed using these three fundamental units Common quantities will be presented in relation to these basic units, and we will denote dimensional analysis with the symbols [L], [M], and [T].
2 Questioning the Universe: Concepts in Physics Momentum: [M][L]/[T]
In the MKS system, quantities are defined with specific names, while the English system measures force in pounds and energy in foot-pounds.
To give some feeling for the relation between Newtons and pounds, 10 Newtons is very close to 2.2 pounds.
Powers of 10
The size of an atom varies depending on the specific element, with hydrogen being the smallest at approximately 0.5 angstroms (0.0000000005 meters) In contrast, the speed of light is a constant at 300,000,000 meters per second.
Expressing large and small numbers can be cumbersome; therefore, using powers of 10, also known as scientific notation, is much more convenient For instance, the size of a hydrogen atom is represented as 5.0 × 10⁻¹⁰ meters, while the speed of light is denoted as 3.0 × 10⁸ meters per second.
When expressing numbers in scientific notation, such as the size of a hydrogen atom, we move the decimal point to adjust its position, which can affect the value For instance, moving the decimal point ten places to the right increases the number by ten powers of ten, necessitating a correction by dividing by \(10^{10}\), which is equivalent to \(10^{-10}\) Similarly, for the speed of light, shifting the decimal point eight places to the left would incorrectly reduce its value by eight powers of ten, so we correct this by multiplying by \(10^8\).
In scientific notation, numbers are typically expressed as a value between 1.0 and 9.999 multiplied by a power of 10 For example, instead of writing 55 × 10², it is more conventional to write it as 5.5 × 10³ This format not only standardizes the expression of numbers but also indicates the precision of the value, reflecting the number of significant figures For instance, the number 35,500,000 is understood to have eight significant figures, while in scientific notation, it would be represented as 3.55 × 10⁷.
The number 3.550 × 10^7 indicates precision to four significant figures, while 3.5 × 10^7 reflects accuracy to three significant figures In this book, we will typically use two or three significant figures for clarity and simplicity.
Physicists explore the entire universe, requiring the use of very large or very small numbers To effectively work with these values, simple mathematical operations such as multiplication and division are often necessary We will first outline these operations in a general sense before providing a specific example for clarity.
For multiplication of two numbers, a × 10 m by b × 10 n , we simply multiply a and b and add the powers of 10 to get (ab) × 10 (n+m)
To give a specifi c example: (6 × 10 4 )*(3 × 10 6 ) = 18 × 10 10 = 1.8 × 10 11 For one more example: (6 × 10 4 )*(3 × 10 − 6 ) = 18 × 10 − 2 = 1.8 × 10 − 1 , which could also be expressed as 0.18.
For division, we divide a by b and subtract the powers of 10.
Using the same numbers as above, we get (6 × 10 4 )/(3 × 10 6 ) = 2 × 10 − 2 and
1 How many meters in a yard?
2 What is the mass, in kilograms, of a 10-pound weight?
3 The speed of light is 186,000 miles/s What is the speed of light expressed in meters/second?
4 A nanosecond is 10 − 9 seconds What distance does light travel in 1 nanosec- ond? Give your answer in both centimeters and feet.
5 Write the following numbers in scientifi c notation: a 1,000 b 1/10,000 c 2,500,000 d .000025 e 1
6 Calculate the following using and giving the answers in scientifi c notation: a .001 × 10 4 b (5 × 10 − 4 )/(2 × 10 − 3 ) c 10 6 /.01 d (3 × 10 35 ) × (7.5 × 10 − 3 ) e (24 × 10 − 5 )/(.8 × 10 2 )
A light year (LY) is defined as the distance that light travels in one year, with the speed of light denoted by the symbol c, which is approximately 3 × 10^8 meters per second or 186,000 miles per second To calculate the value of 1 LY in meters and miles, one must first establish the appropriate equation Additionally, considering that the sun is 93 million miles away from Earth, the time it takes for light to travel from the sun to Earth can be determined by writing the necessary equation before inserting any numerical values.
What Is Physics?
A discipline is defined by its subject matter and methodology In the case of physics, defining it often results in a list of topics However, to capture the essence of physics, a more meaningful definition is needed, despite potential disagreements on its interpretation.
Physics is the search for the basic laws that govern the universe around us
Throughout this book, we will frequently discuss the concept of assumptions, highlighting their significance It's crucial to recognize that we often accept these assumptions without questioning their validity, which can lead to misunderstandings By becoming aware of our assumptions, we can better evaluate their truthfulness and impact on our perspectives.
OK, here is my list (maybe you can fi nd even more):
1 We assume there are Laws to be found (We will talk more about “Laws” a little later).
2 We also assume that we humans are capable of fi nding them.
3 We assume universality This means that the laws we fi nd here on earth are true everywhere else in the universe.
The definition encompasses the entire "universe around us," which includes everything from the tiniest particles to the vastest celestial bodies Consider what you perceive as the smallest and largest objects in existence.
Methodology
The First Scientist
The first scientist, perhaps a caveman, observed the sun rising in the east and setting in the west daily, sparking his curiosity about this consistent phenomenon This observation marked the inception of scientific inquiry, as he began to question the reasons behind the sun's daily journey His initial theory posited that the sun revolved around the earth, providing a straightforward explanation for its movements Excited by his discovery, he eagerly shared his idea with a friend, illustrating the fundamental nature of scientific exploration and the importance of sharing knowledge.
A scientifically minded friend proposes a competing theory to the idea that the sun moves, suggesting instead that the earth spins on an axis from west to east, creating the illusion of the sun moving from east to west This theory would likely spark a debate, particularly regarding the objection that if the earth spins, people would be thrown off, a common argument against heliocentrism in Copernicus's time Ultimately, this objection prevailed for many in the sixteenth century Today, we evaluate which theory aligns with experimental facts and offers predictive power for new observations, a discussion we will revisit soon.
Why Do You Believe?
Many people confidently accept the heliocentric theory, which posits that the Earth revolves around the sun and rotates on its axis every 24 hours However, it's valuable to reflect on the reasons behind this belief Have you conducted personal observations or encountered experimental evidence that supports this view? If not, what influences your conviction? Exploring the foundations of your belief system can provide deeper insights into your understanding of astronomy.
Back to the Questions
We cannot ask any questions we desire; instead, we should focus on specific types that aid our search These are known as process questions, often framed as "how" questions rather than "why" questions While "why" questions suggest a final cause or intelligence, philosophers refer to them as teleological questions.
Not all questions that start with "why" are teleological; many are process-oriented For example, asking "Why is the sky blue?" can be reframed to inquire about the process behind the blue appearance of the sky In contrast, the question "Why was the universe created?" suggests a purposeful intent behind its existence, making it teleological However, asking "How did the universe begin?" is a valid scientific inquiry, which falls under the field of cosmology.
How Do We Answer the Questions?
Science fundamentally relies on observation, which is the essence of its empirical nature It begins and concludes with experimental data, as illustrated by our caveman story where the observation of the sun's cycles led to a theoretical understanding When multiple theories exist, the one that gains acceptance is determined by experimental observations Even a single proposed theory must undergo rigorous testing; if it fails any experimental test, it cannot be deemed entirely accurate, regardless of its initial plausibility.
We have used the word theory several times already without really defi ning it, even though most of you probably have a reasonable idea of what the word means
The experimental method in natural sciences is distinguished by the ability to control a limited number of variables, allowing researchers to change one variable at a time while keeping others constant This control is essential for achieving consistent results across multiple trials In contrast, the social sciences involve complex human behaviors, where individuals may respond differently to the same situation, even at different times While some social scientists may dispute this perspective, it is widely acknowledged that the reliability of the scientific method in natural sciences stems from its capacity to manage variables effectively.
While errors are inevitable in scientific research, independent analysis plays a crucial role in identifying mistakes Scientists, being human, are not infallible, and they often approach new results with skepticism, questioning potential errors This self-skepticism is particularly important, as researchers prefer to uncover their own mistakes before sharing results publicly, rather than having others point them out.
8 Questioning the Universe: Concepts in Physics
The Need to Be Quantitative
Experiments are inherently quantitative, requiring measurements expressed in numerical values such as time, length, or mass This necessitates framing theories and predictions in terms of measurable quantities For instance, the question “Why is the sky blue?” lacks scientific rigor because "blue" is not a measurable quantity, especially considering variations in color perception, such as color blindness A more scientifically valid question would be, “What causes the light from the sky to have maximum intensity at short wavelengths?” This emphasizes the importance of using precise language to ensure clarity and avoid ambiguity in scientific discussions.
Mathematics plays a crucial role in the language of physics, serving as an essential tool for physicists despite not being a science itself While its subject matter and methodology differ from scientific disciplines, the ability to use mathematics is vital for every working physicist Theories in physics are often articulated through mathematical equations, which efficiently convey the laws of nature and facilitate the integration of various theories to derive new insights Throughout this book, we will focus on understanding a variety of equations, rather than performing extensive mathematical manipulations, to explore their implications about the natural world.
Theories
In this article, we will delve deeper into the concept of theories, building on your existing understanding Prepare to be surprised by new insights and information that will enhance your knowledge of this intriguing subject.
A theory is fundamentally an educated guess that connects and explains the relationships between different phenomena For example, early humans proposed various theories to understand the sun's rising and setting, with one suggesting it orbited the Earth, while another theorized that the Earth spins on its axis Both theories offered explanations for the same observations, prompting the question of how to determine which theory to accept.
A theory must be testable, meaning it should generate new predictions that can be empirically evaluated Philosopher Karl Popper emphasized that every theory must be falsifiable, indicating it should have the potential to be proven false if its predictions do not hold true This criterion is essential for distinguishing between valid theories and those that lack scientific credibility Popper's perspective on science highlights the importance of rigorous testing and the ability to refute theories, ensuring that only those that withstand scrutiny are accepted.
Science advances not merely through observation and verification, but through bold conjectures that extend beyond existing data These theories are rigorously tested and refined through falsification and critical experimentation The deliberate pursuit of falsifications, alongside the resilience of theories against them, is essential for the progression of science and the development of objective knowledge.
And along the same line:
Science is not merely a method of proving facts; rather, it represents a systematic approach to debate and inquiry Its strength lies in the recognition that knowledge is provisional and subject to change, as it is built on hypotheses that can be tested and potentially disproven.
In science, no theory can be definitively proven, as it may fail a new test at any time, regardless of its successful past experiments While the scientific community may accept a theory as highly likely to be correct after consistent testing, the inherent uncertainty remains that it cannot be conclusively proven.
The distinction between a theory and a law primarily lies in historical context rather than fundamental differences For example, while Newton's law of gravity has been foundational, it is not entirely accurate, as general relativity provides a more comprehensive understanding of gravitational forces.
Einstein's special theory of relativity is widely regarded as correct, having successfully passed all experimental tests to date A cynical viewpoint suggests that scientific concepts proposed before 1850 are labeled as laws, while those introduced afterward are termed theories Nonetheless, our definition of physics includes the term "law" to describe the fundamental laws of nature that we aim to uncover.
Models
Models play a crucial role in theory development, with some theories relying heavily on models while others do not require them at all A well-known example is the planetary model of the atom, which illustrates the structure of atoms effectively There are various types of models, each contributing to a clearer understanding of the physical systems we aim to analyze.
10 Questioning the Universe: Concepts in Physics
1 The analogy: This is probably the most common type It is a picture we have in our mind to help us a visualize something we cannot directly per- ceive The planetary atom is just such a model Here we think of the atom resembling our solar system, with the nucleus like the sun and the electrons rotating in their orbits like the planets We will discuss the history of the different atomic models in more detail in Chapter 9.
2 The systemic model: This allows us to explain properties of a specifi c sys- tem in terms of the properties of a more generalized system For example, is light (a specifi c system) a wave or particle phenomenon (generalized sys- tems)? Since waves and particles have properties that are quite different, identifying light with one of them implies certain properties for light.
3 The mathematical model (do not worry, we will not do much with this):
Mathematical equations can often bridge different areas of physics, offering valuable insights into new theories For instance, the equation governing the motion of a mass on a spring is identical to the equation that describes electrical current oscillations in radio and television tuners This similarity highlights the interconnectedness of physical concepts and the potential for cross-disciplinary understanding.
Aesthetic Judgments
In the realm of scientific inquiry, a theory is deemed valid when it successfully endures experimental scrutiny However, in practice, theories are also evaluated through subjective aesthetic criteria, where qualities like simplicity, elegance, and beauty play a significant role in distinguishing one theory from another.
In his book *Thematic Origins of Scientific Thought*, physicist and historian Gerald Holton discusses the concept of "themata," which refers to the inherent prejudices or presuppositions that scientists hold about the nature of reality A notable example is Einstein's assertion that "God does not play dice with the universe," illustrating his belief in a deterministic universe, despite the probabilistic nature of quantum mechanics Although Einstein's viewpoint lacked empirical backing at the time, contemporary evidence increasingly aligns with the quantum interpretations he found troubling.
In Holton's book, two notable examples highlight the judgments of Nobel Prize-winning physicists regarding each other's theories In 1926, Heisenberg expressed his disdain for Schrödinger's theory, stating, "The more I ponder the physical part of Schrödinger’s theory, the more disgusting it appears to me." Conversely, Schrödinger conveyed his apprehension towards Heisenberg's approach, admitting, "I was frightened away by it, if not repelled."
Interestingly, the previously mentioned quotes reveal that the theories in question are fundamentally identical, merely articulated through different mathematical representations While terms like "disgusting" and "repelled" lack scientific rigor, such subjective aesthetic evaluations are frequently employed by scientists in their work.
End-of-Chapter Guide to Key Ideas
What is the defi nition of physics, and what assumptions are associated with
What is the basic methodology of science?
What types of questions do physicists ask?
How do aesthetic judgments enter into physics?
1 What do you think are the smallest objects that exist? Answer the same question for the largest objects that exist.
2 Formulate at least three questions you have about the physical (as distinct from biological) universe around you.
3 What is the basic methodology of science?
4 List three scientifi c questions that could begin with why.
5 Is the question “Why is the sky blue?” a good scientifi c question? Explain your answer.
6 What is the important characteristic of experiments in the natural sciences?
7 Which of the following can be considered quantitative scientifi c terms: (a) short, (b) hot, (c) temperature, (d) green, (e) frequency?
11 A criticism of the theory of evolution is that it is only a theory and has not been proven Discuss the validity of this criticism.
12 What does Karl Popper mean when he says that a theory must be falsifi able?
13 What can you say about a theory that has been falsifi ed?
15 Is Newton’s law of gravity correct? Discuss your answer.
16 Is Einstein’s theory of relativity correct? Discuss your answer.
17 Should there be a place for aesthetic judgments in science? Discuss your answer.
Motion is an essential topic in physics because it is a familiar experience that we encounter daily Historically, ancient Greeks were among the first to study motion, distinguishing between the perfect, eternal motion of celestial bodies and the imperfect motion of earthly objects In our modern understanding, we recognize that nearly everything is in constant motion; for instance, the Earth revolves around the sun annually and rotates on its axis daily Additionally, the air is perpetually in motion, whether in organized patterns like wind or in random movement, even on calm days The concept of motion is also relative, as demonstrated by the works of Galileo and Einstein, which will be explored further in later chapters on relativity theory.
Understanding the nature of forces is crucial, as it serves as a key to comprehending the universe, which consists of four fundamental constructs: matter, forces, space, and time While matter is essential for existence, forces enable interaction among matter, highlighting their significance in the cosmic framework.
To comprehend the nature of forces and their measurement, we must first grasp the concept of motion and its effects Objects in motion traverse space and time, which we initially consider to be absolute, meaning the distance between two points remains consistent for all observers, and the time interval for an event is uniform across different perspectives Although this assumption appears straightforward, we will later discover in the chapter on relativity that it is, in fact, flawed For the moment, we can set aside these complexities as we explore the fundamentals of motion.
The term "motion" can be ambiguous, as we often recognize when we are in motion However, to grasp its scientific meaning, it is essential to clearly define the variables that characterize motion Consider exploring these definitions for yourself before continuing.
To accurately define motion, we need to consider several key variables: position, distance, speed, velocity, acceleration, and time It's important to distinguish between these terms; for example, position refers to a specific point in space, whereas distance measures the difference between two positions If we denote two positions as x1 and x2, the distance (d) between them can be calculated using the formula d = (x2 − x1).
Also, velocity and speed are sometimes used interchangeably, but they have different meanings Speed is a measure of only how fast you are moving, while
Velocity is a vector quantity characterized by both speed and direction, distinguishing it from scalar quantities like speed, which only have magnitude While speed represents the magnitude of the velocity vector, understanding the full vector nature is often crucial In notation, the velocity vector is denoted in bold as **v**, while speed is represented as *v* Vectors can be visually represented as arrows, where the arrow's length indicates magnitude and its direction shows the vector's orientation.
Relating the Variables of Motion
Distance is defined as the difference between two positions, while speed quantifies how quickly that position changes over time The formula for speed is given by v = (x2 - x1) / (t2 - t1), where x2 represents the position at time t2 and x1 is the position at time t1 This can also be expressed using the symbols Δx for change in position and Δt for change in time, resulting in the equation v = Δx / Δt, where Δx = x2 - x1 and Δt = t2 - t1.
The equation v = d/T, where d represents distance and T represents time, is a familiar concept often used in everyday situations, such as estimating travel time When planning a trip, you likely calculate how long it will take to reach your destination by considering the distance and your speed This common practice involves rearranging the equation to T = d/v to find the time needed for travel.
The average speed, denoted as v, is calculated over the time interval (t2 - t1) Whether the speed remains constant or varies during this period, the average speed remains consistent, as illustrated by the two plots of position (x) versus time (t) in Figure 3.1.
Acceleration measures the rate at which velocity changes over time, making it a second-order derivative of position This concept can be challenging to intuitively understand since it involves the change of a change, linking it closely to the dynamics of motion.
Acceleration is defined as a = (v2 − v1)/(t2 − t1) = Δv/Δt, indicating that it is a vector quantity In one-dimensional motion, acceleration is solely dependent on the change in speed However, in two or three-dimensional motion, both changes in speed and direction must be considered For simplicity, we will focus on one-dimensional motion, which provides a foundational understanding of acceleration.
Graphs of One-Dimensional Motion
Constant Speed
The four graphs below illustrate the motion of an object moving at a constant speed, with time as the independent variable in each case Before diving into the accompanying text, take a moment to analyze the graphs and determine the object's behavior over time Understanding how motion changes with time is key to interpreting these representations effectively.
FIGURE 3.1 Graph of position as a function of time for two different types of motion
In both straight-line and curved motion, the average speed remains constant While the instantaneous speed in straight-line motion is consistent at every moment, the instantaneous speed in curved motion varies over time However, despite these fluctuations, the average speed for both types of motion is identical, as the total change in position over the same time interval is equal.
16 Questioning the Universe: Concepts in Physics
OK, let us see how you did.
Graph 1: We see the position, which is plotted on the vertical axis (the depen- dent variable), is not changing The object is not moving, so it has zero speed.
Graph 2 mirrors Graph 1 but illustrates a scenario where the dependent variable, speed, remains constant, indicating an object with a steady positive speed To differentiate between directions, we categorize speeds as positive or negative; for instance, objects moving to the right are labeled as positive, while those moving to the left are considered negative This classification is arbitrary, as long as one direction is defined as positive, with the opposite direction being negative To represent a constant negative speed on a graph, one would draw a horizontal line below the time axis.
In Graph 3, the position serves as the dependent variable, which shows a consistent increase over time, indicating that the object is moving at a constant speed This constancy in speed can be confirmed by analyzing the slope of the graph, defined as the rise over run, where the rise represents the change in the y variable (position) and the run signifies the change in the x variable (time).
FIGURE 3.2 Graphs depicting motion with constant speed. rise/run = Δx/Δt
The slope of a position versus time graph represents speed, and in this case, the slope is both constant and positive, indicating uniform motion While this graph provides valuable information about the object's position at any given time, another graph depicting the same motion may only convey speed, lacking details about the object's position.
Graph 4: Since the line has a constant negative slope, the object has a constant negative speed.
Constant Acceleration
The following four graphs illustrate the motion of an object experiencing constant acceleration, featuring both positive and negative values Analyze these graphs to determine the object's motion characteristics.
FIGURE 3.3 Graphs depicting motion with constant acceleration.
18 Questioning the Universe: Concepts in Physics
Graph 1 illustrates a scenario of constant speed, indicating that the acceleration is zero It is essential to differentiate zero acceleration from non-zero acceleration, as zero is a special case The line in the graph is positioned above the origin, signifying that the object is moving with a positive speed to the right Conversely, if the object were moving to the left with negative speed, the line would appear below the origin Regardless of the direction, the acceleration remains zero in both situations.
In Graph 2, acceleration serves as the dependent variable, consistently maintaining a positive value If the line were positioned below the origin, it would indicate negative acceleration The upcoming graphs will clarify the concepts of positive and negative acceleration It is crucial to understand that negative acceleration does not inherently imply deceleration.
In Graph 3, two parallel lines represent identical acceleration This is determined by analyzing the slope, calculated as rise over run, or Δv/Δt, similar to our approach for constant speed in previous graphs.
In our earlier definition, acceleration is represented as a = Δv/Δt, indicating that the slope of the velocity (v) versus time (t) graph reflects acceleration Since both lines on the graph are parallel, they share the same slope and thus the same acceleration However, the distinction lies in their positions: the upper line consistently remains above the origin, indicating a positive speed and movement in a single direction Conversely, the lower line begins with a negative speed, suggesting movement to the left, and while its speed is negative, it gradually decreases until it intersects the time axis.
At zero speed, acceleration begins to increase in the positive direction It's important to note that acceleration is considered positive even when speed decreases, as positive and negative refer to direction rather than the rate of speed change In physics, the term "deceleration" is rarely used; instead, acceleration encompasses both increases and decreases in speed, as illustrated in this example Further insights will be discussed in relation to graph 4.
The graph illustrates a negative slope, indicating negative acceleration, which can represent the motion of a ball thrown straight upward Initially, the ball has a positive speed at its maximum value as it ascends When defined with upward as the positive direction, the ball leaves the hand with peak speed, which decreases until it reaches the highest point, where the speed is zero Subsequently, the ball descends with increasing negative speed Throughout the motion, the acceleration remains constant and negative, evident from the graph's consistent negative slope It's important to differentiate between speed and acceleration; while the speed is zero at the peak, the change in speed (acceleration) is not This concept will be further explored in the next chapter, where we will discuss the forces acting on the object in motion.
Two-Dimensional Motion
In our discussion, we explored the complexities of motion, particularly how an object's velocity can change direction in multiple dimensions, unlike one-dimensional motion where direction is limited In our daily experiences, such as walking or driving, we intuitively understand that our velocity alters as we navigate corners or curves It's important to differentiate between velocity and speed; in one-dimensional motion, a change in direction requires a change in speed, as demonstrated in previous graphs However, in two-dimensional motion, it is possible to maintain a constant speed while moving in a circular path.
In circular motion, while the speed remains constant, the velocity is constantly changing due to the continuous change in direction Focusing on two-dimensional circular motion provides a comprehensive understanding of the principles involved, as introducing three-dimensional motion complicates the analysis without adding significant value.
The diagram illustrates a car moving in a circular path at a constant speed, yet its velocity is not constant due to the changing direction of the velocity vectors at different points around the circle Each arrow in the diagram is of equal length, signifying that the speed remains constant, while the varying directions of the arrows indicate that the velocity vectors differ According to the definition of acceleration, which is the change in velocity over time (a = Δv/Δt), the car experiences accelerated motion despite maintaining a consistent speed.
FIGURE 3.4 Velocity vectors for an object moving in a circle.
20 Questioning the Universe: Concepts in Physics
Let us discuss both the direction and the magnitude of the acceleration in this case of circular motion Neither is obvious from our defi nition of a as given above.
The acceleration vector always points inward toward the center of the circle at any given point along its path For instance, envision a car positioned at the tail of an arrow at the top of the circle; if it were to move in a straight line, it would reach the tip of the arrow shortly However, since the car is following the circular path, it moves inward, closer to the center of the circle, rather than continuing straight.
Centripetal acceleration occurs when a car accelerates inward, as demonstrated by a rigorous mathematical analysis showing that the Δv vector points toward the center of the circle In contrast, centrifugal acceleration points outward, away from the center.
The acceleration magnitude can be expressed as a = v²/R, where v represents speed and R denotes the circle's radius Although this formula differs from the conventional Δv/Δt, analyzing the velocity vectors at the top and bottom of the circle reveals the reasoning behind the inclusion of v squared and the division by the radius in the acceleration formula.
In analyzing the motion of a car moving in a circular path, we define the change in velocity (Δv) as the difference between the velocity at the top of the circle (v2) and the velocity at the bottom (v1), resulting in Δv = 2v The time interval (Δt) required for the car to travel from the bottom to the top of the circle, which is half the circumference, can be calculated using the formula T = d/v Given that the circumference of a circle is 2πR, we find Δt = πR/v Consequently, the acceleration can be expressed as Δv/Δt = (2/π)(v²/R), highlighting the relationship between velocity, radius, and time in circular motion.
The expression (v²/R) illustrates that one factor of velocity (v) arises from the change in the velocity vector, while the second factor relates to the reduced time interval required for a car to traverse between points on a circular path as its speed increases.
The previous argument failed to provide an accurate answer because it calculated the average acceleration over a half-circle, rather than considering the instantaneous acceleration at each point in the motion Since the velocity vector changes direction continuously, it's essential to analyze the change in velocity over a very small segment of the motion This can be achieved through calculus or by examining the motion in smaller intervals rather than over a larger arc of the circle.
At this point we have fi nished our discussion of motion We are ready to go on to consider our next topic: Forces.
End-of-Chapter Guide to Key Ideas
Why begin our fi rst discussion of physics with ideas about motion?
What are the variables needed to describe motion?
How are these variables related?
What is the difference between a scalar and a vector quantity?
Can you read the different graphs of motion and tell what type of motion
Each graph illustrates the motion of a car, providing insights into its position, speed, and acceleration The position graph indicates the car's location over time, while the speed graph reveals how quickly the car is moving at any given moment Additionally, the acceleration graph shows how the car's speed changes, highlighting periods of speeding up or slowing down Together, these graphs offer a comprehensive understanding of the car's motion dynamics.
Why is circular motion accelerated motion?
In circular motion, what determines the acceleration, and which way does
Why does the centripetal acceleration depend on v
1 Why have we begun our discussion of physics with the idea of motion?
2 How are position and distance related?
3 What is the defi ning equation that our old friend d = vT is based on?
4 Defi ne vector Give at least three examples of vector quantities.
5 What is the relation between speed and velocity?
6 Defi ne, both in your own words and by an equation: speed and acceleration.
7 If A has a greater speed than B, does that mean that A has a greater acceleration? Justify your answer in writing and also by a graph.
8 Draw three different graphs (different vertical axes) depicting the motion of an object with (a) positive constant acceleration and (b) negative con- stant acceleration.
To analyze the motion of objects, we can create various graphs: a graph depicting the position of an object moving away from you at a constant speed will show a straight line with a positive slope; conversely, a graph for an object approaching you at a constant speed will present a straight line with a negative slope For an object moving away with constant positive acceleration, the position graph will curve upwards, indicating increasing distance over time In terms of speed, an object moving away with constant positive acceleration will have a linear speed graph that rises steadily When observing an object dropped from a building, its speed graph will increase linearly due to gravitational acceleration, while the acceleration graph will remain constant Finally, the acceleration of an object thrown upward from the ground will show a constant negative value, reflecting the effects of gravity acting against the motion.
The article discusses graphs illustrating the motion of an object along a straight line, categorized into three rows: position (x) versus time, speed versus time, and acceleration versus time It emphasizes that the x-axis intercept is zero on the y-axis For the analysis, it asks for the identification of letters corresponding to graphs that depict nonzero constant acceleration and those that represent nonzero constant speed.
22 Questioning the Universe: Concepts in Physics c Which graphs could represent the same motion? Of the ones you have chosen, which gives the most information about the motion? x v a a d g b e h c f i
11 Can the velocity of an object be zero at the same time that its acceleration is not zero? Explain and give an example.
12 Can you assume a car is not accelerating if its speedometer shows a steady reading of 40 mph? Explain.
13 Does the odometer of a car measure a vector or scalar quantity? What about the speedometer?
14 If you go around a curve at constant speed, is this accelerated or nonacceler- ated motion? If accelerated, what is the direction of the acceleration vector?
15 Will the acceleration be the same when a car rounds a sharp curve at 50 mph as it is if it rounds a gentle curve at the same speed? Explain.
16 What is the average speed of a sprinter who runs the 100-yard dash in 9.8 s? What would be the time for 1,500 m at this pace?
17 At an average speed of 11.8 km/h, how far will a car travel in 175 min?
18 Two cars go around the same curve Car A has a speed of 30 mph, while car
B has a speed of 60 mph Which has the greater acceleration, and what is the ratio of their accelerations?
A boat traveling in still water at a speed of 10 mph takes 24 minutes to cover a distance of 4 miles When navigating downstream in a river flowing at 4 mph, the boat's effective speed increases to 14 mph, allowing it to travel 2 miles in approximately 8.57 minutes Conversely, heading upstream against the current, the boat's effective speed decreases to 6 mph, resulting in a travel time of 20 minutes for the same distance The total time for a round trip of 4 miles, consisting of both downstream and upstream journeys, is about 29.14 minutes Given the initial time calculated for the 4-mile distance in still water, the differences in travel times due to the current should not be surprising, as they highlight the significant impact of water flow on boating speed.
The Fundamental Forces
A force can be defined as a push or pull, but a more comprehensive understanding is that it represents an interaction between two or more objects In our discussion, we will primarily focus on interactions involving two objects.
Forces play a significant role in our daily interactions with the world around us To enhance your understanding, I encourage you to create your own list of forces before reviewing mine This exercise may lead you to discover additional forces that I haven't included.
Gravity Electric Weak nuclear Strong nuclear Centrifugal
Friction Wind force Contact force (between surfaces) Muscular force Chemical Atomic
The arrangement of forces in my list highlights their categorization, beginning with the distinction between centrifugal and centripetal forces, which describe the outward and inward actions during curved motion, respectively Gravity, a fundamental force, anchors us to Earth and maintains our connection to the sun The second column groups various familiar forces under the electromagnetic force, illustrating that electric and magnetic forces are simply different expressions of this single force The stability of atoms arises from the electrical attraction between negatively charged electrons and positively charged protons, while interactions between different atoms are governed by similar attractions and repulsions Thus, all forces in this column stem from the electromagnetic force, including those related to muscle and air atoms.
24 Questioning the Universe: Concepts in Physics
The strong and weak nuclear forces operate at subnuclear scales, influencing our lives indirectly by holding atomic nuclei together and facilitating certain radioactive decays These fundamental forces are crucial for the stability of atoms and play a vital role in powering the sun, which serves as the primary energy source for life on Earth.
The four fundamental forces of nature—gravity, electromagnetic, strong nuclear, and weak nuclear—are believed to have originated from a single primordial force at the moment of the Big Bang As the universe expanded and cooled, this unified force evolved into the distinct forces we recognize today This concept of unification in physics highlights that seemingly different phenomena are often interconnected and represent various forms of the same underlying principle, a notion that extends to the fundamental building blocks of matter, known as elementary particles.
Understanding how a single force can manifest in different forms is essential in physics For instance, the force between atoms can lead to various effects, such as friction, wind, and muscular forces Additionally, the relationship between electric and magnetic forces illustrates that they are both expressions of the unified electromagnetic force A helpful analogy is water, which, at high temperatures, exists solely as steam, demonstrating how one substance can take on different states based on external conditions.
As temperatures decrease, water transitions into distinct states: gaseous and liquid, each with unique characteristics With further cooling, water can also become solid, resulting in three different forms—gas, liquid, and ice—while remaining the same element Similarly, as the universe cools, the primordial force manifests in various forms.
Scientists have made progress in understanding the relationship between the electromagnetic force and the weak nuclear force, recognizing them as different manifestations of a single electroweak force This insight may aid in the ongoing quest to unify all fundamental forces, a pursuit that dates back to Einstein's attempts to connect electromagnetism and gravity through his unified field theory However, gravity remains the most challenging force to integrate with the others Some theoretical physicists propose that string theory, which posits that fundamental particles are tiny vibrating strings in an eleven-dimensional universe, could provide a pathway to this unification.
Figure 4.1 depicts the historical journey of force unification and its anticipated future It highlights Newton's groundbreaking achievement in demonstrating that celestial and terrestrial gravity are fundamentally the same.
Electromagnetism Light Planetary Motion Gravity NEWTON Weakest General Relativity Space Time
Maxwell Electromagnetic Theory Earthly Motion
Weak Force Strong Force Strongest
Fundamental Forces Today FI G U R E 4 1 ( S e e colo r i n se r t f ol lo w in g pa g e 1 0 2 ) T h e u n ifi cat io n o f f o rces
In his groundbreaking work, Isaac Newton demonstrated that the same gravitational force keeping the moon in orbit also causes an apple to fall to the Earth, revolutionizing our understanding of gravity in the seventeenth century Over 230 years later, Albert Einstein further transformed this concept with his theory of general relativity, which unified space and time into the single entity of "space-time." In this framework, gravity results from mass warping space-time, offering a fundamentally different perspective on the nature of gravitational forces.
Maxwell unified electricity, magnetism, and light, as will be explored in the upcoming chapters This unification extends to the electromagnetic force and weak nuclear force, collectively known as the electroweak force The pursuit of merging the strong nuclear force with the electroweak force is referred to as grand unified theory (GUT) Furthermore, the ultimate goal of integrating gravity with these forces is termed the theory of everything (TOE), with some physicists aspiring for string theory to fulfill this role.
Our comprehension of forces extends far beyond mere push or pull descriptions, offering insight into the intricate organization of nature.
A Specifi c Force Law: Newtonian Gravity
Weight
Weight is defined as the force exerted on an object due to Earth's gravitational attraction According to the universal law of gravity, we can express the gravitational force between Earth, with mass denoted as Me, and an object with mass m located near the Earth's surface To accurately calculate this force, we must consider the distance d, which is influenced by the varying proximity of different parts of Earth's mass to the object m When averaging these effects, it is determined that Earth's mass can be treated as if it is concentrated at its center Therefore, we use the Earth's radius, represented as Re, as the distance in our weight calculation.
We will get back to this equation very shortly to discover some very interesting consequences.
How Does Force Affect Motion? Newton’s Second Law
Chapter 3 emphasizes that understanding forces is best achieved by examining their impact on motion, which is articulated through Newton’s second law of motion This fundamental law outlines the relationship between force and motion, providing a framework for analyzing how forces influence the movement of objects.
FIGURE 4.2 Force of gravity as a function of the distance between two masses The farther the masses are apart, the weaker is their gravitational attraction.
28 Questioning the Universe: Concepts in Physics
The relationship expressed in the equation may appear straightforward, but it can be misleading Typically, equations suggest that the variables on the right side influence the variable on the left For example, in Equation 4.2, an object's weight is determined by its mass and other factors However, Equation 4.3 incorrectly implies that acceleration causes force, when in fact, it is the application of force that results in acceleration To accurately represent this concept, Newton’s second law should be expressed as a = F/m (4.4).
The concept of force (F) can often be misunderstood, particularly when considering everyday situations For example, while sitting in a chair, one might assume that gravity is the only force at play, leading to the conclusion that one should be accelerating downward However, this is misleading, as the symbol F represents the sum of all forces acting on an object In this case, two forces are at work: the downward pull of gravity and the upward force exerted by the chair These forces counterbalance each other, resulting in no net acceleration, which explains why you remain stationary while seated.
So, the sum of these forces (sometimes we call the sum of all forces the net force) is zero Equation 4.4 then predicts that the acceleration should indeed be zero.
The symbol F represents the sum of all external forces acting on an object, highlighting that only these external forces can influence its motion Internal forces within an object do not affect its motion, emphasizing the distinction between internal and external influences For instance, if three different external forces act on an object, they can be combined to determine the overall effect on the object's motion.
The total force acting on an object, denoted as F, is the sum of all external forces, represented by the equation F = (F1)ext + (F2)ext + (F3)ext Here, the subscript "ext" emphasizes that we are only considering forces that originate from outside the object itself.
In Chapter 3, we discussed the concept of constant acceleration for a ball thrown into the air, highlighting that the acceleration remains unchanged throughout its motion This was supported by the observation that the velocity versus time graph forms a straight line, indicating a constant slope Since the slope represents acceleration, it confirms that acceleration is indeed constant However, I recognize that this graphical-mathematical explanation may not have been convincing for everyone.
According to Newton's second law, when the net force on an object remains constant, the acceleration must also be constant In the case of a ball in the air, the only force acting on it is the gravitational pull of the Earth, as we can disregard air resistance Within approximately 100 miles above the Earth's surface, this gravitational force is consistent and directed downward, resulting in a constant downward acceleration for the ball.
Newton, the Apple, and the Moon
Isaac Newton is credited with the first major unification in physics by recognizing that there is only one type of gravity, which applies both to terrestrial objects, like a falling apple, and celestial bodies, such as the moon in its orbit This groundbreaking realization—that the same gravitational force acts on both the apple and the moon—represents a significant moment in history, illustrating the creative genius of linking everyday phenomena to the cosmos.
Newton's discovery that gravitational force decreases with the square of the distance (1/d²) was pivotal in understanding celestial motion He recognized that the Earth's gravitational pull causes both the apple and the moon to accelerate, with the moon experiencing centripetal acceleration due to its orbit By calculating this acceleration and comparing it to the acceleration of objects near the Earth, as measured by Galileo and others, Newton demonstrated that the ratio of these accelerations corresponds to the square of the ratio of the Earth's radius to the distance to the moon This finding confirmed that gravitational force diminishes with distance, reinforcing the 1/d² relationship.