Physics is at the heart of our understanding of physical reality
Physics is the fundamental science that describes the physical reality, encompassing everything from the infinitesimal quarks and leptons that form protons and neutrons to the vast structures of stars, galaxies, and the universe itself It serves as the foundation for other sciences, particularly chemistry and biology, but emergent properties in complex systems indicate that physics alone cannot fully explain chemical and biological phenomena The study of physics is organized into various scales, including the subatomic scale, which deals with elementary particles like protons and neutrons (approximately 1 femtometer), and the atomic and molecular scale, which pertains to atoms and molecules (around 1 nanometer).
Nanotechnology operates at the scale of the smallest human-engineered structures, typically ranging from 1 to 100 nanometers, and overlaps with the atomic and molecular scales In contrast, the human scale reflects everyday dimensions, with an average height of 1 to 2 meters The astronomical scale encompasses celestial bodies, measuring from megameters to zettameters, while the cosmological scale represents the universe's largest structures, typically around 10^26 meters Understanding the behavior of small-scale entities like atoms and particles can illuminate the functioning of larger systems Remarkably, insights gained from studying galaxies and the universe's evolution also enhance our understanding of elementary particles and their interactions A significant development in modern physics is the collaboration between cosmology and particle physics, driven by data from various space agencies and research institutions.
The ultimate goal of physics is to establish a comprehensive theory, often referred to as a Theory of Everything (TOE), that explains the universe in its entirety, although we have yet to achieve this Reductionism posits that understanding the interactions of elementary particles—specifically, up quarks, down quarks, and electrons—will lead to a complete understanding of the universe These quarks form protons and neutrons, the nucleus's building blocks, while electrons orbit around them Chemistry reveals how atoms combine to create molecules, and physics provides the underlying principles governing these processes As cell biologists and microbiologists uncover the molecular mechanisms of cellular function, we see that molecules assemble into cells, which in turn form organisms, suggesting that life emerges from fundamental particles Emergent properties arise from complex systems, exemplified by crystal structures in physical systems and life in biological systems Although physics serves as the foundational science, comprehending these emergent properties necessitates insights from other disciplines, including social sciences and history This course will utilize mathematics and demonstrations to explore the core principles of physics, which elucidate both natural and technological phenomena, ultimately forming the groundwork for our understanding of the universe The curriculum encompasses classical mechanics, waves and fluids, thermodynamics, electromagnetism, optics, and modern physics, structured into six sections for comprehensive learning.
0 because it is an introduction to physics—consists of the ¿ rst
Cosmology explores the universe's structure and evolution, while particle physics investigates the fundamental constituents of nature An emergent property refers to a higher-level characteristic that arises from micro-level interactions within complex systems Among the elementary particles are quarks, which are essential components that combine to form protons and neutrons There are six types of quarks: up, down, charm, strange, top, and bottom, each with a fractional charge.
Lecture 1 explores the concept of reductionism, which posits that complex systems can be comprehended by examining their individual components and their interactions Additionally, it introduces the theory of everything (TOE), a hypothetical framework aimed at unifying all established branches of physics, such as classical mechanics, relativity, and quantum theory.
Rex and Wolfson, Essential College Physics (ECP), chap 1.
Wolfson, Essential University Physics (EUP), chap 1.
At the close of the 19th century, many scientists believed that physics had nearly achieved a complete understanding of the fundamental principles of physical reality, assuming future work would only involve exploring finer details and applications However, this perspective was profoundly misguided Today, physicists are on a quest for a "theory of everything" that aims to explain the entire universe through a single interaction This raises the question: would such a theory signify the culmination of physics as a grand pursuit for understanding, reducing it to the simplistic exploration of details akin to 19th-century expectations?
The principles of physics fundamentally underpin the sciences of chemistry and biology, as they govern the behavior of matter and energy at all levels Chemistry relies on physical laws to explain chemical reactions and interactions, while biology incorporates these principles to understand processes such as metabolism and cellular function However, both chemistry and biology also possess their own unique principles and theories that operate independently of physics, allowing them to explore complex systems and phenomena specific to their fields Ultimately, while physics provides a foundational framework, the distinct principles of chemistry and biology are essential for a comprehensive understanding of their respective domains.
Physics is mathematical, but its verbal language is just as important
Physicists refine everyday language to convey precise meanings and new concepts, establishing a framework for understanding complex physical phenomena The mathematical aspect of physics succinctly articulates principles that would be cumbersome in natural language, capturing both numerical values and the relationships between physical quantities Before exploring physics itself, it is essential to grasp the specific language and mathematical expressions that describe physical reality Scientific terminology often differs from everyday usage; for instance, a scientific theory is a well-verified framework that unifies a body of knowledge, contrasting with the common notion of a mere guess The public often struggles with the inherent uncertainties in science, which stem from three main sources: the potential inaccuracy of theories, numerical uncertainties in measurements and models, and the fundamental uncertainties highlighted in quantum physics.
The Heisenberg uncertainty principle highlights that uncertainty is a fundamental aspect of nature at the quantum level While mathematics primarily deals with numbers, it also encapsulates profound relationships and interactions essential for understanding the universe Accurate and precise numerical values are crucial for addressing quantitative questions, as they represent the sizes of physical quantities Additionally, scientific notation is vital for expressing a wide range of values, such as 650,000,000 watts, ensuring clarity and utility in scientific communication.
650 million watts or 650 megawatts, either of which can be written more easily as 6.5 × 10 8 W.
Iridium is one of the chemical elements used to construct the International Prototype Meter Bar
Courtesy NIST x Throughout the course, we will almost exclusively use the International System of Units (SI) For example, kilo stands for
1000, which means that a kilogram is 1000 grams and a kilometer is
The International System of Units (SI) is essential for measuring both large and small quantities, with prefixes like "milli" for thousandths and "micro" (ȝ) for millionths In physics, physical quantities are measured using units, which are human-created systems that must be included in calculations and results The three fundamental quantities in mechanics are length, mass, and time, represented by the meter, kilogram, and second, respectively Over time, standards for these units have evolved, aiming for a universally reproducible benchmark in laboratories worldwide Mathematical equations play a crucial role in expressing key physical relationships, highlighting the importance of accurate measurements and units in scientific discourse.
10 í24 yotta zetta exa peta tera giga mega kilo hecto deca
— deci centi milli micro nano pico femto atto zepto yocto
In Lecture 2 of the Physics course, while it's possible to participate without a strong math background, engaging with the mathematical concepts will enhance your understanding A solid grasp of high school algebra and trigonometry is expected, with minimal focus on calculus You should be able to solve basic equations, such as determining that x equals 3 when given 2x equals 6 Additionally, understanding the relationship in equations, like x equals b/a when ax equals b, is essential Trigonometry, which involves the functions of angles—sine, cosine, and tangent—is crucial, with the Greek symbol theta (θ) representing angles Lastly, familiarity with reading graphs and interpreting the relationship between axes is important for overall mathematical comprehension in this course.
The Heisenberg uncertainty principle establishes a fundamental limit on the precision of simultaneously measuring a particle's position and velocity; precise measurement of one leads to uncertainty in the other A theory is a broadly accepted principle that aligns with observable facts and experimental data Trigonometry, on the other hand, is a mathematical discipline focused on the relationships among the components of triangles.
Rex and Wolfson, ECP, chap 1.
Physics is the science that elucidates the fundamental principles of the physical universe In contrast, the role of mathematics raises questions about its nature—whether it is a discipline rooted in physical reality or primarily a human invention.
The second is defined operationally based on the wavelength of light emitted by specific atoms, while the kilogram is defined by the mass of a physical prototype stored in Paris Operational definitions, such as that of the second, are often preferred over standard objects like the kilogram because they offer greater precision and consistency in measurement, independent of physical artifacts that may change over time.
3 Since 1983, it’s been impossible to measure the speed of light Why?
Motion is essential to the universe, influencing everything from atomic electrons to galaxies It is defined by changes in position, with velocity representing the rate of this change and acceleration describing how velocity itself varies This lecture focuses on the description of motion through words and mathematics, emphasizing two key concepts: space (measured in meters) and time (measured in seconds) We will simplify our analysis to one-dimensional motion For instance, if you walk 1.2 kilometers to an ice cream stand in 20 minutes and return at the same speed, your total distance traveled is 2.4 kilometers Your speed on the way to the stand is calculated as 1 meter per second, and your average speed for the entire trip, covering 2400 meters in 50 minutes, is derived from the total distance divided by total time.
During your trip to the ice cream stand, you traveled at a speed of 1 m/s, but your overall velocity is not specified since we only know the direction was toward the stand Despite moving, your displacement is zero because you returned to your original position This leads us to understand that position, represented by the variable x, indicates your location relative to an arbitrary origin, defined as position zero In one-dimensional motion along a straight line, positions can be visualized on a number line, where they are classified as positive or negative based on the direction of travel from the origin.
In physics, displacement (ǻx), represented by the Greek letter delta, indicates a change in position, calculated as the difference between final (x2) and initial positions (x1) Unlike distance, which measures the total ground covered and is always positive, displacement can be positive or negative and depends solely on the starting and ending points, making the path taken irrelevant Speed, defined as distance divided by time, is also always positive, while velocity, which is displacement divided by time, can indicate direction based on its sign Average speed and average velocity are calculated as total distance and displacement divided by the time interval, respectively Instantaneous velocity, a calculus concept, represents the slope of the position versus time graph and varies continuously, while instantaneous speed refers to its magnitude Acceleration, the rate of change of velocity, is determined by the change in velocity over time, and any change in velocity indicates acceleration Instantaneous acceleration is similarly defined as the slope of the velocity versus time graph.
Walking from home to an ice cream stand and back results in a covered distance, yet there is no displacement since your starting and ending points are identical.
Lecture 3: Describing Motion x The values of the motion concepts of position, velocity, and acceleration are not related; it’s the rates of change that relate these
Acceleration refers to the rate at which an object's velocity changes over time, calculated as distance divided by time Displacement measures the net change in an object's position from its starting point to its final location Distance indicates how far apart two objects are, while speed represents the rate of change in an object's position, also measured as distance over time Lastly, average velocity is determined by dividing the total distance traveled by the time taken to cover that distance, typically expressed in units such as miles per hour.
Rex and Wolfson, ECP, chap 2.
An hour after the minute hand of a clock points at 12, it returns to the same position, indicating that its displacement is zero However, the distance traveled by the minute hand during that hour is significant, as it completes a full circle around the clock face Thus, while the displacement remains unchanged, the distance traveled is the circumference of the clock, highlighting the distinction between these two concepts in physics.
2 An advertisement boasts that a car can go from 0 to 60 mph in 7 seconds What motion quantity is this ad describing?
3 Throw a ball straight up; at the top of its trajectory, it’s instantaneously at rest Does that mean it’s not accelerating? Explain.
It’s a 3-D World!
Understanding an object's initial position, velocity, and constant acceleration enables precise predictions of its future location, a concept rooted in Newton's clockwork universe theory While calculus typically addresses the mathematical challenges of predicting motion, the simpler case of constant acceleration can be described using straightforward algebraic equations An essential example of this is the motion due to gravity near Earth's surface This lecture will build on previously learned concepts to forecast future motion, emphasizing the deterministic nature of Newtonian mechanics—where knowledge of motion allows for predictions, although not all future movements are predetermined By focusing on constant acceleration, which changes uniformly over time, this course simplifies the analysis, as nonconstant acceleration requires more complex calculus The lecture will utilize equations to determine the velocity of an object experiencing constant acceleration from a specific starting time.
In Lecture 4 on free fall, we explore how velocity changes over a time interval (t) as it transitions from an initial value (v0) to a final value (v) This change in velocity is determined by the product of acceleration and time (at), since acceleration represents the rate of change of velocity Therefore, the final velocity can be expressed as the initial velocity plus the change in velocity (v = v0 + at).
This article discusses predicting future velocity under constant acceleration It highlights that the average velocity of an object from time 0 to time t can be calculated as the mean of the initial and final velocities, represented by the formula v avg = (1/2)(initial velocity + final velocity).
In scenarios involving nonconstant acceleration, the average velocity may not simply be the midpoint between initial and final velocities; it can vary based on the specific changes in acceleration and velocity Understanding an object's position as a function of time is essential for predicting future locations, a concept central to the clockwork universe theory For constant acceleration, the position (x) can be calculated using the equation x = x₀ + v₀t + (1/2)at², where x₀ represents the initial position, v₀t denotes the displacement due to initial velocity, and (1/2)at² accounts for the displacement resulting from acceleration.
The acceleration due to gravity near Earth's surface varies with an object's height For instance, consider an airplane landing at a speed of 80 m/s and decelerating at a constant rate of 4 m/s² Given a runway length of only 900 m, we need to determine if the airplane can stop before reaching the end At touchdown, we set the initial position to 0, with the positive direction towards the runway's end The initial velocity (v₀) is 80 m/s, and the acceleration (a) is -4 m/s², indicating the plane is slowing down To find the stopping time, we use the equation v = v₀ + at, setting v to 0, which gives us t = -v₀/a Substituting the values reveals that the plane will take 20 seconds to stop To calculate the distance traveled during this time, we apply the equation x = x₀ + v₀t + (1/2)at², using the initial parameters.
The acceleration due to gravity near the Earth's surface is 9.8 m/s² By solving the equation for the plane's position when it stops, we find that it comes to a halt at 800 meters, ensuring a safety margin of 100 meters.
In Lecture 4, we explore the concept of free fall, defined as a state where gravity is the sole force acting on an object, resulting in gravity's acceleration This phenomenon does not always involve falling downward We also discuss constant acceleration, which refers to acceleration that increases uniformly over time, in contrast to nonconstant acceleration, where the rate of increase varies Additionally, we touch on determinism, the idea that future events are entirely dictated by the present state of the universe, including the precise positions and momenta of all particles.
Rex and Wolfson, ECP, chap 2.
When two identical objects are dropped from rest, one from a height of h and the other from a height of 2h, the time it takes for the second object to fall is not twice that of the first Instead, it takes less than twice as long to reach the ground due to the effects of gravitational acceleration, which causes both objects to fall faster as they descend.
In this lecture, we explored the calculation of the minimum runway length needed for an airliner to land If the landing speed of the airliner is doubled, the required runway length does not simply double; instead, it is more than twice the original length due to the physics of deceleration Additionally, the time it takes for the plane to come to a stop will also be affected, necessitating a longer distance and time for safe landing and halting.
In our 3-dimensional world, motion typically involves multiple dimensions, necessitating the use of vectors—quantities defined by both magnitude and direction Unlike one-dimensional motion, where direction is limited to a single line, three-dimensional movement allows for greater complexity and a wider range of phenomena To effectively describe this motion, vectors become essential mathematical tools, enabling us to add, subtract, and multiply them In a two-dimensional space, the relationship between an origin point (O) and another point (P) is represented by a vector, with the vector's length indicating the distance from O to P The concept of displacement is exemplified by the paradigm vector, highlighting the significance of directionality in multidimensional motion.
Lecture 5: It’ s a 3-D W o rld! x If we add another point, Q, a different vector describes the displacement from P to Q—the vector ǻr, which is the change in position when you travel from P to Q.
The vector from point O to Q represents a direct journey and is expressed as the sum of vectors r and ǻr (r + ǻr) Vectors, which have both magnitude and direction, differ from scalars, which possess only magnitude without direction For instance, the vector 2v is aligned with v but is twice its length, while the vector -v maintains the same length as v but points in the opposite direction To determine the vector difference r2 - r1, one must add the difference vector ǻr to reach the endpoint of r2 starting from r1 In physics, vectors are essential for illustrating quantities related to motion, with the velocity vector specifically indicating the rate of positional change.
The position vector (ǻr) indicates an object's location relative to a specific origin, while the velocity vector (v) represents the change in position over time (ǻr/ǻt) The average velocity during a time interval (ǻt) is calculated by the object's displacement (ǻr) Acceleration, denoted as the acceleration vector (a), measures the rate of change of velocity Average acceleration during a time interval is determined by the change in velocity (ǻv) divided by that interval (a = (v - v₀)/ǻt) Any alteration in velocity signifies acceleration, whether it involves a change in speed, direction, or both Notably, acceleration and velocity can differ in direction; acceleration aligned with velocity increases speed, while acceleration in the opposite direction decreases speed without altering direction When acceleration occurs at a right angle to velocity, it solely changes the direction, still classifying as acceleration.
Acceleration can occur even when speed is constant, provided that the direction of motion is changing Projectile motion, influenced by gravity, follows a curved trajectory In our coordinate system, the horizontal x-axis and vertical y-axis allow us to analyze motion, as the horizontal movement is entirely independent of the vertical movement.
In Lecture 5, we explore motion in a 3-D world, focusing on two key equations that describe motion under the influence of gravity in two dimensions The first equation outlines horizontal motion, where the position (x) is determined by the initial x position and initial x velocity, without any acceleration term since gravity acts vertically In contrast, the vertical motion (y) incorporates constant gravitational acceleration, represented as the initial y position plus the initial y velocity minus half the product of gravity and time squared The second equation illustrates the vertical position as a function of horizontal position, detailing the trajectory of an object launched from the origin with an initial speed at a specific angle Key terms include the paradigm vector, which describes displacement, the position vector that measures position from an origin, and distinctions between scalars and vectors, with vectors indicating quantities that possess both magnitude and direction, such as the velocity vector that indicates the rate of change of position.
Rex and Wolfson, ECP, chap 3.1–3.4.
1 We generally consider the acceleration due to gravity to be constant near Earth’s surface What 2 factors make this assumption only approximately correct?
2 A line drive in baseball follows an almost straight, horizontal path from the batter to the out¿ eld Is a truly horizontal line drive possible? Explain.
The ideal trajectory of a projectile is a perfectly symmetric parabola, where the ascending path mirrors the descending path However, significant air resistance would disrupt this symmetry, causing the trajectory to become asymmetrical The rising segment would be affected differently than the falling segment, likely resulting in a steeper descent compared to the ascent.
Circular motion is a fundamental aspect of two-dimensional accelerated motion, characterized by a constant change in direction, even when speed remains constant, known as uniform circular motion This type of motion is prevalent across various systems, including technological devices, natural phenomena, and cosmic structures For instance, when a car navigates a curve, both the vehicle and its wheels experience acceleration due to the change in direction, despite maintaining a steady speed Circular motion can occur in complete or partial circles, and acceleration is defined as the rate of change of velocity over time, measured in m/s² Importantly, even if speed does not vary, any alteration in velocity—whether in direction or magnitude—constitutes acceleration The velocity vector, which is tangent to the circular path, continuously changes direction, illustrating the dynamic nature of circular motion.
The velocity vectors of the balls are of equal length, indicating that they are moving at the same speed However, their different directions signify that the velocities are distinct.
In uniform circular motion, the distance from the center of the circular path remains constant, defined by the radius (r), while the velocity vector is always tangent to the circle and perpendicular to the radius The change in velocity (Δv) is determined by positioning the two velocity vectors (v1 and v2) tail-to-tail and connecting them, allowing us to calculate the final velocity by adding Δv to the initial velocity As time progresses, the velocities change, and the acceleration can be calculated by dividing Δv by the time interval (Δt), expressed in m/s² This acceleration, resulting from the change in velocity during the time interval, can be approximated using the formula a = v²/r, derived from calculus Since the direction of acceleration points towards the center of the circle, it is termed centripetal acceleration In contrast, nonuniform circular motion involves changes in speed while maintaining circular motion, and its analysis combines concepts of straight-line acceleration with the circular path's acceleration magnitude, which is also v²/r.
• Nonuniform circular motion can be due to a changing path curvature or, possibly, to changing speed
• Circular motion is never motion with constant acceleration It may have constant speed, but it never has constant velocity.
When discussing circular motion, it's essential to avoid using constant acceleration formulas, as they apply to situations with constant magnitude but variable direction Centripetal acceleration refers to the acceleration experienced by an object moving in a circular path around another object or point In contrast, nonuniform circular motion involves changes in the speed of the object as it moves along its circular trajectory, while uniform circular motion describes a scenario where the object maintains a constant speed throughout the circular path.
Rex and Wolfson, ECP, chap 3.5.
1 You round a curve on the highway, and your speedometer reads a steady
50 mph Are you accelerating? Explain.
The hour hand of a clock is half the length of the minute hand, leading to a significant difference in the acceleration of their tips To demonstrate this, we can analyze the angular velocities and the relationship between the lengths of the hands The minute hand completes a full rotation in 60 minutes, while the hour hand takes 12 hours for the same rotation Consequently, the minute hand's tip experiences a much higher linear velocity due to its longer length and faster rotation When calculating the centripetal acceleration, it becomes evident that the tip of the minute hand accelerates at a rate 288 times greater than that of the hour hand's tip, highlighting the dynamic differences in their movements.
Over 2000 years ago, Aristotle mistakenly believed that a force was necessary to maintain motion, suggesting that pushing or pulling caused movement However, Galileo and Newton revolutionized this understanding by recognizing that uniform motion—moving at a constant speed in a straight line—is a natural state that does not require a force to sustain it Instead, forces are responsible for changes in motion, whether in speed or direction Galileo’s thought experiments led to the formulation of the law of inertia, which states that an object in uniform motion remains in motion unless acted upon by a net force Newton expanded on Galileo's ideas, developing his three laws of motion: the first law reaffirms inertia, the second law quantifies the relationship between force and change in motion, and the third law asserts that for every action, there is an equal and opposite reaction Newton's second law, which relates the rate of change of momentum to the net force acting on an object, highlights the distinction that it is the change in motion that requires a force, not motion itself.
Lecture 7 discusses the causes of motion, highlighting Newton's second law, which states that the net force (F net) acting on an object is equal to the mass (m) multiplied by its acceleration (a), expressed as F net = m a This principle indicates that the change in an object's motion is directly proportional to the net force and inversely proportional to its mass For instance, a 1-kg mass experiencing an acceleration of 1 m/s² experiences a net force of 1 newton The net force is the vector sum of all forces acting on an object; if two forces act in the same direction, their net force is the sum of their magnitudes, while equal forces acting in opposite directions result in a net force of zero Additionally, normal forces, which act perpendicular to surfaces, exemplify everyday forces, such as the floor pushing upward against the downward pull of gravity.
Aristotle's discovery that forces cause motion laid the groundwork for understanding various types of forces, including invisible ones like gravity, friction, and electromagnetic force Physicists have identified three fundamental forces in the universe, which they believe may eventually be unified as a single force Electromagnetism and electroweak forces are often grouped together, while the strong force may merge with them to create a grand unified force Gravity remains the least understood force, and its integration with other forces could lead to a comprehensive theory of everything Key concepts include force, which causes acceleration; Newton's law of inertia, stating that an object remains in uniform motion unless acted upon by a force; momentum, the tendency of an object to maintain its state of motion; and net force, the sum of all forces acting on an object, measured in newtons (N) within the International System of Units.
Lecture 7: Causes of Motion thought experiment: A highly idealized experiment used to illustrate physical principles.
Rex and Wolfson, ECP, chap 4.1–4.2.
1 Why is asking, “What causes motion?” the wrong question to ask? What’s the right question?
When you pull your suitcase across the airport floor at a constant speed in a straight line, it indicates that the net force acting on the suitcase is zero This means that the force you exert while pulling the suitcase is balanced by the opposing forces, such as friction As a result, the suitcase maintains its constant velocity, demonstrating Newton's first law of motion, which states that an object in motion will remain in motion unless acted upon by an unbalanced force.
3 In what sense is Newton’s ¿ rst law really a special case of the second law?
Newton’s second law provides the link between force and acceleration
Gravity is a crucial force that causes weight, which is directly proportional to an object's mass, resulting in the same gravitational acceleration for all objects When weighing yourself in an elevator, your apparent weight changes based on the elevator's acceleration Newton's laws apply only to observers in inertial reference frames, where the law of inertia is valid, and Earth approximates such a frame under normal conditions Newton’s second law (F = ma) allows us to determine mass by calculating the force acting on an object and its acceleration Mass represents the amount of matter in an object and is better understood as a measure of inertia, indicating resistance to changes in motion Weight, the gravitational force on an object, is often mistakenly equated with mass, which is a fundamental property related to inertia.
In the SI system, mass is measured in kilograms (kg) while force is measured in newtons (N), contrasting with the English system where force is in pounds and mass is in slugs, a rarely used unit Mass and weight are often confused due to their direct proportionality; an object weighing twice as much as another also has twice the mass However, weight varies based on location because it depends on gravitational strength, such as the difference between Earth's 9.8 N/kg and Mars' 3.7 N/kg, whereas mass remains constant regardless of location Consequently, all objects on Earth experience the same acceleration due to gravity, allowing mass to cancel out in many gravitational scenarios.
Astronauts in orbit experience a sensation of weightlessness due to their spacecraft accelerating at the same rate as their surroundings However, they are not truly weightless; the gravitational force in space is simply weaker than that on Earth.
NASA astronauts experience apparent weightlessness in a space station due to the equal acceleration of themselves and their surroundings while in a changing circular orbit around Earth Although gravity is slightly weaker at the space station, it still plays a crucial role Consider an elevator with a total mass that includes both the elevator and its passengers, accelerating upward The cable tension must exceed the force required to merely support the elevator's weight, as it must counteract both gravity and provide additional force for the upward acceleration The forces acting on the elevator include the upward cable tension (T) and the downward gravitational force (mg), necessitating that T be greater than mg during upward acceleration This scenario can be analyzed using Newton's second law, expressed as the vector equation F = ma.
The equation T + mg = ma illustrates the relationship between tension (T), mass (m), and acceleration (a), where the upward tension is considered positive and the downward gravitational force (mg) is negative By simplifying this equation, we find that T can be expressed as T = ma + mg, indicating that the total tension is the sum of the mass times acceleration and the weight of the object.
In Lecture 8 on Newton's Laws and 1-D Motion, we explore the relationship between tension and weight in an elevator scenario When acceleration (a) is zero, the tension (T) in the cable equals the gravitational force (mg), indicating that the elevator is in a state of equilibrium Conversely, when the elevator accelerates upward (a > 0), the tension in the cable increases to support both the weight of the elevator and the upward acceleration.
T is greater than mg, and the tension is greater than the weight
To accelerate an elevator, the tension in the cable must exceed the weight of the elevator When acceleration equals gravitational acceleration (g), tension becomes zero, resulting in free fall, where gravity is the only acting force In an elevator, your weight as measured by a scale can differ from your actual weight due to varying forces When the elevator accelerates upward, a net upward force is required, leading to a greater scale reading than your normal weight, as the scale force must surpass the gravitational force acting on you Apparent weightlessness occurs in freely falling environments, like orbiting spacecraft, where all objects experience the same acceleration and appear weightless relative to their surroundings An inertial reference frame is one in which measurements adhere to Newton's first law.
Important Terms mass: A measure of an object’s material content or an object’s tendency to resist an acceleration. weight: The force of the gravitational pull on a mass.
Rex and Wolfson, ECP, chap 4.3.
2 Astronauts are often described as being “weightless” or “in zero gravity.” Are these terms strictly correct? Why or why not?
3 You’re standing on a scale in an elevator Why, as the elevator starts moving upward, is the scale reading greater than your weight?
Newton's third law is often misinterpreted due to its outdated phrasing, "action and reaction." This law actually defines the relationship between forces exerted by interacting objects, stating that the force one object applies to another is equal and opposite to the force applied back It is essential to understand that this law is not an isolated principle but works in conjunction with Newton's second law to offer a coherent explanation of motion in classical mechanics.
Newton's third law emphasizes the mutual interaction between two objects, where equal and opposite forces act upon each other, rather than distinguishing between active and passive agents This principle is essential for a coherent understanding of motion and underpins conservation laws in both quantum mechanics and relativity For example, when a 1-kg block and a 2-kg block are placed on a frictionless surface and the 1-kg block is pushed with a force of 6 N to the right, the total mass of the system is 3 kg By applying Newton's second law (F = ma), we find that the acceleration of the entire system is 2 m/s² This illustrates how forces and masses interact according to the laws of physics.
According to Newton's second law, the net force acting on a 2-kg block with an acceleration of 2 m/s² is calculated as F = ma, resulting in a force of 4 N directed to the right Consequently, Newton's third law states that this 2-kg block exerts an equal and opposite force of 4 N to the left on a 1-kg block, effectively addressing our initial inquiry.
The Horse and Cart Dilemma
In the horse and cart dilemma, the horse exerts a force to pull the cart, while the cart simultaneously applies an equal and opposite force back on the horse This raises the question of how the system can initiate movement despite these opposing forces.
According to Newton's third law, the forces involved in an interaction occur between two different objects and do not cancel each other out For instance, when a horse pulls a cart, the horse exerts a force on the cart while the cart simultaneously exerts an equal and opposite force on the horse However, these forces do not sum to zero because they act on separate entities.
The net force on the cart is determined by the force exerted by the horse, assuming there is no friction The horse pushes against the ground, which in turn pushes back on the horse If this ground force exceeds the force the cart applies on the horse, the net force on the system is directed to the right, causing movement.
Lecture 9: Action and Reaction x Friction is a force opposing the relative motion between any
Friction occurs between two surfaces in contact due to electrical forces between their atoms and is approximately proportional to the normal force acting between them The frictional force can be expressed as F_f = μN, where μ (the coefficient of friction) is a constant specific to the surfaces involved, and N is the normal force Coefficients of friction typically range from near zero to over one; for instance, a waxed ski on snow has a low coefficient of about 0.04, while rubber on dry concrete, essential for vehicle stopping, can be close to 1.
1 at 0.8 x There are 2 types of friction: kinetic friction and static friction
Kinetic friction occurs between moving objects and is always less than static friction, which acts on stationary objects The reason static friction is stronger is that when two surfaces are at rest, the atomic bonds between them can solidify, creating a greater resistance to motion.
Sir Isaac Newton wrote Mathematical
Friction, while often seen as a nuisance that reduces efficiency in machinery, plays a crucial role in everyday activities such as walking and driving When walking, the foot in contact with the ground pushes backward, while the ground exerts an equal force forward, demonstrating Newton's third law of motion This interaction illustrates how friction acts as a force opposing relative motion between two objects in contact, ultimately enabling movement.
Rex and Wolfson, ECP, chap 4.2.
1 For every action there is an equal and opposite reaction Explain this common statement in the context of Newton’s third law and the concept of force.
When a tractor trailer starts, the cab exerts a force on the trailer, while the trailer simultaneously pulls back on the cab with an equal force, as stated by Newton's third law of motion This raises the question: how does the rig manage to begin moving despite these opposing forces?
3 You’re pushing a trunk across the À oor when a child hops onto the trunk Why does it become harder to push?
Newton’s Laws in 2 and 3 Dimensions
Force and acceleration are vectors with both direction and magnitude, as described by Newton's second law, which states that acceleration is proportional to net force This principle is crucial for understanding two-dimensional motion, helping to clarify various physical phenomena In this context, we analyze how a car must achieve a specific velocity to successfully navigate a loop-the-loop track, where the curvature radius at the top resembles a circle During circular motion, acceleration points toward the center, indicating that at the top of the loop, the net force acting on the car is directed downward To apply Newton's second law effectively, we identify the forces at play: gravitational force (mg) and the normal force (N) exerted by the track This leads to the equation mg + N = ma, illustrating the relationship between these forces and the car's acceleration.
To successfully navigate a loop, a car relies on the presence of a normal force between the vehicle and the track, as its absence would result in instant loss of contact The top of the loop represents the minimum condition for completing the circuit, where the normal force is at its lowest point, nearly reaching zero However, once the car passes the top, the normal force begins to increase, re-establishing a secure connection with the track, at which point gravity becomes the sole acting force.
In Lecture 10, we explore Newton's Laws in two and three dimensions, focusing on circular motion The acceleration in circular motion is defined as the square of the speed divided by the radius, directed toward the center (a = v²/r) By substituting this acceleration into the equation mg + N = ma, we derive the relationship N = m(g - v²/r), which simplifies to v² = gr, demonstrating that all objects experience the same gravitational acceleration regardless of mass This implies that if a vehicle travels at speed √(gr) at the top of a loop, it will barely maintain contact Exceeding this speed results in a nonzero normal force, keeping the vehicle on the track, while traveling slower causes it to lose contact and follow a parabolic trajectory Additionally, we examine the banking angle (θ) required for cars to navigate a banked track without relying on friction The net force, directed horizontally toward the center of the turn, ensures that vehicles can safely maneuver through the curve.
In analyzing the forces acting on a car navigating a turn, we identify the gravitational force (mg) and the normal force (N), which acts perpendicular to the road at an angle (ș) To prevent the car from sliding, the vertical component of the normal force must counterbalance gravity, while the horizontal component provides the net force necessary for the car's acceleration toward the center of the turn According to Newton's second law, the equation can be expressed as mg + N = ma, with the positive direction defined toward the circle's center In circular motion, the acceleration is calculated as the square of the velocity divided by the radius.
Lecture 10: Newton’ s Laws in 2 and 3 Dimensions
Trigonometry is essential in analyzing the components of the normal force in this problem, which includes both horizontal and vertical components that share the same angle, ș The horizontal component, represented as Nsin(ș), relates to the centripetal force, while the vertical component is expressed as Ncos(ș) balancing the gravitational force, leading to the equation N = (mg)/cos(ș) By substituting this expression for N into the horizontal component equation, we derive tan(ș) = v²/(gr) Notably, the mass cancels out, indicating that the banking angle is consistent for vehicles of different sizes Therefore, if the desired velocity and turn radius are known, the banking angle can be calculated using tan(ș) = v²/(gr) Constructing this banking angle allows vehicles to navigate the turn without relying on friction, as the normal force alone will maintain their path.
Rex and Wolfson, ECP, chap 4.4–4.5.
1 What holds a satellite up? After all, gravity is pulling it toward Earth.
In a vertical circular flight path, a stunt pilot experiences varying forces on their seat due to the effects of gravity and centripetal acceleration The seat exerts the greatest force on the pilot at the lowest point of the circular path At this position, the combined forces of gravity pulling downward and the centripetal force required to maintain the circular motion result in the maximum force exerted on the pilot by the seat.
In physics, work occurs when a force acts on a moving object, provided the force is not perpendicular to the object's motion Energy, a fundamental concept in physics, includes kinetic and potential energy, with the principle of conservation of energy stating that an object's total energy remains constant when only conservative forces are acting Work is defined more specifically in physics than in everyday language, calculated as the product of the force component in the direction of motion and the object's displacement If no movement occurs, despite the application of force, no work is done When the force aligns with the motion, work equals the force multiplied by the distance moved; if at an angle, only the force component in the direction of motion is considered Energy, measured in joules (J), represents the capacity to perform work, with one joule defined as the work done by a force of one newton moving an object one meter In scenarios where force varies, such as with gravity or springs, the relationship between force and position is illustrated graphically, highlighting how force changes with position.
In the interval of position ǻx, the work done is calculated by applying a force F1, resulting in an approximation of work equal to F1 multiplied by ǻx This method can be repeated for subsequent forces, such as F2, allowing for the calculation of work between each position The area of each rectangle, representing force multiplied by displacement, signifies the work done from one position to the next, and these values can be summed to estimate the total work accomplished.
In the study of work and energy, the work done can be represented as the area under the force-versus-position curve, expressed mathematically as W = F x Δx A notable example of a position-dependent force is the force exerted by springs, which increases with the amount of stretch applied This relationship is quantified by the spring constant, k, which indicates the force required for a specific stretch; stiffer springs have a higher spring constant compared to more flexible ones According to Hooke's law, the force of an ideal spring is directly proportional to its stretch, and the work needed to stretch a spring from its equilibrium position is represented by the area of the triangle under the force curve, calculated as (1/2)kx² This work is stored as potential energy, which signifies that while it is not currently in use, it has the potential to perform work Additionally, forces can be categorized into conservative forces, such as gravity and elastic forces, which store the work done against them as potential energy, and nonconservative forces, which do not have this property.
Nonconservative forces, such as friction, air resistance, and fluid viscosity, do not return the work done against them, instead converting it into heat rather than potential energy For instance, when lifting a mass \( m \) a distance \( h \), the force applied equals its weight \( mg \), resulting in gravitational work of \( mgh \), which is the energy stored as gravitational potential energy in the lifted object, denoted as \( U \).
In the study of work and energy, elastic potential energy is defined as the energy stored in a spring when stretched, represented by the equation ǻU elastic = (1/2)kx² The symbol ǻU indicates the change in potential energy, measured from a defined zero point at the equilibrium position Generally, the change in potential energy can be expressed as ǻU = -W conservative, where the negative sign reflects the work done against conservative forces, such as gravity when lifting an object When an object experiences a constant net force and moves a distance ǻx, changing its speed from v₁ to v₂, the net work done on the object can be calculated using W net = F net ǻx By applying Newton’s second law, we find that F net = ma = m(ǻv/ǻt), which allows us to substitute into the net work equation for further analysis.
The net work done on an object can be expressed as W net = (1/2)mv₂² – (1/2)mv₁², leading to the definition of kinetic energy (K) as K = (1/2)mv², which changes only when net work is applied This relationship stems from Newton’s second law and is encapsulated in the work-energy theorem, stating that the change in kinetic energy (ΔK) equals the net work done (W net) The conservation of energy principle asserts that energy can transform but cannot vanish, extending beyond Newtonian physics When only conservative forces act, the work done equals the change in potential energy (ΔU), resulting in the equation ΔK + ΔU = 0, which reflects the conservation of mechanical energy In systems governed by conservative forces, the total mechanical energy, comprising kinetic and potential energy, remains constant, allowing for energy conversion between forms Additionally, elastic potential energy, defined by ΔU elastic = (1/2)kx², represents the energy stored when stretching an object, while gravitational potential energy relates to an object's position relative to others, emphasizing the various forms of energy in physical systems.
Hooke's law states that in an ideal spring, the force exerted is directly proportional to its stretch The joule (J), a unit of work in the International System of Units (SI), represents the work done when a force of 1 newton is applied over a distance of 1 meter, named after physicist James Prescott Joule Kinetic energy, defined as the energy of motion, is calculated for a particle of mass m moving at velocity v using the formula K = (1/2)mv² The law of conservation of energy asserts that in a closed system, energy cannot be created or destroyed but only transformed Potential energy refers to the energy stored in an object due to its position or chemical configuration Work is the application of force over a distance, and according to the work-energy theorem, the change in kinetic energy of an object equals the net work done on it (ΔK = W_net).
Rex and Wolfson, ECP, chap 5.1–5.4.
In a restaurant setting, when a waitress skillfully carries a heavy tray of food across a horizontal floor, it raises the question of whether she is performing work in the physics sense According to physics, work is defined as the force applied to an object multiplied by the distance over which that force is applied Since the tray is being moved horizontally without any vertical displacement, the waitress is not doing work in the strict physical definition, despite the effort and skill involved in her task.
2 Distinguish conservative from nonconservative forces, and give an example of each.
When stretching an unstretched spring by 1 inch, a specific amount of work is done However, if you stretch the spring an additional inch, the work required is not the same; it actually increases This is due to the nature of springs, where the force needed to stretch them grows with the distance stretched, meaning more work is done for each subsequent inch.