Introduction(, Waves, Electricity, Magnetism, Quantum, Mechanics)Etc. Check Description., Study notes of Quantum Physics

Download Introduction(, Waves, Electricity, Magnetism, Quantum, Mechanics)Etc. Check Description. and more Study notes Quantum Physics in PDF only on Docsity! Thermodynamics & Waves Thermodynamics: Understanding Heat Energy and Work In physics, thermodynamics is the study of heat energy and the behavior of systems when heat is added to them. The goal is to determine if we can convert heat energy into useful work. Matter, such as gas, contains energy in the form of moving atoms. This energy is measured as temperature. When heat is added to a gas, the motion of the particles increases, leading to a rise in temperature. We want to figure out how to make the system, like a piston, do work for us. Adding heat energy to a gas expands and pushes the piston up, which is considered work. This concept is used in various applications, such as steam engines and jet engines, where heat energy is converted into motion and work. However, it's important to note that we can't get more work out of a system than the heat energy we put in. This is known as the first law of thermodynamics, which states that the work output can never exceed the heat input. The second law of thermodynamics also states that systems tend to increase in disorder over time. Without any outside stimulus, a gas in a container will naturally spread and fill the entire space. This law highlights the tendency of systems to move towards greater disorder. Introduction to Physics, Part 3 (Electricity, Magnetism, Quantum Mechanics & Relativity) Electric fields can be compared to peanut butter and jelly; if electric charges are the peanut butter, the electric field is the jelly. Imagine a proton with a positive charge surrounded by an invisible field with radial arrows that emanate from the charged particle instead of closed loops. The relationship between electricity and magnetism was combined into a single concept called electromagnetism in modern physics. A proton is pushed in the same direction as the field lines, while another charged particle moving through the field is pushed tangentially to the magnetic field. Theories of relativity and quantum mechanics were developed by many scientists in the early 20th century, yet even today we struggle to fully understand these theories. There remain many unsolved problems in both relativity and quantum mechanics. We can demonstrate that time travels at different rates for different people. For instance, by using highly accurate atomic clocks onboard airplanes, we observed that when we returned to Earth, clocks no longer agreed as time takes varied time depending on how quickly you move. Quantum Mechanics predicts that when we have a proton in an atom's nucleus, electrons surrounding the nucleus do not behave like a solar system going round and round as one might learn in a Chemistry classroom. The human transformation has evolved from basic survival instincts like learning to make shelter and make a fire to developing technology to communicate, advance medical imaging, calculate solutions, and even go into space. Without Quantum Mechanics, computer chips would not exist, and neither would computer screens like Androids or iPhones. Review of Essential Trigonometry (Sin, Cos, Tangent - Trig Identities & Functions) Review of Essential Trigonometry In this lesson, we will be reviewing essential trigonometry, which is critical for solving physics problems. Physics problems often involve angles, such as wedges, planes, or forces at an angle. However, you will never be explicitly told which trigonometric function to use. Instead, you will need to draw a diagram and apply your knowledge of trigonometry to solve the problem. The Right Triangle A right triangle is a key component in trigonometry. It consists of a 90- degree angle and two other angles, one of which we will label as theta. The sides of the triangle are also labeled:  Opposite (opp): The side opposite to the angle theta.  Adjacent (adj): The side adjacent to the angle theta.  Hypotenuse: The longest side of the right triangle. In any right triangle, the Pythagorean theorem can be applied. The theorem states that the hypotenuse squared is equal to the sum of the squares of the other two sides. This equation allows us to find the length of the third side of the triangle. Fundamental Trig Functions There are three fundamental trigonometric functions that are commonly used in physics problems:  Sine (sin): The sine of an angle theta is defined as the opposite side divided by the hypotenuse.  Cosine (cos): The cosine of an angle theta is defined as the adjacent side divided by the hypotenuse.  Tangent (tan): The tangent of an angle theta is defined as the opposite side divided by the adjacent side. Remembering these three trig functions, and understanding their definitions, will allow you to solve many physics problems. Most of the other concepts we will discuss in this lesson will involve manipulating these functions to solve problems. Please draw the triangle below: Let's solve a quick problem to illustrate the concepts. We have a triangle with sides measuring 3, 4, and 5 meters. We know this is a right triangle because it satisfies the Pythagorean theorem (c^2 = a^2 + b^2). Now, let's find the values for sine, cosine, and tangent of the angle:  Sine of theta: opposite side (3) divided by hypotenuse (5) = 3/5 = 0.6  Cosine of theta: adjacent side (4) divided by hypotenuse (5) = 4/5 = 0.8  Tangent of theta: opposite side (3) divided by adjacent side (4) = 3/4 = 0.75 Notice that sine, cosine, and tangent are always less than 1. The angle theta can be determined based on the lengths of the sides of the triangle, but we need more information to calculate the exact value. There are multiple ways to calculate the angle of a triangle. One method is to use the sine of the angle. To find the angle, we can use the inverse sine function, also known as arcsine. For example, if the sine of the angle is 0.6, then the angle is 36.87 degrees. We can also use the cosine of the angle to calculate it. To find the angle using cosine, we can use the inverse cosine function. If the cosine of the angle is 0.8, then the angle is 36.87 degrees. Similarly, we can use the tangent of the angle. To find the angle using tangent, we can use the inverse tangent function. If the tangent of the angle is 0.75, then the angle is 36.87 degrees. It is important to note that when using these trigonometric functions to find the angle, the calculator will always give the positive angle. In some cases, you may need to adjust the angle based on the orientation of the triangle. In addition to these trigonometric functions, there is a useful way to express the relationships between the sides of a triangle. If we have an angle theta, an opposite side, an adjacent side, and a hypotenuse, we can use the sine and cosine functions to express the lengths of the sides. The opposite side is equal to the hypotenuse times the sine of the angle, and the adjacent side is equal to the hypotenuse times the cosine of the angle. These equations are useful in physics when dealing with vectors. Understanding Projectile Motion When an object is thrown at an angle, its motion can be broken down into horizontal and vertical components. The horizontal component represents the speed in the horizontal direction, while the vertical component represents the speed in the vertical direction. By using trigonometry, we can calculate these components.  The vertical speed can be found using the equation: vertical speed = hypotenuse * sine(angle)  The horizontal speed can be found using the equation: horizontal speed = hypotenuse * cosine(angle) In the given example, a baseball is thrown at 10 meters per second at an angle of 35 degrees. Using the above equations, we can calculate the vertical and horizontal speeds: How Does Scientific Notation Work? Let's use an example to illustrate scientific notation. Suppose we want to write the number 316,000,000. Instead of writing out all the zeros, we can write it as 3.16 x 108. The key is to write the actual nonzero numbers, place a decimal point after the first nonzero digit, and then multiply it by the appropriate power of 10. Step-by-Step Guide 1. Write down the nonzero numbers: 3, 1, 6. 2. Place a decimal point after the first nonzero digit: 3.16. 3. Write it as a product of the number and a power of 10: 3.16 x 10 8. By using scientific notation, we can easily represent both large and small numbers without writing out all the zeros. This makes calculations and problem-solving much more manageable. Now that you understand the basics of scientific notation, you'll be able to confidently tackle problems in physics. Good luck! Physics Units And The Metric System Physics Units and the Metric System In this lesson, we will discuss physics units and the metric system. We will provide an overview of the basic units used in physics and talk about conversion factors. Please note that this is not an exhaustive list of units, but rather an introduction to the core units and metric system prefixes. SI Units The most common SI (International System of Units) units are the base units upon which everything else is built. The first base unit is time, which is measured in seconds (s). In physics, we use seconds as the unit of time, even though other units like hours or days may be used in everyday life. It is important to convert these units to seconds when working with physics problems to ensure consistency. Remember, the rule of thumb is to always use the base units as much as possible. If you encounter a problem with units that are not in the base units, you will need to convert them to the base units before proceeding. Unit Conversions Unit Conversions in Physics In this section, we will be discussing unit conversions, which is an incredibly useful skill in physics, engineering, and chemistry. Understanding unit conversions is essential because every problem in these fields involves calculations with units. If you don't convert the units correctly, you won't be able to solve the problem. In this article, I will teach you a method that will make unit conversions easy and foolproof. Let's get started! The Importance of Unit Conversions Every problem in physics, engineering, and chemistry requires working with units. Whether it's speed, energy, or time, you need to have the correct units to perform calculations. Without proper unit conversions, you won't be able to solve the problems effectively. Additionally, unit conversions can be used to solve specific problems in physics and chemistry. Mastering this skill will greatly enhance your problem-solving abilities. The Simplest Conversion Let's start with a simple example to demonstrate the concept. Suppose we want to convert 2 feet to inches. We know that there are 12 inches in a foot, so we can easily calculate that 2 feet is equal to 24 inches. This is a straightforward multiplication, but not all conversions are this simple. Sometimes, it can be challenging to determine whether to multiply or divide by the conversion factor. The Foolproof Method I will now teach you a foolproof method for unit conversions. Start by writing down what you know, which in this case is 2 feet. Draw a horizontal line under it and a vertical line next to it. On the next line, write down the conversion factor, which is 1 foot equals 12 inches. Now, here's the crucial part: write the conversion factor in two ways. First, write it as 1 foot equals 12 inches. Then, write it as 12 inches equals 1 foot. This step ensures that you don't get confused about whether to multiply or divide. Learn Distance and Displacement in Physics (Displacement Formula Vs. Distance Formula) Distance and Displacement We this lesson, we will be discussing the concepts of distance and displacement. These concepts are essential in understanding physics and are found in every physics book. Let's start by understanding what distance and displacement mean. 1D Motion Before we dive into the details, let's clarify what we mean by 1- dimensional motion. In 1D motion, objects can only move back and forth along a line, represented by the x-direction. We will be focusing on this type of motion for now, disregarding any other dimensions. Reference Point When discussing motion, it is crucial to establish a reference point. This point serves as the starting point for any measurements. In our case, the reference point is the origin, denoted as x=0 on our number line. For simplicity, we will measure distances in meters. Displacement is a vector quantity as it includes both magnitude and direction. Distance, on the other hand, is a scalar quantity as it only represents magnitude. What is the Definition of Average Speed & Velocity in Physics? (Speed Formula & Velocity Formula) Average Speed and Average Velocity in Physics we will be covering the topics of average speed and average velocity. While these two concepts may seem similar, they are not exactly the same thing. Just like distance and displacement in the last section, speed and velocity are closely related but have distinct differences. A Recap on Vectors In the previous lesson, we learned that displacement is a vector quantity because it not only tells us how far we moved but also the direction in which we moved. On the other hand, distance is a non-vector quantity because it only tells us the magnitude of how far we moved, without any information about direction. Defining Average Speed Now, let's define the concept of average speed. Think about the speedometer in your car. In the USA, it shows miles per hour, while in other countries, it's usually kilometers per hour. Notice that the units of speed are always distance over time. So, the equation for average speed is simply the distance traveled divided by the time it takes to travel that distance. Example: Let's say you traveled 10 meters in 2 seconds. To find the average speed, you would divide the distance (10) by the time (2), resulting in a speed of 5 meters per second. It's important to note that distance is always a positive number, as it does not convey direction. Similarly, the time interval is always positive, as time only moves forward. Since we are dividing two positive numbers, the average speed is always positive. Therefore, average speed is not a vector quantity, as it does not indicate direction. An Example on Average Speed Calculation Let's consider a simple example to understand the calculation of average speed. Suppose you start at position x = -2 and move to position x = 2 in one second. The distance traveled is 4 units to the right. By dividing the distance (4) by the time (1), we find that the average speed is 4 meters per second. Remember, average speed does not tell us the direction of motion. It only provides information about how fast an object is moving. Average Velocity We have previously defined average speed, now let's talk about its cousin, average velocity. The formula for average speed is distance over time. For average velocity, we use displacement over time interval. Displacement can be positive or negative, while time interval is always positive. Therefore, average velocity can be positive or negative, indicating the direction of travel. Average velocity is a vector quantity, represented as: Average Velocity = (change in x) / (change in t) Here, change in x represents the final value of position minus the initial value of position, and change in t represents the final value of time minus the initial value of time. It is important to understand the difference between speed and velocity. Speed tells us how fast we are moving, while velocity includes the direction of travel. Average velocity is useful in determining the directions of collisions or movements. Example: Average Velocity Calculation Let's consider an example where you move from an initial position of -4 meters to a final position of 2 meters in 3 seconds. To calculate average velocity, we use the formula: Average Velocity = (final position - initial position) / (final time - initial time) Plugging in the values, we get: Average Velocity = (2 - (-4)) / (3 - 0) = 6 / 3 = 2 meters per second The average velocity in this example is 2 meters per second. Average Speed To calculate average speed, we take the absolute value of the average velocity. In this case, the average speed is also 2 meters per second. Calculating Average Velocity and Average Speed In this problem, we are given a number line and we start at position 0 and move to position -4 in 6 seconds. The initial position is 6 meters and the final position is -4 meters. We can calculate the average velocity and average speed using the formulas: Average Velocity = (Final Position - Initial Position) / (Final Time - Initial Time) Average Speed = |Average Velocity| Using these formulas, we can calculate the average velocity and average speed as follows: Average Velocity = (-4 - 6) / (6 - 0) = -10 / 6 = -5/3 m/s Average Speed = |-5/3| = 5/3 m/s world. In the next few sections, we will delve deeper into the concepts of vectors and scalars and learn how to work with them. Acceleration can be positive or negative. Positive acceleration means speeding up, while negative acceleration (deceleration) means slowing down. Acceleration is a vector quantity because it has both magnitude and direction. Other examples of vector quantities include magnetic field and electric field. These fields have strength and direction at every point in space, represented by field lines. Scalar quantities, on the other hand, do not have direction. Examples include speed, which only has magnitude, and force, which has both magnitude and direction. Vectors are often represented with an arrow on top of the variable. For example, velocity is represented as v with an arrow on top, while acceleration is represented as a with an arrow on top. The Representation of Vectors in Physics When dealing with vectors in physics, it is important to understand how they are represented graphically. In most textbooks, vectors are indicated by an arrow on top of the variable. For example, a vector representing velocity might be denoted as v with an arrow on top. However, some books may use a straight bar instead of an arrow. The length of the arrow represents the magnitude or strength of the vector, while the direction of the arrow indicates the direction in which the vector is acting or moving. For instance, if we have a ball and draw an arrow pointing to the right with a length of 10 meters per second, we are representing the velocity of the ball. On the other hand, if we draw an arrow pointing downwards with a length of 15 meters per second, we are representing a different velocity. It is important to note that the length of the arrow corresponds to the magnitude of the vector. A longer arrow indicates a stronger or larger magnitude, while a shorter arrow represents a weaker or smaller magnitude. Additionally, the direction of the arrow indicates the direction in which the vector is acting or moving. Examples Let's consider another example with force. If we have a vector representing a force with a length of 16 newtons, we can draw an arrow to represent it. Similarly, if we have another force vector with a length of 30 newtons, we would draw a shorter arrow to indicate a smaller magnitude. Graphically representing vectors as arrows helps us visualize their magnitudes and directions. However, in solving real physics problems, we will eventually move away from relying on graphical representations and use equations instead. The use of graphical representations is similar to learning to add negative numbers using a number line as a visual aid. It is a helpful tool to start with, but as we become more familiar with vectors, we won't need to rely on it as much. Adding Two Vectors Graphically in Physics (Vector Sum & Resultant Vectors) Adding Vectors Graphically A vector is a quantity with both magnitude and direction, represented as an arrow. Adding vectors is more complex than adding numbers because we must consider both magnitude and direction. We will start by using simple examples to understand how to add vectors together graphically. Example 1: Adding Vectors in the Same Direction Let's consider a box with two vectors acting on it. The first vector, represented as F1, is a force pushing the box to the right. The second vector, represented as F2, is another force pushing the box to the right. Both vectors have the same magnitude and direction. To add these vectors together, we simply add the magnitudes, resulting in a total force twice as strong as each individual force. Example 2: Adding Vectors in Opposite Directions Now, let's consider a box with two vectors acting on it. The first vector, represented as F1, is a force pushing the box to the right. The second vector, represented as F2, is a force pushing the box to the left. Both vectors have the same magnitude but opposite directions. In this case, the vectors cancel each other out, resulting in a total force of zero. Adding Vectors at Angles While the previous examples were simple cases, adding vectors can become more complicated when the vectors are at angles. In the next lessons, we will explore a method to add vectors together that works for all cases, taking into account both magnitude and direction. 's the answer that 's what results in factor me how do i do it first before we jump into that let me give you a couple of other examples that i think you can get the answers to without doing any math without doing any actual you know formal procedure let 's give you a couple of things and just see if you can kind of predict what the vector addition right might might be what if i have some ball or something right here and let 's say i 'm acting on it wifx a force here that i 'm calling f1 and on the same ball i'm acting horizontally with another guy it 's called f2 now these are 90 degrees apart so i 'm pushing up and also my friend or whoever is standing next to me also touching the ball is pushing horizontally what do you think is gon na happen to a box or crate i 'm pushing this way and somebody underneath is pushing up what 's gon na happen well you kind of know that it 's not just gon na go up and it 's not just gon na go this way it 's gon na go and This method can be used for any vectors, not just the ones shown in the example. By adding the vectors together, we can find a single resultant force that represents the combined effect of multiple forces acting on an object. This simplifies problem-solving by reducing multiple forces to a single force. Examples Example 1: Consider a ball with two forces acting on it: f1 and f2. By adding these forces together, we can find a single resultant force that represents the combined effect of both forces. This allows us to treat the ball as if it were being acted upon by a single force. Example 2: Let's take another example where a ball is being acted upon by two forces: f1 and f2. In this case, f1 is not horizontal and has a specific angle and length. f2 is slightly shorter and at a slightly different angle. By adding these forces together, we can find the resultant force that represents their combined effect on the ball. You see how d1 and d2 here are two different vectors. This one is longer and this one starts from a different point, so it has a different distance and direction. But the sum of these two vectors actually gives me exactly the same thing as this one. One more quick example: let's do the same thing with x and y. Let's say 1, 2, 3 for x and 1, 2, 3, 4, 5, 6 for y. The mailbox is my destination, which is basically over here. So let's draw the resultant first. There are many ways to get there, so instead of cutting across here, let's go all the way directly across from the mailbox. We'll call that d1. I'll look both ways across the street and then go straight up, which we'll call d 2. So d2 is slightly different than d1; it's pointed differently and it's longer. But the same result is true for them. The moral of this story is that for displacement vectors, or in general vectors, when you have a resultant, there are many ways to add to get that resultant. Two vectors can be added differently, just like forces. If I know that I'm applying a force of 6 newtons on a box and I know that I can get that force by adding two vectors together, there's an infinite number of ways I can do that. I can have the two vectors lined up exactly with half of the force, or I can add them at a little bit of an angle and adjust their magnitudes to give me that same force. Just like I can pick lots of different paths to get to the end. So that's just a lesson that I want you to keep in the back of your mind and it's also a good practice in vector addition. Now, let's briefly talk about velocity as a vector. Nothing changes, everything is added the same way. Think about a spaceship in space. The spaceship is shaped like a bullet. It has a magnitude and a direction. Let's say there's an engine pushing it in one direction, v 1, and another engine pushing it in another direction, v2. We call it delta v. How will the ship behave? We add them vectorially, graphically. Take v 1 and leave it alone, then draw v2 anywhere you want and put it tail to head. That's the resultant vector. The final velocity you get is the sum of those two velocities. It doesn't matter the order that you add these vectors. Addition of vectors is commutative. If you add vector v 1 to vector v2, you will get a resultant. If you separately take v2 and add it to v1, you will get the same resultant. So it doesn't matter the direction or the order, you get the same answer. You don't have to stress out about which way to arrange vectors for addition. Just put them on paper, connecting the tail of one vector to the head of the next. The order doesn't matter, as vector addition is commutative. In the next lesson, we will learn how to add three or more vectors graphically. However, we will also show you a method to break vectors into chunks and add them together numerically, as drawing arrows can become cumbersome. Calculating Vector Components in Physics, (Component form of a Vector) In this lesson, we will be learning about calculating vector components. We will break this lesson up into several parts to give you plenty of practice. Calculating vector components is important because it allows us to simplify problems involving vectors. Instead of dealing with curved paths, we can break vectors into vertical and horizontal components, making problem-solving much easier. There are two ways to represent a vector. The first way is by drawing an arrow to represent the magnitude and direction. The second way, which we will focus on in this lesson, is by using numbers. To represent a vector as numbers, we need to specify the magnitude and the angle it makes with the positive x-axis. The magnitude is the length of the arrow, while the angle determines the direction of the vector. To represent the magnitude of a vector, we use absolute value signs and write the number. For example, if the magnitude of a vector is 10 newtons, we write |10|. The angle of the vector is represented by the symbol theta, and it is measured from the positive x-axis. By using these two numbers, we can represent any vector in a two- dimensional plane. The magnitude determines the length of the arrow, while the angle determines the direction. For example, a vector with a magnitude of 5 newtons and an angle of 45 degrees would be represented by |5| and theta = 45 degrees. Let's look at another example. Suppose we have a vector that is pointing in a different direction, like this. We can represent this vector by specifying its magnitude and angle. The magnitude might be 10 newtons and the angle might be 60 degrees. By changing these two numbers, we can represent vectors of different lengths and directions. Example Problem Let's consider a vector that represents the distance to a mailbox. The vector is 10 meters long and makes an angle of 30 degrees with the horizontal axis. We want to find the components of this vector in the x and y directions. Solving for dy To find the vertical component, we can use the sine function. We know that the opposite side to the angle is dy, and the hypotenuse is 10. So, dy = 10 * sin(30). Using the value of sin(30) being 1/2, dy = 5 meters. Solving for dx To find the horizontal component, we can use the cosine function. We know that the adjacent side to the angle is dx, and the hypotenuse is 10. So, dx = 10 * cos(30). Using the value of cos(30) being 0.866, dx = 8.66 meters. Pythagorean Theorem Check We can use the Pythagorean theorem to check our calculations. According to the theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, 10^2 = 8.66^2 + 5^2. Calculating, we get 100 = 74.99 + 25, which is approximately equal to 100. This confirms that our calculations for dx and dy are correct. Importance of Vector Components Understanding and calculating vector components is crucial in solving physics problems. Breaking down vectors into their components makes problem-solving easier and more manageable. It allows us to analyze the different parts of a vector separately. By decomposing vectors into their components, we can solve complex problems with ease. It is a fundamental concept that every student should practice and understand. Now that we've covered the basics of vector components, we will move on to more examples and explore motion in two dimensions. Study Hack: Mastering Vector Components in Physics If you're struggling with understanding vector components and how to calculate them in physics, don't worry - you're not alone. However, with a little bit of practice, you can master this important concept. In this article, we'll explore some helpful tips and tricks to help you improve your understanding of vector components. What are Vector Components? In physics, vectors are used to represent quantities that have both magnitude and direction. Vector components are the individual parts of a vector that contribute to its overall magnitude and direction. In other words, they break down a vector into its horizontal and vertical components. Calculating Vector Components To calculate vector components, you'll need to use trigonometry and the given information about the vector's magnitude and direction. Here's a step-by-step guide: 1. Identify the magnitude and direction of the vector. 2. Choose a coordinate system, typically using the x-axis and y-axis. 3. Use trigonometry to find the horizontal and vertical components of the vector. 4. Apply any additional formulas or calculations, if necessary. Practice Makes Perfect Like any skill, mastering vector components requires practice. Here are some tips to help you get more practice:  Work through textbook problems that involve vector components.  Create your own practice problems and solve them.  Seek out online resources, such as videos or interactive simulations, to further your understanding.  Collaborate with classmates or join a study group to discuss and solve problems together. Summary Understanding and calculating vector components is an essential skill in physics. By breaking down vectors into their horizontal and vertical components, you can better analyze their magnitude and direction. Remember to practice regularly and seek out additional resources to strengthen your understanding of this concept.