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Important Physical Units
Important Physical Units
First and foremost, physical quantities have units. Whether you're measuring the side of a table, the mass of an atom, or the energy the sun emits, all of these quantities have some unit of measure. We will be using the International System of Units in this course, which tells us the basic units of measure for a bunch of important quantities. While the exact units are important to calculate the right values in computations, it's important to also look at the dimensions of each quantity. We will focus on the units of length, time, energy, temperature, and mass. We will typically use the following units for these quantities:
Length: meters or Mega-parsecs (m or Mpc)
Time: seconds or Giga-years (s or Gyr)
Energy: Mega-electron volts (MeV)
Temperature: Kelvin (K)
Mass: Kilo-grams or Mega-electron volts per speed of light squared (kg or MeV/c2)
The prefix 'Kilo' tells you to multiply the base unit by one thousand, 'Mega' by one million, and 'Giga' by one billion. That means the unit of Giga-years is translated to one billion years. Some units you may not have heard of include Mega-parsecs and Mega-electron volts. Using units with smaller values can be cumbersome to carry around, especially when measuring really large or really small things. A Parsec is a useful unit of distance for galaxy-sized measurements, and an electron-volt is a unit of energy useful for measurements at the size of atoms.
We can write one unit of measurement in terms of another. Some useful conversions include:
Length, Area, and Volume
Length, Area, and Volume
In this course, we will need to understand the difference between length, area, and volume measurements. Simply put:
A length measurement tells us the distance between two points of an object.
An area measurement tells us how much space a surface of an object takes up.
A volume measurement tells us how much total space an object takes up.
Keep in mind, though, that a length measurement is not always a straight line and an area measurement is not always of a flat surface. We often times need to calculate distances along or around an object or the area along the surface of an object. We provide figures illustrating these distinctions as well as some common formulas here.
Important Physical Constants
Important Physical Constants
There will be a few fundamental constants in our course that you may not have seen before. We will define them here for reference and define their applicability as we progress through the course.
Important Physical Concepts
Important Physical Concepts
Position, Velocity, and Acceleration
Position, Velocity, and Acceleration
The position, x, of an object tells us the distance from a zero point in 3 spatial dimensions. We organize the three distance measurements as a list of values to indicate the magnitude in each direction, called a vector. The velocity, v, is also organized as a vector and describes the speed (how fast an object is traveling) and the direction of motion for an object in each spatial dimension. The velocity describes how the position changes with time. Relating back to our background module in math, we can write the velocity as the time derivative of the position: v = dx/dt.
The acceleration, a, of an object is also organized as a vector and tells us how the velocity is changing in each spatial direction. This can also be written as the time derivative of the velocity: a = dv/dt. This also means that the acceleration is the double time derivative of the position: a = dx2/d2t.
I'll do the math for you to get an expression, called the kinematic equation, on how we can relate the position of an object that is undergoing constant acceleration, a, to its initial position,x0 , initial velocity, v0, and time, t.
Mass and Density
Mass and Density
The mass, m, describes how much matter an object has, while a mass density describes how much mass there is in a given volume of space. The term density can be widely applied to mean how much of something exists in a given volume. We will often use energy densities to describe the amount of energy in a volume and number densities to describe the number of particles in a volume. A density then has the units of something divided by the units of volume.
Force, Energy, and Momentum
Force, Energy, and Momentum
A force F, as stated by Newton's Second Law, is defined as the mass of a particle times its acceleration, or F = ma. When a force acts on a particle, it will generally create a push or pull on it which will change the particle's motion.
There are a few kinds of energy that we will concern ourselves with the first is kinetic energy, which tells us the energy related to the motion of a particle, and is defined as KE = ½ mv2. Another is potential energy, which tells us the energy that an outside force imparts on a particle to affect its motion. The final energy we concern ourselves with is the rest-mass energy of a particle. As Einstein once said: E = mc2. Mass, itself, is a form of energy!
The momentum of an object, p, is defined as the mass of a particle multiplied by its velocity, p = mv, and tells us how resistant a particle is to having its motion changed.
We group these definitions together because they are all related to one another. A force acting on an object acts to change its motion; a force should then change the momentum and energy of an object. Recalling that the acceleration is the rate of change of the velocity over time, we can relate the influence of a force to an object's change in momentum as F = m dv/dt = dp/dt.
Likewise, a force changes the potential energy, U, of an object. Forces typically have a source and aims to bring an object to the lowest potential energy possible. The potential energy then tells us how much potential the force has to change a particle's motion. If we have an attractive force, like gravity, acting on an object that is very far away, we will have a greater potential energy since the force can influence this object's motion substantially over time. For a repulsive force, though, the opposite is true! The direction of the force then informs us where high and low potential energy exists. We can then write the relationship between the force and the potential energy as -dU/dr = F.
Power, Temperature, and Pressure
Power, Temperature, and Pressure
Power tells us how much energy is outputted given a length of time and has units of energy/time.
Pressure refers to the amount of force exerted on an area. Pressure can be exerted through many day-to-day phenomena. For instance, by simply leaning on a table your hand exerts pressure on the surface of the table. Pressure has units of force/area.
The temperature measurement of a bunch of particles will tell about us their average kinetic energy. The units of temperature we see day-to-day, like Celsius and Fahrenheit, are actually related to units of energy through the Boltzmann constant.
Luminosity, Magnitude, and Flux
Luminosity, Magnitude, and Flux
The luminosity of some object regards the energy output per time of some object in the universe. Luminosity, then, is a measure of power. Because sources in the universe can be very bright or very dim, we usually talk about Luminosity in terms of Magnitude, which the negative log of the luminosity or the exponent of the luminosity's value. More luminous objects will have a negative magnitude and less luminous objects a positive magnitude.
Flux tells us how much power goes through an area. Light sources only emit a set number of light particles. When we observe light sources from very far away, the distribution of light becomes more diffuse. This means that the amount of light we see from a distant source is much less than if we were right next to it. On Earth, this would be like looking at a flashlight that is in your hand as opposed to the opposite side of a field. The flashlight far away from you will appear dimmer because your eyes are receiving less light from it. Flux then has units of Luminosity/Area.
Check-in: Which of these quantities are described in the form of a vector? Which are described in the form of a single value (a scalar)?
Check-in: Which of these quantities are described in the form of a vector? Which are described in the form of a single value (a scalar)?
Conservation of Mass-Energy
Conservation of Mass-Energy
Conservation of Momentum
Conservation of Momentum
Conservation of Mass-Energy simply states that the total energy you have before an interaction should be equal to that after an interaction, which includes any energy in the form of the mass. However, this concept is not a law and can be broken. In the context of this course though, this concept can be safely applied to most systems.
Conservation of Momentum simply states that the total momentum you have before an interaction should be equal to that after an interaction. This concept will be particularly useful later on in Unit 3. However, this concept is not a law and can be broken. For instance, we know that having a force can change the momentum. in the context of this course, we can apply this principle so long as there is not an outside force acting on the system.
Properties of Light
Properties of Light
Light is really interesting. As human beings, we only see a tiny fraction of the light spectrum that exists. Click this link to investigate the other kinds of light and their properties. Light in the vacuum of space travels at a constant speed, c, which is about 300 million meters per second: really fast. In fact, it's the fastest speed anything can travel! Light particles, known as photons, are also massless and travel through space as a repeating pattern of energy known as a wave.
Light waves repeat their pattern over and over again, oscillating between a high and low amplitude. The time it takes for the pattern to repeat once is called the period and has units of time per cycle. The period is related to a quantity called the frequency, which tells us how often the pattern repeats in a set amount of time, having units of cycles per time. If we set T to represent the period and f to represent the frequency, the relationship between the two is described as f = 1/T .
The size of the repeating segment is called the wavelength. Using these quantities, we can figure out the speed of a traveling wave. The period tells us how much time it takes for the wave to travel one wavelength. We know that speed tells us how far something travels in a a certain amount of time. Defining the wavelength as λ and the period as T, the speed of the wave is then λ/T. Using the relationship between the period and the frequency, we see the speed of the wave is defined as λf. If the light wave we're looking at travels in the vacuum of space, we know the speed that it's traveling at: c!
We have then defined a set relationship between the wavelength and the frequency of light traveling through space:
The energy associated with a certain wavelength of light is a little harder to come by through intuition and is actually found through quantum mechanics. However, scientist Max Planck determined that the energy of light is proportional to its frequency through the self-named Planck's constant, h.
It also turns out that light has momentum. Thinking about it earlier, though, we defined the momentum to be the velocity of a particle multiplied by its mass. But photons are massless! How could a massless particle have momentum?? This delves into a little more quantum mechanics investigated by Einstein. You've heard of E = mc2, but this isn't the entire picture of the TOTAL energy of a particle: especially light.
In general, anything traveling close to the speed of light is called relativistic and obey laws of physics that appear to be different than what we see in our day-to-day lives. The total energy of these very fast particles also depend on how fast they're moving in addition to their mass. The general form of this relationship is given as follows:
For a massless photon, we can take the mass term to be zero and find that E = pc. So despite being massless particles, light still has momentum. This also means that light can exert a force or pressure on other objects. Scientists are beginning to take advantage of this fact for space travel in the form of a solar sail. Check out Bill Nye's explanation of how this works!
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