Intro to Vectors & Scalars

Vectors & Scalars Intro to Vectors & Scalars: Vector: has both a magnitude and a direction (i.e. 5 meters to the right - displacement) Scalar: only has a magnitude (i.e. 5 meters - distance) Visualizing Vectors in 2 Dimensions: When you add two vectors together, you place them head to tail to one another Vectors can be broken down into their horizontal and vertical components (helps when adding them together!) Absolute value bars (|a|) show a magnitude and not necessarily a direction You can use trigonometry in order to solve for components of a vector (x and y directions) Remember, there are 3, 4, 5 triangles Unit Vector Notation: 𝑥 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛: 𝑥,, 𝚤̂ (Unit vector notation allows summation to be MUCH easier) 𝑦 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛: 𝑦,, 𝚥̂ (Just multiply the quantity/magnitude by the unit vector) 𝑧 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛: 𝑧̂ , 𝑘4 (When placing two vectors together, just add their components together) IMPORTANT: A scalar multiplied by a scalar will produce a scalar. A vector multiplied by a vector will produce a vector. A vector multiplied by a scalar will produce a vector (dot product) Memorize the Unit Circle! Acceleration, Speed & Velocity Calculating Avg. Speed & Velocity: Remember: average acceleration = CHANGE in V / CHANGE in t Acceleration at a Glance: Acceleration is simply the change in velocity over the change in time Just because you have negative acceleration does not always mean you are going backwards, you could just simply be slowing down → Only works if acceleration is constant 2 2 Also, vf = vi + 2aΔd Remember, the area under a v vs. t graph will give you displacement Jerk: change in acceleration over change in time Airbus 380 Take-Off Distance: With constant acceleration you can do average velocity multiplied by change in time to get a displacement Why Distance is Area under Velocity vs. Time Line: Remember, if you integrate whatever plot, you got back one (integrate acceleration to get velocity, integrate velocity to get displacement) Average Velocity for Constant Acceleration: Remember, area under the velocity vs. time curve is displacement You can only do average velocity (vi + vf) when acceleration is constant (if the slope is constant then the acceleration is constant as well) Newton’s Laws & Equilibrium More on Newton’s First Law of Motion: Newton’s First Law of Motion: every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed (body at rest, tends to stay at rest) You really can’t tell if you are moving at a constant velocity or not moving at all Applying Newton’s First Law of Motion: An unbalanced force does not always change an object’s speed (but it does with regards to their velocity) Inertia: a property of matter by which it continues in its existing state of rest or uniform motion in a straight line, unless that state is changed by an external force Newton’s Second Law of Motion: Newton’s Second Law of Motion: F = ma When you double the force, you double the acceleration An object acted on by an unbalanced force will always accelerate in the direction of that unbalance force (or rather the net force) Newton’s Third Law of Motion: Newton’s Third Law of Motion: to every action, there is always an equal and opposite reaction, or the force of two bodies on each other are always equal and directed in opposite directions Center of Mass: If a force is applied to an object at its center of mass it will move in the same way as if all of its mass was concentrated on a point If you apply a force to an object on a location other than its center of mass, it will rotate around its center of mass When you apply force that results in rotational motion as opposed to translational motion, this is known as Torque Equation: Center of Mass = - !1 "2 #!2 "2 !1 #!2 Introduction to Torque: Torque Equation: T = rFsinθ (this only gives the magnitude of torque, it is a vector quantity so you would have to do a dot product) Take into account the force that is perpendicular to the object Even though work and torque have the same units, they are not the same thing because for work you want the force component in the same direction as displacement while with torque you want the perpendicular component When rotating in the clockwise direction, torque is negative and when rotating in the counterclockwise direction, torque is positive Uniform Circular Motion: motion in a circular path at a constant speed (not velocity) due to a centripetal force towards the axis of rotation Normal Forces & Tension Balanced & Unbalanced Forces: Basically objects at rest or with constant velocity have balanced forces Normal Force & Contact Force: Normal force is the force keeping stuff from falling through the surface and continuing to accelerate towards the center of the earth (always perpendicular to the surface the object is standing on) Introduction to Tension: Memorize the basic angles (sin30, sin45, sin 60, cos30, cos45, and cos60) Tension is determined by the weaker of two forces. A helpful way to think about it is considering that if only one side of the rope is pulled, the tension in the rope is zero. It’s only by having a weaker opposing force that tension is created. Forces on Inclined Planes Inclined Plane Force Components: Need to break down force of gravity into x and y components in order to calculate normal force and other forces as well Use the correct angles when you break forces down into their component parts Force of Friction Keeping the Block Stationary: Remember, friction force is simply µN (can be either the static coefficient of friction or the kinetic coefficient of friction) → If angle of incline is increased then N is decreased The friction force always occurs opposite to the direction of acceleration if there was no friction present The coefficient of static friction is simply the ratio between the force needed in order to overcome friction and normal force of the two objects Ice Accelerating Down an Incline: Normal force is what keeps the object from moving into the object/incline that it is stacked on top of often called the contact force Force of Friction Keeping Velocity Constant: Friction force keeps matching the force in the opposing direction (static friction) until you apply enough force to overcome it Coefficient of kinetic friction applies when the object is in motion (sometimes it can be different from coefficient of static friction) → CAN be lower than the coefficient of static friction (for some reason when something is moving, it is lower than when it is at rest) Intuition of Static and Kinetic Friction Comparisons: Basically, contact forces are simply the repulsion of atoms/molecules onto one another (their electron clouds don’t like one another) The reason that µs ≥ µk is likely because once something is already moving, it doesn’t really settle into the ruts of an uneven surface while something at rest will typically settle into these little divots (5:55 in the video) Remember, once you budge the object, µs is used in order to find the overall force but IMMEDIATELY AFTER, you must use µk Also, remember that when they are for gravitational force literally just do Fg = mg (that’s it, no dividing into its components) Work & Energy Introduction to Work & Energy: Work is energy transferred by a force (W = F ᐧ d or Fdcosθ → this is a dot product) and is in units of Joules (J) which is simply Nm or m2kg/s2 The cosθ is there because the only component of the force that does work is the one that lies along the same direction as displacement (the perpendicular component does NO work) If work a positive then the force is giving the object energy (force points in same direction as displacement), if work is negative then that force is trying to take away energy from that object (force points in opposite direction as displacement) Energy is the ability to do work It seems kind of circular, but that’s because they ARE related to one another There is a brilliant sort of relation between work and kinetic energy shown in this video (8:31) 1 Kinetic energy is the amount of work needed in order to get an object from rest to its current velocity (Ek = 2 𝑚𝑣 2 ) Potential energy is the potential to do work (Ep = mgh) and doesn’t have energy due to motion like an object with kinetic energy would Work Energy Principle simply states that Wnet = ΔEK Remember, if something moves with constant velocity then W is 0 because there was no ΔEK of the object Introduction to Springs & Hooke’s Law: Relationship between the amount of force applied to a spring and its compression is actually linear (within reasonable tolerance) Hooke’s Law: F = -kx (where k is the spring constant for that particular spring) → this equation gives the restorative force showing the direction of the force vector with regards to the spring) Potential Energy Stored in a Spring: Work for a spring is simply the area under the curve of a F vs. x (displacement) graph and therefore follows the formula W = 1 2 - 𝑘𝑥 2 Total work necessary in order to compress the spring by a distance of x Remember, centripetal acceleration is 𝑎$ = %2 & (can use this equation in order to find the speed at the top of a loop) You can always find the work done by a force on an object if you know the energy that the force gives or takes away from the object - - Conservative Forces: Conservative forces are any force where the work done by that force only depends on the initial and final positions of the object (path independent) and therefore conserve mechanical energy (Ek + EP) Examples include gravitational force, spring force, An example of a nonconservative force is friction and air resistance (since its path is important in determining its work) Also, it’s important to know that conservative forces have potential energy associated with them while nonconservative forces do not This basically means that nonconservative forces can not be transferred back and forth from U and K Power: Power measures the rate at which someone/thing does work 𝑃 = - ' ( (units are in Watts or J/s) This only gives average power since you would need calculus in order to give the instantaneous power If power is constant the entire time, then average power and instantaneous power are equal to one another If you know the instantaneous velocity, then you can figure out the instantaneous power by using 𝐹 = 𝐹𝑣𝑐𝑜𝑠𝜃 Introduction to Mechanical Advantage: Fulcrum is where the lever pivots and the mass is the object sitting on the lever If you have a lever, you can push twice as force for half as much force in order to move an object with double the amount of force (mechanical advantage is the factor of which you are using less force → such as using 5N to move an object of 10N, MA = 2 )*&$+ *) *-.+$( and if you use 20N to move an object of 10N, MA = ½ → 𝑀𝐴 = )*&$+ /0+1 (* !*%+ (ℎ+ *-.+$( = - 2! 2" ) Mechanical advantage is also proportional to the ratio between the two differences on either side of the fulcrum and can be summed up as F1d1=F2d2 and is actually related to the moment of inertia (cross product) For inelastic collisions, you can use conservation of momentum (p = mv) Also for finding MA of a ramp, do 𝑀𝐴 = ℎ34*(+5/0+ ℎ+67ℎ( Fluids at Rest The Buoyant Force does not get smaller as you sink: On earth, surface pressure is about 101 kPa (or 1 atm). 1 Pa = 1 N/m2 and 1 atm = 101.325 kPa Unlike with air, pressure rapidly increases with water (since it is much more dense than air) Pressure increases as you dive deeper into water because there is more and more water sitting on top of you (weighing - down on you → on another planet the pressure could be different) Buoyant force is the force that allows something to stay afloat The buoyant force is constant no matter the surrounding pressure (since water is incompressible) Archimedes principle shows the equation that gives you the size of the buoyant force 𝐹- = ⍴89(+& 𝑉0/-!9&65+ 𝑔 ⍴89(+& is around 1 g/cm3 or 1000 kg/m3 Also, the buoyant force on an object equals the total weight of water that it pushes out of the way Pressure & Pascal’s Principle: A fluid is anything that takes the shape of its container (gas or liquid) The difference between a liquid and a gas is that a gas is compressible while a liquid is incompressible (technically compressible but by only such a small degree that we don’t even consider it) Pressure is defined as a force per given area Pascal’s Principle states that any external pressure that is applied to a fluid, that pressure will be distributed throughout the fluid equally (P = F/A) Pressure at a Depth in a Fluid: Density = mass/volume (kg/m3) 𝑃 = ⍴ 𝑔ℎ where h is the depth in the liquid and rho is the density of the liquid (gives in units of Pa) Finding Height of Fluid in a Barometer: Specific gravity is the ratio of how dense that substance is to water (mercury is 13.6 meaning its 13.6 times denser) Archimedes Principle & Buoyant Force: Remember, pressure is acting in every single direction when submerged, it’s just that some vectors cancel out with one another, where only the upwards/downwards ones end up mattering The pressure at a lower part of the submerged object will be higher than the pressure on top 𝐹5+( = ⍴ 𝑔𝑉 where V is the volume of the liquid displaced (also the volume of the object) V and ⍴ can both me replaced by mass, leaving you with mass of the liquid displaced time gravity to give you the net force, stating that the weight of liquid displaced is the net force acting on the object The fact that there is more pressures on the bottom of an object than the top is why things float and is known as buoyant force To find how much of an object is submerged, use the formula 𝑉𝑠𝑢𝑏𝑚𝑒𝑟𝑔𝑒𝑑 𝑉𝑜𝑏𝑗𝑒𝑐𝑡 = ⍴𝑜𝑏𝑗𝑒𝑐𝑡 ⍴𝑙𝑖𝑞𝑢𝑖𝑑 Specific Gravity: Specific gravity of an object is simply - ⍴𝑜𝑏𝑗𝑒𝑐𝑡 ⍴𝑤𝑎𝑡𝑒𝑟 (a ratio between the density of the object to water → can even be used as a ratio to other types of liquids such as oil) and can be used in order to find the fraction of submerged object in th liquid (as shown in the equation two above) No units because they end up canceling each other out For something that floats, the specific gravity tells you the fraction of that object that is submerged Anything over 1, then the object will not float (it’s density is greater than that of water’s) Can use a ratio of: 𝐹𝐵 𝐹𝑔 = ⍴ ⍴ 𝑙𝑖𝑞𝑢𝑖𝑑 𝑜𝑏𝑗𝑒𝑐𝑡 (??) Extra Notes: Absolute pressure is simply the atmospheric pressure plus the gauge pressure 𝑃 = 𝑃= + ⍴𝑔ℎ where PO is atmospheric pressure Fluids in Motion Volume Flow Rate & Equation of Continuity: Laminar Flow is when a fluid has no resistance and is not viscous, meaning it flows freely (low velocity or when fluid is very viscous) Equation of Continuity states that the same amount of volume coming into the pipe must leave the pipe as well → Aivit = Afvft (time can be taken out since it is the same on both sides of the equation) The volume per second coming in is the same as coming out which is Volume in per unit time (also known as flux) Eddies are the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime Bernoulli’s Equation Derivation: You can derive this by starting with the idea of conservation of energy (Wi + PEi + KEi = Wf + PEf + KEf) Bernoulli’s Equation Pi + 𝜌ghi + 0.5𝜌vi2 = Pf + 𝜌ghf + 0.5𝜌vf2 Mass is not present because it can be canceled out on both sides since mass in is the same as mass out P = 0 if it is done in a vacuum (air above the fluid has nothing in it) - Viscosity & Poiseuille Flow: The viscous force (FV) only has to do with the area of the lid in contact with the fluid (kind of like friction but with objects being dragged on a liquid) and also the velocity by which you drag it along the fluid, inversely proportional to the depth of the fluid, and ɳ (coefficient of viscosity of the fluid in units of Pa*s → Ns/m2) Sometimes people is Poise which is equal to 1/10 Pa*s The coefficient of viscosity is very dependent on temperature Gases have viscosities even less than that of liquids > FV = ɳ𝐴 1 Newtonian fluids are those that have a constant coefficient of viscosity, regardless of the speed > Poiseuille’s Law is ( = 𝑄 = - ?4@& 4 8ɳA Δp refers to the difference in pressures at the start of the tube and at the end (pressure must be greater at start in order for it to flow) r is simply the radius L is the length of the tube Turbulence at High Velocities & Reynold’s Number: Once you get to a certain speed, fluids start exhibiting turbulent flow which is highly unpredictable and very difficult to describe Reynold’s Number gives you a way to predict what the critical speed is that you cross when the fluid begins to show turbulence Cɳ (𝑉$&6(6$9B = 2D&) - The R in this case is Reynold’s Number NOT RADIUS and is a unitless, dimensional quantity Surface Tension & Adhesion: Floating only occurs when the object is LESS dense than the liquid If it does float when the object is MORE dense than the liquid, then surface tension is counteracting this action Water molecules have attractions to each other due to H bonding and this is known as cohesion (same molecules attract each other) Water molecules at the surface are much more able to form interaction because they don’t have any liquid above them to pull them in every single direction Adhesion is when one set of molecules are attracted to another set (such as water molecules attracted to the container) and can lead to the formation of a meniscus (which is the curved upper surface of a liquid in a tube) Capillary action describes the fact that if you place a tube into the surface of a liquid, the liquid will rise due to the adhesion forces between the tube and the liquid Venturi Effect & Pitot Tubes: The flow rate through every section of the pipe must remain constant ( - 𝑉1 𝑡1 = 𝑉2 𝑡2 ) This means, the velocity of the fluid will likely mean that the fluid flows faster through a constricted portion of tubing If the velocity of the liquid increases, then the pressure must decrease according to Bernoulli’s equation Bernoullie’s Principle states what was said above, that when the pressure decreases within a fluid, that the fluid therefore increases its speed Two Circulations in the Body: Red Blood Cells don’t have a nucleus, mitochondria, etc. and therefore don’t really use oxygen (just carry them to a location) The heart does use blood that it pumps out (one of the first vessels that come from the heart) The lungs also has its own blood vessels in order to get oxygen Arteries vs. Veins - What’s the Difference?: Arteries carry blood away from the heart and veins carry blood towards the heart Capillaries are the very small vessels where there are a lot of exchanges of gases between the vessels and their surroundings Pulmonary Artery carries deoxygenated blood to the lungs for refuel and returns it through the Pulmonary Veins Arteries have high pressure/low volume and therefore would spurt out a lot more blood compared to a low pressure/high volume vein Resistance in a Tube: Why is it more difficult to blow through a straw with a smaller radius (same length)? 8Aɳ Poiseuille’s Equation 2.0? → 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (𝑅) = @& 4 Arterioles are covered in smooth muscle that can either be contracted (vaso-constriction) or relaxed (vaso-dilated) in order to generate different resistances based on what the body needs at the specific time - Putting it all Together: Pressure, Flow, & Resistance: 𝛥𝑃 = 𝑄𝑅 where Q = blood flow rate (volume/min) and R is resistance Another way to find Q is finding the stroke volume (which is the volume of blood per one beat) and multiple by the heart rate (beats per minute) and an example of this is shown to the right Power Output is simply Pressure multiplied by the flow rate (Power = PQ) Gas Phase Absolute Temperature & the Kelvin Scale: Temperature is the measure of the average kinetic energy of the particles within the system Kinetic Energy is known as the energy of motion 9 The °C to °F equation → TF = 5TC + 32 The Kelvin scale never has any negative values (0 is absolute zero meaning there is no kinetic energy) Pressure & the Simple Mercury Barometer: Remember, Pressure is simply a force per unit area (P = F/A) 1 atm is the same as 760 mmHg/Torr or 1.01x105 Pa (N/m2) Definition of an Ideal Gas, Ideal Gas Law: Pressure and Temperature are directly related Pressure and Volume are inversely related Pressure and moles of gas are directly related Ideal Gas Equation is PV = nRT (R is a constant) → USE KELVIN, NOT CELSIUS/FAHRENHEIT BECAUSE YOU 1. 2. 3. DON’T WANT NEGATIVES R = 0.0821 atm L/Kmol = 8.314 J/Kmol = 8.314 Pam3/Kmol An Ideal Gas is a gas that: No intermolecular forces (if there was then this would inhibit the idea that all the KE is directed towards the pressure) Molecules occupy no microscopic volume (point masses) All collisions are perfectly elastic Standard Temperature & Pressure (STP) occurs at a temperature of 0°C and 1 atm Remember, one mol of gas takes up about 22.4 L Also, remember that 1m3 = 1000L Boyle’s Law: If the number of moles and temperature is constant, then P1V1 = P2V2 Although they are multiplied together, remember that they are inversely proportional Charles’s Law: If the number of moles is constant and the pressure doesn't change, then V1/T1 = V2/T2 Although they are divided by one another, they are directly proportional Avogadro’s Law: 1 mol = 6.02x1023 molecules (or atoms) = 22.4 L @STP 1 L = 0.041 mol V/n = a (a is SOME number that is a constant which is similar to R) V1/n1 = V2/n2 The van der Waals Equation: Modification of the ideal gas law in order to take into account how REAL gases work You need to subtract how much volume the actual particles take up and how much their forces take away/give to the pressure ← Van der Waals Equation Partial Pressure: They will give you percentage of mass and you want to know how many moles there are and then you can figure out the partial pressure by inserting it into the ideal gas law Total pressure should equal a summation of ALL of the individual partial pressures Kinetic Molecular Theory of Gases Boltzmann’s Constant: For something to have a high temperature, then the average kinetic energy of the molecules are high as well PV = NkBT (N is simply the number of molecules/atoms of that substance and kB is Boltzmann’s constant) kB = R/NA = 1.38x10-23 J/K - Heat Capacity at Constant Volume & Pressure: For a monatomic ideal gas (only the noble gases), the internal energy is simply: KEtot = Uint = (3/2) PV = (3/2)NkT = (3/2)nRT In order to increase internal energy, heat up the system or increase the pressure on the system (do work) Change in internal energy is: ΔU = Q + W (Q is heat and W is work) and is actually the first law of thermodynamics which states there are only two ways in order to add energy to a system (If the gas expands then energy is decreased → -W and if - gas is compressed then energy is increased → +W) W = PΔV and only works if pressure remains constant (basically you need to heat up the system in order to keep pressure the same) Heat capacity is when heat is added, how much does the temperature increase? (C = Q/ΔT) You can do this for constant volume (CV) or constant pressure (CP) CV is constant for any monatomic ideal gas (CV = (3/2)NkB = (3/2)nR) CP is constant for any monatomic ideal gas (CP = (5/2)nR Therefore, CP - CV = nR Kinetic Molecular Theory of Gases: Macroscopic components of a substance such as pressure, volume, temperature have to do with the microscopic components, such as positions and speed Force is also: F = ΔP/Δv (where P is momentum) KEavg = (3/2)kBT Also, remember that V = Ah (Area times height) Electrostatics Triboelectric Effect & Charge: Triboelectric Effect is a type of contact electrification on which certain materials become electrically charged after they are separated from a different material with which they were in contact The charge of a proton/electron is 1.6x10-19 C (just opposite magnitude for a proton vs. an electron) Coulomb’s Law: Coulomb’s Law describes the electric force between two charges: 𝐹! = His law is very similar to Newton’s Law of Gravity which is: 𝐹& = "|$1 $2 | %2 &'1 '2 %2 The electrostatic force occurs at a much smaller scale than the gravitational force, also gravity does not repel like electrostatic forces can The electrostatic constant k is approximately 9x109 Nm2/C2 (in order to make sure Newtons are left over) Conservation of Charge: As long as no charge enters or leaves a closed space, then the charge within that area will be remain constant even if they come together and convert to other things This law works even in principles such as decay and the likes Conductors & Insulators: Although not 100% true, you can basically boil down materials into two categories: conductors and insulators In both solids, the positively charged nucleus does not move, but electrons in conductors can move around freely and not in insulators The electrons don’t move freely, you need to generate some sort of electric field If you set up an insulator to a battery, the electrons and protons can shift, even though they don’t necessarily move If you add extra electrons (charge) to a conductor, the electrons will migrate to the furthest edge in order to get as far away from one another as possible Also, taking away electrons is sort of like adding a positive Examples of insulators: glass, wood, plastics, etc Examples of conductors: gold, copper, silver (basically the metals) Charge by induction occurs when charged rod approaches a non charged rod and therefore end up having positive and negative charges attracting the two rods (don’t touch physically though) → connect to a ground which will give the second rod a net charge (cut the ground and therefore keep the net charge on the new rod) - Electric Field: Fe/q = kQ/r2 = E (just a manipulation of Coulomb’s Law) → Electric Field is in units of N/C Therefore, Fe = Eq The electric field lines radiate outward from a point charge (higher the density the stronger the field) and they never cross The direction of the field lines has to do with the direction a positively charged particles would move if placed within the electric field Electric Potential: Electric potential energy is the energy that is needed to move a charge against an electric field (work done to get it somewhere) Electric Potential (Voltage) is the difference in potential energy per unit charge between two locations in an electric field (J/C) and is therefore independent of the size of the particle (find this in Voltage video if you need further explanation) PEe = Ue = qV → this is because you need to cancel out the coulomb value to get back to potential energy) V = kQ/r so therefore Ue = (kQq)/R (Big Q is what is generating the electrical field and little q is what is being placed into the field!) Also, voltage is the DIFFERENCE in electric potentials and can be referred to in units of volts Electric Potential Energy: Remember, W = Fd which can be also be expressed as W = (Eq)d Mass of a proton is 6.67x10-27 kg Mass of an electron is 9.1x10-31 kg ` Current & Resistance Current & Resistance: Electrons actually move pretty slowly through direct current circuits Electrons don't really move in alternating current circuits - Introduction to Circuits & Ohm’s Law: Ohm’s Law connects the ideas of voltage (V), current (I), and resistance (R) → V = IR Current is simply the amount of charge per unit time (Q/tr C/s) and is measured in units of amps Conventional current is the flow of positive charge (positive terminal to negative terminal) Resistance is how much charge flow is impeded in its path and is measured in units of ohms Resistors in Series: The higher the voltage, the more that the electrons want to get to the positive terminal of the battery Assume all wires (without any resistors present) have no resistance Current is the same throughout a loop in series When in series, RT = R1 + R2 + R3 Resistors in Parallel: In parallel, IT = I1 + I2 For resistors, (RT)-1 = (R1 + R2)-1 Once you find total resistance in parallel, you can just combine them as one and make the resistance equal to the total resistance across that area that is in parallel - Resistivity & Conductivity: Resistance is a constant if it is an ohmic resistor (since if you increase voltage, current increases proportionally) Increasing distance (L) of the resistor will increase resistance Increasing the cross sectional area (A) of the resistor will decrease resistance The material that the resistor is made of can also change the resistance of the resistor (symbol for resistivity is 𝜌 in units of ohms*m) Therefore, R = 𝜌L/A Remember resistivity is difference from resistance Conductivity refers to how much something allows current to flow (shown by the symbol σ) Resistivity and conductivity are related by 𝜌 = 1/σ Therefore if something is a good conductor, its a bad resistor, and if it's a good resistor, then its a bad conductor Electrolytic Conductivity: You want to use plates in order to find the conductivity because they have a defined cross sectional area Voltmeters & Ammeters: Place the voltmeter ends on either side of a resistor in order to find the resistance drop across that resistor (in parallel) → have very high resistances For an ammeter you do NOT hook it up in parallel, you need to hook it up in series (must disconnected/break circuit in order to measure the current) → have very little resistance Extra Notes: For power, P = VI = V2/R = I2R Capacitors Capacitors Article: Capacitors store energy by holding apart pairs of opposite charges Therefore, charge on the plates Q = CV (where C is the capacitance, which is how much of a charge difference the capacitor holds when a certain voltage is applied) 1 E = 2 𝐶𝑉 2 Capacitors do not store charge, they store an imbalance of charge (net charge of both plates is always 0) The further two plates are, the less the other plate’s electrons feel the repulsion (capacitance of plates will be inversely related to plate separation If the area is larger, then adding charge to the capacitor makes it easier and is therefore capacitance is directly related to area There: 𝐶 = (0 ) * where ε0 = 1/(4kπ) A is the area of the plates, ε is the permittivity constant (determines how easily the air between the plates allows an electric field to form) Dielectrics Article: Dielectrics are materials that don’t allow current to flow (often called insulators) and even though they don’t flow When you place a dielectric substance in the gap of a parallel-plate capacitor, it increases the total capacitance (Cwith dielectric = kCwithout dielectric) The higher the k value is, the better a material functions as an insulator and therefore increases the capacitance of the capacitor, decreasing the electric field inside of the capacitor Remember, Q stays constant if the battery is disconnected first and therefore the introduction of the dielectric material will decrease the V by whatever factor the Cwith dielectric is Remember, V stays constant if the battery is still connected to the capacitor, and therefore the introduction of the dielectric will, the difference in charge will increase by the same factor as the Cwith dielectric went up by Capacitors & Capacitance: Even if the capacitor plates are different sizes, they both store equal and opposite charges Capacitance tells you how good that capacitor is at storing charge C = Q/V which is in units of C/V or F (Farads) Capacitance is stationary because you need to alter the distance or area between the two plates in order to change it Energy of a Capacitor: When a capacitor is charged up, it not only stores charge but also energy (form of electrical potential energy) which can be used to generate current (and light up a lightbulb) 1 1 F2 E = 2 𝑄𝑉 = 2 𝐶𝑉 2 = 2G for the potential energy of a capacitor and is because as additional electrons go back to the positively - charged plate, the voltage drops over time (use the ones where you KNOW two variables) If you have C in the formula you can “see” the V2 (helpful way to remember) V is only the same as the battery if the capacitor was fully charged (so a whole bunch of capacitors in sequence/parallel, you need to find the voltage across that specific capacitor Capacitors in Series: (Ceq)-1 = (C1 + C2 + ...)-1 ← For capacitors in series For capacitors in series, charge on every capacitor in series is the same Sum of the voltages of capacitors in series will equal the voltage of the battery Vtot = Q/C1 + Q/C2 Capacitors in Parallel: The voltage drop across capacitors in parallel have the same voltage drops Charges for capacitors in parallel are not the same Ceq = C1 + C2 + ... Also, you can add up the charges on the capacitors in order to come up with Qtot Dielectrics in Capacitors: A dielectric material, like paper, will be placed in between the two capacitor plates in order to 1. Keep them from touching and 2. Increases the capacitance of that particular capacitor and therefore decreases the voltage (since charges in the dielectric are canceling some of it out) If you leave the battery connected, you can supercharge the capacitor plates since the voltage decreased, the battery keeps charging the capacitor plates Dielectric constant for an insulating material is always greater than 1 since it always increases the capacitance Extra Notes: When the battery and dielectric material are taken out of a circuit and the charges are allowed to redistribute, then the charges will equilibrate on all capacitors, even though some had the dielectric material originally Magnetism Using the Right Hand Rule: Once charges start to move, they begin to experience the force of the magnetic field (which is not in the same direction as that of the magnetic field) Right hand rule: velocity is in the thumb, field is in the direction of the fingers, and magnetic force is coming out of the palm Magnetism - Part 1: A magnet always has a North and South Pole EVEN if you cut the magnet in half (conventionally called this way because it oriented itself towards geographic north) So technically, the south pole is what the north is pointing towards The magnetic field lines go from the north pole to the south pole A key difference between magnetism and electrostatics → for magnets there is always a dipole while in electrostatics there doesn’t have to be this duality - Magnetism - Part 2: The pointer of a compass needle is tangential to that of the field lines (so the compass’s north pole is facing the south pole) Magnetic field is simply the electrostatic field in motion The force acting on a moving charge is: 𝐹 = 𝑄(𝑣𝑥𝐵)→ if parallel, then the value is 0 (this is why you use 𝐹 = 𝑄|𝑣||𝐵|𝑠𝑖𝑛𝜃) B is in units of Ns/Cm or 1 Tesla (T) Remember, v goes in thumb and field is the fingers Magnetism - Part 3: If your units for magnetic field force are in C, T, and C, you’ll get newtons are your final units Magnetism - Part 4: FC = mv2/r ← centripetal force and can be used in order to find the radius of a circle that a magnetic force makes the proton go through a uniform magnetic field Remember, for an electron the magnetic force is in the opposite direction of that of a proton (could technically use the left hand rule) Also, the radius would likely be smaller because the mass of an electron is smaller than that of a proton Extra Notes: Alpha particles is basically the nucleus of a He atom (2 protons and 2 neutrons) → doesn’t do very much damage because its big and can’t penetrate things like the skin very well Beta particles are just electrons and positrons (which also emit an electron antineutrino) Diamagnetic materials create an induced magnetic field in a direction opposite to an externally applied magnetic field due to paired electrons (don’t really generate a magnetic field) Paramagnetic materials are those that create an induced field in the same direction as the external field due to unpaired electrons Electrochemistry - Electrochemistry: Galvanic/Voltaic cells traditionally are used as sources of DC (direct current) electrical power and derives its energy (current) from spontaneous redox reactions “Red Cat An Ox” shows that reduction occurs at the cathode and oxidation occurs at the anode (negative terminal) In order to go from reduction potential to oxidation potential, you just flip the sign ° = −𝑛𝐹𝐸° where n is the moles of electrons transferred, E° is the standard state EMF (V or J/C), and F is Faraday’s 𝛥𝐺 constant (105 C/mol e-) ° = −𝑅𝑇𝑙𝑛𝐾𝑒𝑞 allows us to find the equilibrium constant K where R = 8.314J/molK 𝛥𝐺 CH Therefore, 𝐸 * = 52 𝑙𝑛(𝐾+I ) ° − 𝑅𝑇 𝑙𝑛(𝑄) where Q = 𝑛𝐹 Nernst Equation allows us to use current concentrations with the law of mass action 𝐸 = 𝐸 [4&*1/$(0]5 [&+9$(95(0]6 and helps show the instantaneous cell potential at different points during a reaction (leave out solids in Q) → As Q increases, E decreases At 25°C, RT/F = 0.0257 J/C or V For other equations similar to this, look at the image to the right → Logarithm Approximate Value Actual Value log(1) 0 0 log(3) 0.5 0.48 log(5) 0.7 0.70 log(7) 0.9 0.85 log(10) 1.0 1.0 - Electrolytics cells an electrical current is used to drive a nonspontaneous redox reaction This cell can be used for the decomposition of compounds such as water, NaCl, into simpler compounds In aqueous solution, at the cathode, if a metal is more reactive than hydrogen, the reduced metal reacts with water to produce H2 gas and if the metal is less reactive than hydrogen, then the reduced metal does not react with water Electroplating, which occurs at the cathode, means we can calculate the amount of metal plated on the cathode from charge/current that passes through the cell → Qcharge of e- from battery = Qcharge of e- for reduction where It = nF where I is current, n is moles of electrons, t is time, and F is Faraday’s constant Redox Reaction from Dissolving Zinc in Copper Sulfate: Think electronegativities when comparing whether the atom wants electrons or to given them up When dealing with ionic compounds the oxidation state is the same as the charge of the ion Introduction to Galvanic/Voltaic Cells: Once a metal loses/gains electrons, it can now dissolve into water since it is now polar A salt bridge allows for neutralization of the generation of positive/negative charge forming in solution in order for current to keep flowing through the wire (positive will flow into the negative liquid and negative will flow into the positive liquid, effectively neutralizing the liquid - Electrodes & Voltage of Galvanic Cell: Negative terminal (anode) is where electrons are coming from/oxidation is occurring and positive terminal (cathode) is where electrons are going to/reduction is occurring (“Red Cat An Ox”) Standard electrode potentials is relative to hydrogen reactivity and measures different ions (gives you values based on reduction half reactions) Add up the potentials of the terminals in order to come up with the voltage of the entire battery E° = Ered + Eox Standard reduction potential of hydrogen is 0 V Positive standard reduction potentials lead to stronger oxidizing agent and negative standard reduction potentials lead to strong reducing agents Shorthand Notation for Galvanic/Voltaic Cells: Zn(s) | Zn2+(aq) || Cu2+(aq) | Cu(s) ← anode is the left and cathode is the right (a comes before c) Voltage as an Intensive Property: ΔG° changes based on how many moles you have (extensive property) but voltage does not change (same voltage to form 2 moles of a substance as it is to form 1 mole of a substance and is therefore an intensive property) → NEVER MULTIPLE VOLTAGE BY 2 IN THE STANDARD ENERGY EQUATION EVER If E° is negative, then ΔG° is positive, and therefore nonspontaneous Concentration Cell: When the concentrations of liquid are different, the concentrations want to equilibrate and become equal, which is enough to generate x amount of current When concentrations are equal, then E = 0 Introduction to Electrolysis: For an Electrolytic cell, an electrical current is used to drive a nonspontaneous redox reaction (E° < 0) and therefore needs a battery in order to drive the reaction (you need a voltage greater than the E°) Reduction occurs at the cathode and oxidation occurs at the anode (always) Cathode in this case is negative and the anode for this case is positive (opposite for galvanic cells) Quantitative Electrolysis: Remember, A = Q/t in units of C/s and you can find the charge, then use Faraday’s (C/mol e-) constant in order to figure out the moles of electrons gained/lost Sound Sound is a Longitudinal Wave: The energy from the source of a sound is what travels to your ears, not the original particles that made the sound Basically, a density vs time graph is the same as a pressure vs time graph Sound waves only travel in space if there are enough particles around to transmit the energy in the wave from the source to the listener The speed of sound is 340 m/s Production of Sound: Remember, sound is simply the transfer of energy across particles in space that eventually get to your ear (disturbance of air) Waves can transport energy through a medium without transporting the medium itself Sound Properties: Amplitude, Period, Frequency, & Wavelength: If you increase amplitude, volume increases If the period (T) is decreased, then the frequency (f) increases, leading to a higher pitch (f = 1/T and is in units of 1/s, which is also known as a Hz) Humans can hear frequencies from 20 Hz to 20,000 Hz Speed of Sound: Transverse wave has particles that are displaced perpendicular to the direction the wave travels Longitudinal wave has particles that are displaced parallel to the direction the wave travels (i.e. sound) 𝑠𝑝𝑒𝑒𝑑 = 𝛌 H = 𝛌𝑓→ works for all kinds of waves, like those on the electromagnetic spectrum (increasing/decreasing wavelength/frequency won’t alter the speed of sound because the other variable will just compensate for the change) The only way to change the speed of sound is to alter the medium with which the sound is traveling in (i.e. you can also change temperature, humidity, density of air or change the composition of the air entirely) Changing amplitude of the sound wave ONLY changes the volume NOT its speed Relative Speed of Sounds in Solids, Liquids, & Gases: The stiffer the medium, the faster the sound wave will travel through it The denser the medium, the slower the sound wave will travel through it (i.e. hot air is fast vs cold air is slow) Speed of sound is almost always faster through solids, then liquids, and then gases Decibel Scale: The decibel scale tells you how loud something is with equation: 𝛽 = 10𝑙𝑜𝑔10 ( M 10712 '/!2 ) where β is the # of decibels, I is the intensity of the soundwave (Power/Area), and the 10-12 W/m2 is the threshold of human hearing (the softest sound you can hear) Why do sounds get softer?: People closer to the speaker hear the sound as louder because of higher intensity (power/area) → area is a lot smaller than if you - - are farther from the speaker (your ear only measures a sample of the sound dispersed evenly throughout the area) Intensity is proportional to 1/r2 Also, as distance is covered, energy is lost and therefore the amplitude decreases over distance (AKA attenuation) Ultrasound Medical Imaging: Ultrasound has to do with frequencies above 20,000 Hz Ultrasounds use high f and low wavelengths because there is less diffraction (lower amount of waves spread out which would curve waves and confuse the transducer) → also since you can’t hear it, then the patient is more comfortable The lower the liquid in a tube is, the lower the pitch will be Doppler Effect Introduction: The Doppler Effect is an increase (or decrease) in the frequency of sound, light, or other waves as the source and observer move toward (or away from) each other. The effect causes the sudden change in pitch noticeable in a passing siren, as well as the redshift seen by astronomers Equations for source moving towards you → Equations for source moving away → Extra Notes: Inverse square law: intensityA/intensityB = (distanceB/distanceA)2 Light & Electromagnetic Radiation Electromagnetic Waves & the Electromagnetic Spectrum: Changing electric fields can induce a magnetic field and magnetic fields can induce an electric field Light is an example of an electromagnetic wave The speed of a wave equation: 𝑐 = 𝜆𝑓 (sometimes frequency is shown as 𝓿) - Polarization of Light, Linear & Circular: Polarized light is light where the electromagnetic waves are only oscillating in one direction Polarizers let light through but in only one direction (you can orient it in basically any angle you want) When light reflects off a surface, it is typically partially polarized in the orientation of the surface it bounced off of Diffraction & Constructive/Destructive Interference: Constructive interference occurs when two waves line up perfectly and combine to make a wave with the sum of the amplitudes Destructive interference is the opposite and occurs when two waves that are not in phase with one another come together (makes amplitudes smaller) A laser is basically just a bunch of light waves with the same wavelength that are all lined up with one another - Photon Energy: Energy of a photon: Ephoton = hf (where f is sometimes called v) h is Planck’s constant, which is 6.626x10-34 Js Infrared & Ultraviolet/Visible Spectroscopy Introduction to Infrared Spectroscopy: Wavenumber (v with a tilde over it) is: 𝑣 = - - 1 + → inverse to wavelength c = 𝜆v (wavelength x frequency) In an IR Graph, the left half (<1500 cm-1) is known as the diagnostic region, meaning it can be indicative of certain functional groups 2100 cm-1 shows the presence of a triple bond Fingerprint region is unique to each molecule, but is much harder to decipher Only vibrations that produce a change in the dipole moment are observed as signals on an IR spectrum Bonds as Springs: For frequency of a spring, use the equation to the right → µ is the reduced mass (use if there are two masses) Double bonds need to double the k value (approximately) IR Spectra for Hydrocarbons: 3300 cm-1 shows up for sp hybridized Carbons, 3100 cm-1 shows up for sp2 hybridized carbons, and 2900 cm-1 for sp3 hybridized carbons Signal Characteristics - Intensity: About 1700 cm-1 is a carbonyl bond stretch (if the signal is strong) For symmetrical compounds, the dipole moments cancel out with each other and therefore do not often show on the spectra Signal Characteristics - Shape: OH groups typically show broad peaks If there is steric hindrance for hydrogen bonding, then there won’t really be broad signals 1° Amines will have one peak at about 3300 cm-1 while 2° amines will have two peaks, an extra at 3400 cm-1 (same thing can happen with other molecules, just need the option of symmetric and asymmetric stretches) UV/Vis Spectroscopy: E = hv = hc/λ Increased conjugation means increased absorbed wavelength of light