Physics equations and problem types, Study notes of Physics

Download Physics equations and problem types and more Study notes Physics in PDF only on Docsity! Translational Motion I. Vectors vs Scalars a. Scalars: length, time, mass, distance, speed, (kind of: energy) b. Vectors: displacement, velocity, acceleration, force II. Motion Formulas: a. vav = Δx / Δt b. x = vavt c. vav = (vf + vi)/2 d. a = Δv / Δt = (vf - vi)/t e. vf 2 = vi 2 + 2aΔx f. Δx = vit + ½at2 III. Types of problems: a. Trajectory of ball b. Trajectory off cliff c. Free fall Forces I. Newton’s 3 Laws a. 1 – law of inertia – object in motion stays in motion, object at rest stays at rest unless acted upon by external force b. 2 – F = ma c. 3 – every action has an equal an opposite reaction i. Understand that things can move because action reaction pairs are acting on different objects ii. Rockets work in vacuum because if push off II. Gravity a. F = GMm/r2 b. Weakest of the fundamental forces (‘four’ces because there are four). Only accounts for attractive force between bodies. c. Normal force is actually electrostatic force pushing back d. Terminal velocity i. When the force of air resistance is equal to the weight of the object in free fall, the object is said to reach terminal velocity because it stops accelerating. That object is now in equilibrium and no forces are acting on it. This is the highest velocity it will go. III. Uniform circular motion a. Diameter = dπ = 2πr b. Radian -> 2π = 360 degrees c. a = v2/r d. velocity is the tangential velocity at any given point e. Centripetal force: ma = mv2/r i. When thinking about centripetal force, think of it as an outward centrifugal force for calculations when you are balancing forces. Centripetal force isn’t actually a new force, it’s just the sum of the inwardly facing forces of a circular path. ii. Ball on string in horizontal direction: T = mv2/r ; there is no other force acting on the ball so all of the inward force comes from the centripetal force. But if it isn’t completely horizontal and it is at an angle with the horizontal – then the vertical component of the Tension = mg and the horizontal component of the Tension = mv2/r iii. Ball on vertical string: At what point is the tension the highest? When you are the top of the loop, the sum of the inward forces is the Tension + mg. So mv2/r = FT + mg or FT = mv2/r – mg. When at the bottom of the loop, Tension points up while mg points down, so they oppose each other (and since you are still moving inward toward the circle, T is bigger) so mv2/r = FT – mg or FT = mv2/r + mg. Tension is highest at the bottom. 1. It may help to picture the centripetal force as a centrifugal force from the reference point of the object. When at the top of the circle, mv2/r is the outward force and opposes the two inward forces (T and mg). When at the bottom of the circle, mv2/r and mg are outward forces opposing T. That’s why T is bigger iv. Normal force: Where is your weight the highest? Equator or North Pole? At the North pole, the only force acting on you is mg, so the normal force (and your weight) is mg [N = mg]. At the equator, since you are spinning with r = the radius of the earth, the sum of forces, mv2/r = mg – N because technically you are accelerating toward the center along the circular path. As such, your weight is actually [N = mg - mv2/r] or slightly smaller than mg. So you weigh less. To conceptualize, think about the centrifugal force pushing you away from the earth slightly so you are slightly less heavy f. Frequency = amount of revolutions in 1 second g. Period = Time it takes for one revolution h. Angular velocity. If you travel inward toward the center of the circle, your r decreases so your tangential velocity decreases (why you don’t get thrown out of a merry go round on the inside), but your angular velocity (the amount of angle you traverse in an amount of time) does not decrease. If a 90 degrees = π/2 in radians and represents ¼ of the arclength of a circle (1/4 * 2πr = r* π/2), then S arclength = r*θ or θ = S/r. i. ω = Δθ / Δt = the angular velocity, so ΔS/r * 1/t = ω. ΔS/t = v, so ω = v/r ii. This explains why you feel less acceleration at the center of a merry go round. a = v2/r would make it seem like a goes up when r goes down, but in reality, your b. a single upward tension here means that there is no mechanical advantage c. in comparing the two pulleys, B has two upward tensions so it requires half the force (but twice pull length) the move the weight d. count the number of upward tensions. This should be (6) and so you would only need 500N of force to pull this up. Momentum, Torque, Equilibrium I. Center of mass a. Usually treat things as point masses, but if the problem has torque then the center of mass becomes relevant. Center of mass for simple objects are at the geometric center (such as a sphere), but for a donut it is not in the shape at all (in the hole). b. For problems that have center of mass being relevant, the density or shape is nonuniform. Simple problems will likely include hanging ruler with weights or seesaw. Massless objects do not matter, but you may need to take into account the center of mass of a massed object. That will generally be at the geometric center for these types of problems. c. Center of mass formula: xcenter = (m1x1 + m2x2 + … ) / (m1 + m2 + …) II. Equilibrium a. Something is in equilibrium when no net forces acting on it. For translational motion, that means sum of the forces = 0, a = 0. The object can be at rest or constant velocity b. Note that if something is moving at a constant velocity with a frictional force of 50N opposing the motion, the constant velocity still means net a = 0, so there is no net force acting on it. c. 3 types of equilibrium > static: no velocity/movement; translational: Fnet = 0; rotational = torque = 0 i. Translational equilibrium > when the object’s center of mass is moving at a constant or zero velocity, it is in translational equilibrium. If you had a fulcrum NOT at the center of mass for an object and it rotates, then the center of mass is accelerating and is NOT in translational equilibrium III. Torque a. Torque is the measure of how much a force rotates an object about a fixed point. Since there is a force making it rotate, note that this force will make the object accelerate rotationally. It is measured in Newton meters Nm b. Torque equation: τ = rFsinϴ i. Usually the equation is r x F (cross product) since you are really only considering the Force that’s perpendicular that would cause it to rotate. If the Force were applied parallel, no rotation. c. Equation can also be given by τ = F * l where l = the lever arm. Lever arm is the shortest distance between the pivot point and the line of action. If they give you the lever arm distance, then use that instead. d. Notice that the further you apply the force, the less force you need (or the greater the force exerted if using the same force) to make rotation (should be intuitive), but you need to move the point further. IV. Momentum a. Important concept is that when things collide, the momentum is conserved. These are vector quantities so make sure you don’t mix up your signs. Otherwise intuition serves well in this area. b. Momentum equation: p = mv c. Impulse measures the change of momentum of an object. It has the same units as momentum (kg m/s). Since Force = kgm/s2, you can multiply by t to get J (impulse) = Δp = FΔt = mΔv = mvf - mvi d. Conceptualize for MCAT: Injuries occur when you experience a massive force. If a two runners with the same mass and velocity fall and one tumbles but the other hits the ground, they both have the same impulse (mv = mv). While FΔt is the same for both parties, since Δt is much smaller for the second runner, he experiences a much higher force. e. Conservation of momentum – for any action reaction pair, the total momentum is conserved, so change in momentum = 0. Imagine radioactive decay where He is ejected. The parent nucleus must recoil with a momentum equal to the ejected particle. Momentum is only conserved in the absence of other forces (for instance it is not conserved if friction is taken into account) f. Collisions: there are three important types of collisions – note that momentum is always conserved in collisions – only way to have no conservation is if you lose force to friction i. Elastic collision: Kinetic energy is also conserved. The energy transfer is perfect and lossless. Think of two rubber balls bouncing off each other. Never assume this is the case unless it is specified (doesn’t occur in real world) ii. Inelastic collision: There is some loss of energy from deformation/heat loss. Think of a car hitting a bike and denting both (work done to dent is energy lost). iii. Perfectly inelastic: loss of energy and objects are stuck together afterward and move together. Think of a truck crushing a car and they keep driving g. Conservation of momentum: m1v1i + m2v2i = m1v1f + m2v2f > note that for perfectly inelastic, the right half is (m1+ m2)vf c. Note that the restoring force is F = -mgsinϴ d. IV. Wave Properties a. Transverse wave: direction of propagation of wave is perpendicular to the wave’s motion. Any single point on the wave is moving up and down, but the wave propagates to the right. Think of a rope that’s fixed. If you wiggle up and down, the wave travels down the rope but at any point on the rope it is moving up and down. i. Important types: Light, emf, string b. Longitudinal wave: direction of propagation of wave is parallel to the vibrations. Think of a slinky – if you pull a slinky, any point on the slinky is moving in the same back and forth direction as the propagation of the wave. i. Important types: sound, pressure, earthquakes c. v = fλ i. Concept: Speed of wave is determined by medium, not frequency. This is why when you change f, you change λ, but not v. Think about sound – speed of sound isn’t faster for 20Hz for 20,000Hz. ii. Concept: When a wave hits a new medium (only thing that affects its speed), the speed is affected but not the frequency. iii. v on a string: v = √(Tension/linear density). Probably won’t need to know, but you can see how the properties affect wave speed. d. Interference i. Waves can interfere. When peaks meet peaks, you get constructive interference. You just add the waves at any given position. Full constructive is when they are completely in phase. Full destructive is when they are completely out of phase (180 or pi out of phase). ii. When frequencies are not identical, they will not fully interfere. But there will be regions where they have troughs and peaks together. The troughs are soft and form beats. This is called beat frequency and happens when you tune something. fbeat = |f2 – f1| e. Standing waves i. Standing waves look like they’re standing still and have fixed troughs and peaks. ii. Nodes are the A = 0, antinodes are the A = max. f. Resonance – at specific frequencies (dictated by the object’s natural frequency) if you apply a vibration to it you can make it oscillate with greater amplitude at those frequencies. You can impart a lot of vibrational energy (enough to shatter a wine glass, or heat up a carbonyl bond to show on an IR) g. Harmonics of standing waves i. Know first/fundamental harmonic, second harmonic/1st overtone, etc ii. Fixed on both ends so you have two nodes. Fundamental harmonic is half of a full wavelength, so L = ½λ or λ = 2L. Second harmonic is a full one so L = λ. [Will be the same as both ends open, just with nodes and antinodes reversed] iii. fixed on one end so first harmonic is L = ¼λ, or λ = 4L. Second one is L = ¾λ (notice it’s never a full wavelength from peak to peak) Fluids I. Properties a. ρ = mass/volume = kg/m3 or g/cm3 b. specific gravity = density compared to water i. ρwater = 1kg/L or 1g/mL ii. specific gravity is useful because it tells you how much of something will be submerged in water (ice has a specific gravity of .917 so 91.7% of ice is submerged in water) c. Pressure = Force/Area. Pressure = N/m2 = 1 Pa (pascal). i. 1 atm = 100kPa ii. Force: F = mg; Density ρ = m/V 1. F = ρVg 2. Pressure = F/A = ρVg/A; V/A = length (or depth in a fluid) so the pressure a liquid exerts on an object is Pgauge = ρgD iii. Gauge pressure vs total pressure 1. Gauge pressure will tell you the pressure of the liquid (which is ρgD) 2. Ptotal = P(system) + Pgauge a. If the system is exposed to air (like a sea diver) then the total Ptotal = Patm + Pgauge b. If the system is closed, the pressure of the gas above the liquid provides the P(system) which could be 0 if there is no air above the tank or if there is a vacuum. iv. Note that pressure for an object submerged in a liquid depend only on the density of the liquid (ρliquid) and depth – it doesn’t matter what shape the container is or how much total water is above your head. d. Surface tension – attraction between molecules of a fluid/solvent create surface tension. Because the molecules are pulling on each other, higher surface tension means lower surface area. II. Buoyancy a. Archimedes principle – “the magnitude of the buoyant force is equal to the weight of the fluid displaced by the object” i. In other words, the buoyant force is given by the weight of the liquid that is displaced, not the weight of the object. ii. FB = ρliquidVsubmergedg iii. For Buoyancy, remember that mg (or ρobjectVobject) is your downward force while ρliqVsubdg = is your upward force. If mg > ρliqVsubdg then the object will sink. iv. Apparent weight: weight – FB v. Float: FB = weight Sink: weight > FB Rise: FB > weight III. Hydrostatic Pressure a. Pascal’s Law: Pressure in = Pressure out b. Since the pressure exerted on the liquid becomes the pressure on the other end, then F1 = A1/A2 * F2. If A1 < A2 then F2 > F1. Or in other words, a small input force on a smaller area creates a higher output force on the place with the larger area c. However, because the total work ends up being the same (F1d1 = F2d2) then d2 < d1 . That means that you raise the side with the greater area much less (the total volume moved is the same) IV. Hydrodynamics a. Flow rate = the volume of liquid that passes a particular point per unit time i. Know the difference between flow rate and flow speed. Flow speed = how fast the water moves out of the hose when it’s open (slower) vs pinched (faster) but the flow rate is the same (liquid output per second) ii. Expressed in m3/s iii. Flow rate can be calculated by knowing how fast a liquid is travelling and the cross sectional area of the point it passes through, or f = Av iv. Since the flow rate cannot change (the water in a pipe isn’t stopping for the water in front that’s going through a smaller opening) so a smaller opening means a faster flow: A1v1 = A2v2 b. Bernoulli’s Equation i. Bernoulli’s equation describes the conservation of mechanical energy for a flowing liquid ii. Bernoulli’s equation only applies to ideal fluid flow! There are four requirements: 1. Fluid is incompressible 2. Negligible viscosity 3. Laminar (non-turbulent) flow 4. Steady flow rate iii. Equation: P + ½ρv2 + ρgh = constant or P1 + ½ρv1 2 + ρgh1 = P2 + ½ρv2 2 + ρgh2 iv. Essentially KE + PE = KE + PE where gravity is providing the potential energy and the flow is providing the kinetic energy c. Bernoulli/Venturi Effect i. By the Bernoulli equation, if h1 = h2, then the side which is faster much have lower pressure. Thus, if a fluid has a faster flow speed v, then the pressure is lower. Think of a shower, when you turn on the water (and the air starts moving faster) then the pressure drops so the curtain pulls into the shower. d. Viscosity and Poiseuille’s Principle i. Viscosity is a measure of the friction within fluids. A highly viscous liquid (like honey) would oppose the flow of the liquid. Viscosity is higher for colder fluids ii. Poiseulle’s Principle – flow rate is affected by viscosity, length of tube, and radius of tube. iii. Important takeaway – know that Q (flow rate) ∝ r4 Q ∝ 1/L Q ∝ 1/viscosity And especially for blood flow, that r = radius of blood vessel and L = length of tract. Narrow vessels mean much less blood flow REMEMBER – a larger radius gives a much larger flow rate, but much smaller flow speed. If r goes to ½r, then flow speed increases by 4 but flow rate decreases by 16 times. e. Turbulent vs Laminar flow i. Laminar flow is streamlined flow while turbulent is the chaotic flow ii. Reynolds number is used to predict turbulent flow iii. Think of white river rafting for turbulent flow – what causes turbulent flow? 1. Obstruction (rocks/plaques in arteries) 2. Flow speed (faster flow increases the likelihood of turbulence) a. Av = Av so if you decrease the area, you will increase the flow speed. So decreased area also increases turbulence 3. Decreased viscosity (honey will have a more laminar flow than water if at the same speed) V. Solids a. When subjected to various forces, solids can also change shape b. Stretching forces, Compressing Forces, Bending Forces – these forces can strain an object until it reaches an elastic limit, at which point it gets permanently deformed. c. Stress: pressure exerted on an object, which is given by σ = F/A d. Strain: the change as a result of the stress ε = ΔL/L0 e. Shear: bending force that f. Young’s modulus is a constant like the spring constant for an object until it permanently deforms, at which point it no longer applies. Y = Stress/Strain g. Stress = Modulus * Strain (or shear) h. F/A = Y * ΔL/L0 i. Think of it in terms of Hooke’s Law. F = kx, F = Y(A/L0)ΔL. Since Y is constant, Area is constant, L0 is constant a. Source charges exert a force on nearby charges. So if you have a source charge that has electric field pointing outward, and you place a positive charge near it, it will push that charge. There is a potential energy much like gravity, and an object will move from higher potential energy to lower potential energy (and pick up kinetic energy) b. Electric potential: V = kQ/r i. This is described as the voltage, or the electric potential between two points. Charges will flow from higher potential to lower potential (or higher voltage to lower voltage). Volts are given in J/C c. Electric Potential energy: ΔPE = qΔV d. V = Ed e. A dipole in an electric field will align itself with the field (MRI) i. IV. Induction a. V. Gauss’s law – just understand that a point charge will send out electric field lines. If you choose an arbitrary surface area around that charge and measure the electric field lines coming out of it (the flux), you can determine the amount of charge in that space. Magnetism I. Properties a. Magnetic field given by B and measured in Tesla (T) b. Magnetic field lines go from North Pole to South Pole c. F = qvB sinϴ = ILB sinϴ d. The force is always perpendicular to the magnetic field and the velocity e. For a positive charge, use the right hand rule. Thumb is the velocity of the positive charge, index is the direction of the magnetic field, rest are in direction of force. f. If charge in wire, the direction of current is the thumb and the electric field wraps around it in a circle like your fingers in your right hand. i. qv becomes IL (or current times length of wire) because I = q/t and v = L/t g. If charge in a solenoid, then your wrapping fingers are the current and the thumb is the B field II. Practical a. Velocity selector – if you pass a charged particle through an electric field capacitor, then the particle (let’s say positively charged) will move in the direction of the electric field. You can then set up a magnetic field such that the force pushes the opposite way so the charge is deflected by the magnetic field as well. Since F = qvB = qE, if the forces are equal, the particle isn’t deflected at all and goes straight through the middle. So you can select for v = E/B b. A mass spec first uses a velocity selector so that you know all the particles entering the mass spec have the same velocity. Since that is constant, then you can deflect them in a third chamber. If qvB = mv2/r and the initial velocity of these was constant, then a bigger m means a bigger r and you can separate particles by their mass based on how big the r was in their deflection path. III. Flux Circuits I. Current a. I = ΔQ/ Δt or amount of charge flowing through a cross section of a conductor b. Conventions: would be direction of positive flow i. Even though the actual charges that move are electrons c. Units: Ampere (A) or C/s II. Voltage/emf a. Not really force, but a potential difference with the units of voltage b. Positive charges flow from high potential (higher voltage) to lower potential (lower voltage) c. Batteries are a source of emf. They have a positive cathode and negative anode (by convention – this is because positive charge flows from positive to negative while negative charge flows from the anode to cathode. Because oxidation occurs at the anode (loss of electrons) it actually gets positively charged and will attract anions – think of a gel) d. Batteries have internal resistance such that the true potential difference in a battery is emf – voltage drop due to internal resistance. Together this is called terminal potential (potential between the terminals) III. Resistance a. V=IR b. R = ρL/A i. ρ is the resistivity of a material, which is a constant based on the material. A bigger cross sectional area will let me charge through, so it reduces resistance. A longer wire will be more material to traverse so it increases resistivity c. Power is dissipated by a resistor and given off as heat d. Circuit problems i. RULES: 1. Junctions – The sum of all currents entering a junction must equal the sum of all currents leaving a junction (currents don’t change unless they hit a junction) 2. Loops – the sum of potential changes around any closed path must be 0. Emf raises the potential up and as it goes around the circuit it must end at 0. e. Resistors in series i. RT = R1 + R2 +… Resistors in a series add up. 1. Series increase resistance 2. Larger resistances = larger total resistance ii. All resistors in a series share same current (no junction to split current) iii. Each resistor dissipates voltage. Since I doesn’t change across resistors, the voltages must be different f. Resistors in Parallel h. Charge/Discharge of a capacitor i. When the circuit is first closed, you have electrons flowing from the negative anode of the battery to the negative plate. This will repel electrons from the positive plate to move toward the positive cathode of the battery, inducing the charge buildup on the capacitor. It’s easier at first to build up charge but soon the negative buildup resists more electrons flowing toward the negative plate and the movement of electrons slows down. Using appropriate conventions, it means the positive current going from the positive battery cathode to the positive plate is fast at first, and ultimately slows down as it is harder to buildup charge. It experiences exponential decay. Voltage or charge (however you want to think about it, since building up the charge induces the voltage potential across the plates) climbs quickly at first and builds up, but once flow is opposed, it slows down until it hits the max voltage. ii. When the circuit is open to the resistor that is going to be using the stored charge of the capacitor, you have a large initial burst of energy as the electrons rush to the opposite side plate (or current rushes to the negative plate) through the resistor, but soon the attractive force is reduced because the total charge on the plates is reduced, so it also decays exponentially. iii. Current goes through the path of least resistance first, so it will preferentially go to the capacitor first rather than the resistors. The current will drop off quickly to the capacitor as it charges and then start going to the resistors. Voltage across the capacitor increases gradually as charge is built up in the capacitor. VI. Conductivity a. Conductivity is the inverse of resistivity. An object or materials conductivity is high when there is low resistivity (and thus increases with higher area and shorter length). b. Conductivity is driven by electrolyte concentration. There is an optimal point where ions are mobile and there are enough charges to elicit conductivity. Too few electrolytes means low conductivity, too many electrolytes means too much crowding and low movement c. Temperature control i. Metals – conductivity decreases as temperature increases (cold metals are better conductors) ii. Semiconductors – conductivity increases as temperature increases iii. Superconductivity – some materials have superconductivity at incredibly low temperatures and have no resistance (current loops forever) iv. In a solution, a capacitor will discharge because the solution or medium is conducting some charge. You can measure the conductivity of a solution this way. VII. AC vs. DC a. Every formula prior was for a direct current. When you alternate a current by constantly switching the direction of the flow, you create alternating current. b. You need to use root mean squared formulas for current and voltage i. Irms = I / √2 and Vrms = V / √2 = .7 VIII. Ammeter and Voltmeter a. Ammeter measures the current. Objects in a series experience the same current (I isn’t split). So you plug it in a series and set the resistance insane low so it doesn’t affect the resistance of the entire circuit (and thus the current). b. Voltmeter measures the voltage drop. Resistors drop the voltage (closed loop drops voltage down to 0) so you don’t want to connect it in a series. You want to connect it in parallel, since parallel resistors do not drop the voltage, just split the current. So you connect it in parallel, but since you don’t want much current to be diverted through it, you put it at very high resistance. This way, the 1/R value is so low it doesn’t change the resistance of the whole circuit Light I. Electromagnetic Spectrum a. Changing electric field induces a changing magnetic field which induces a changing electric field etc… They are always perpendicular to each other and then perpendicular to the direction of propagation. These together create light b. Light exists in a wave particle duality so it can act like a quantized photon and also like a wave c. d. Formulas i. v = fλ ii. c = 3x108 m/s iii. E = hf II. Polarization a. Light from sources have waves in all directions that is not polarized i. b. Polarizers are filters that only allow beams that are oriented in one direction i. c. Based on the angle and material, if light is reflected off something, it becomes polarized. That’s why polarized sunglasses are good at reducing glare Optics I. Reflection a. Reflection occurs when a light wave strikes a boundary between two media. There will be some amount of light that is reflected from the plane surface such that the angle of incidence equals the angle of reflection. Note that the angle of incidence and reflection are measured from perpendicular to the plane, not to the plane! i. b. Mirrors are a special case of reflection where all light is reflected from every angle II. Refraction a. When electromagnetic radiation goes from one medium to another, it is slowed down (only light in a vacuum travels at c) b. The index of refraction is given by n where n = c/v and is always greater than 1. c. Recall that the frequency of a wave does not change when it enters a new medium. The velocity changes here, and so does the wavelength, but not the frequency III. Snell’s law a. When light strikes the boundary of the medium at an angle, it will be refracted into the new medium with a different angle. The relationship is governed by Snell’s Law i. n1 sinϴ1 = n2sinϴ2 ii. b. Apparent depth – thought exercise IV. Dispersion a. White light through a prism will split the light into a rainbow because each index of refraction varies by wavelength. i. If v = fλ such that v is directly proportional to λ, and n = c/v such that v is inversely proportional to n, then n and λ are inversely proportional. That means that a bigger λ means a smaller n and a smaller λ is a bigger n. So red has a smaller n and is refracted less. V. Total internal reflection a. When you go from a medium of higher index of refraction to a lower index of refraction (such as from water into air), there exists a critical angle where the refracted ray is incident at 90o and does not appear to leave the water. Any greater angle of incidence would be reflected internally instead of refracted. i. If you think about it, the refracted angle cannot be greater than 90 or it will just go back into the water. ii. iii. Remember that Total Internal Reflection can only occur when you go from a greater index of refraction to a lower one. 1. Critical angle: n1sinϴc = n2sin90o 2. Once you have the critical angle, anything greater than ϴc gets reflected at the same angle as the incident angle VI. General Optics Rules a. Real vs Virtual image i. For mirrors, same side as object is real, opposite side is virtual. For lenses, opposite side is real and same side is virtual. ii. UV-IR = upright virtual, inverted real b. 1/f = 1/o + 1/i c. m = -i/o d. Conventions: if m>0, upright, if m<0 inverted e. If |m|>1 enlarged, if |m|<1 reduce VII. Mirrors a. Plane mirrors reflect perfectly, such that i = -o. There is no focal point, the image is always upright, virtual, and the same size. This is because m = -i/o = 1 (same size, upright) and since you see the image on the opposite side of the mirror but the light rays are reflecting back toward you, the image is virtual. b. Spherical mirrors – there are two types of spherical mirrors you need to know. i. Concave mirrors – these mirrors are concave and also converging since they converge a beam. The focal length is always positive since it’s always on the same side as the object. ii. Convex mirrors – these mirrors point light rays away and thus are diverging mirrors. Focal length is always negative as it is always on the opposite side of the object (or appears to be). These images are ALWAYS upright and virtual and cannot be projected onto a second image. iii. f = R/2 iv. For concave mirrors, image depends on location of object. Using 1/f = 1/o + 1/i in 5 cases: If you had an image at twice the focal length, then 1/f = 1/2f + x, so x must equal 1/2f (because 2/2f = 1/f). so 1/f = 1/R + 1/R 1. Object beyond R. If object is beyond R, then 1/o got smaller so 1/i must get bigger. That must mean that i must get smaller (but 1/i still has to be smaller than 1/f so i>f) a. Real, Inverted, Smaller; i between f and R 2. Object at R. If object is at R, then i must also be R. Since m = -i/o, the image must be inverted, real, and same size. 3. Object between R and f. If o gets smaller, then i/o got bigger, which means that 1/i needs to get smaller. That means that i gets bigger. a. Real (-i/o is negative), Inverted (-i/o is negative), enlarged and farther (i>o) 4. Object at f. 1/f = 1/f + 1/i. for this to be true i has to equal infinity, so the rays converge at infinity/never converge. No image formed. 5. Object inside f. Since o is smaller than f, 1/o is bigger than 1/f, which means that 1/i must be negative and subtracted from 1/o. If i is negative, the image forms on the right of the mirror. a. Virtual (-i/o is positive), upright c. Lenses