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)/ d. a = Δv / Δt = (vf - vi)/t e. vf^2 = vi^2 + 2aΔx f. Δx = vit + ½at^2 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/r^2 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 = v^2 /r
d. velocity is the tangential velocity at any given point e. Centripetal force: ma = mv^2 /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 = mv^2 /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 = mv^2 /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 mv^2 /r = FT + mg or FT = mv^2 /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 mv^2 /r = FT – mg or FT = mv^2 /r + mg. Tension is highest at the bottom.
- 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, mv^2 /r is the outward force and opposes the two inward forces (T and mg). When at the bottom of the circle, mv^2 /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, mv^2 /r = mg – N because technically you are accelerating toward the center along the circular path. As such, your weight is actually [N = mg - mv^2 /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 = v^2 /r would make it seem like a goes up when r goes down, but in reality, your
- For a pendulum, remember that tension is always highest at the bottom because Tension has to match centrifugal force AND gravity. Treat the tension as the axis and the gravity breaks into components. VI. Mechanical Advantage a. Mechanical advantage is reducing the amount of force but not reducing the amount of work (like a gear turning). You increase the distance so you have to apply less force to get it done. If you had to lift 100kg up 5m, it would require 1000N of force and 5000J. But if you push it up an incline plane, you could apply a force of 100N over 50m and get to the same place. b. Mechanical advantage = weight of object/applied force c. Machines you will have to understand: Lever and fulcrum, incline plane, pulley, hydraulic lift. For this section, only incline plane and pulley are relevant. VII. Incline plane problems a. Just remember that gravity component = mgsinθ and N = mgcosθ. b. If you get a pulley system question, split the force diagram and remember that a 1 = -a 2 and Tension is equal in both equations. In a lot of these questions, plug in extremes using limiting cases (such as even weight or m=0) to try to solve for the equation if possible. VIII. Pulley problems a. The key to understanding pulleys is to count the number of upward forces of tension are supporting the weight being lifted.
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.
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/s^2 , 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: m 1 v1i + m 2 v2i = m 1 v1f + m 2 v2f > note that for perfectly inelastic, the right half is (m 1 + m 2 )vf
Work and Energy I. Work: W = Fdcosϴ a. Work is only done when there is a displacement. Work is measured as the force applied in the direction of displacement. If you push a box downward on the floor with 100N, it doesn’t go anywhere (direction is perpendicular to movement direction) so no work is done. b. WORK IS PATH INDEPENDENT c. Measured in Joules, or kg*m^2 /s^2 II. Kinetic Energy: KE = ½mv^2 III. Potential Energy (gravity): PE = mgh a. Space/orbiting: U = -GMm/R IV. Potential Energy (spring): PE = ½kx^2 V. Power – measured in watts where 1 Watt = 1 J/s a. P = ΔW / Δt = change in work per change in time
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 = |f 2 – f 1 | 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)
Sound I. Properties of sound a. Sound is produced by vibrations in a medium b. Sound waves are longitudinal c. Sound waves must have a medium (cannot be created in or travel through a vacuum) d. Vibrations whose frequency is too low is infrasound (below 20Hz), too high is ultrasound (above 20,000Hz) II. Speed of sound a. Speed of sound is dependent on the medium it travels in b. Media with greater intermolecular forces have greater restoring force so vibrations can more rapidly compress for a new wave. As a result vsolid > vliquid > vgas i. Think of someone banging on a hammer on a train track. If you ear is to the track, you hear the sound faster than if you are listening from the air c. Speed of sound greater in stiff than compressible objects d. Speed of sound is faster in less dense medium, slower in denser media. Gases are less dense than solids, but they are so much more compressible that sound travels slower e. Speed of sound in hotter objects > colder objects i. Speed of sound at 0oC = 331m/s. ii. Speed of sound at 20oC = 340m/s. iii. Relates to speed of sound based on density III. Intensity of sound a. Intensity of sound is how we distinguish loudness. Since it is a measure of energy, it is affected by the Amplitude of the wave b. Intensity = energy per area and time. I = Power / Area or Watts per m^2. c. I = P/A, measured in W/m^2 d. We measure sound levels in decibels (β), which is just the log form of Intensity. e. β = 10 log I/Io i. I 0 is 10-12^ W/m^2 and is the minimum intensity of sound that can be heard. ii. I 0 would have an intensity of 0 dB (Intensity = 10-12^ W/m^2 ) iii. 100 I 0 would be a whisper. 10log 100 = 102 = 20dB iv. Every factor of 10 times the intensity of sound, the decibel raises by 10dB. So 10,000 times louder, or 10^4 times louder, you have an increase of 40dB. f. Intensity of sound is inversely proportional to r^2 , so it drops off pretty fast at larger distances. That’s because the area the wave travels through is so much larger and Intensity is inversely proportional to Area. g. Attenuation is the gradual loss of intensity as it moves through a medium i. Attenuation is greatest for soft, elastic, viscous, less dense IV. Pitch a. Pitch is the human perception of the frequency of sound.
b. Higher frequency = lower wavelength = higher pitch (measured in Hz) c. Humans can hear between 20Hz and 20,000Hz V. Doppler Effect a. Movement of the speed and listener will impact perception of frequency. Movement toward each other will always result in higher perceived frequency b. Moving Observer -> Top formula. You move toward the sound at speed vobserver, while sound wave is traveling toward you at speed vsource so you perceive the wavelengths to be shorter. i. fo = fs (v + vo)/v c. Moving source -> bottom formula. Even though the waves are the same frequency apart, because the source is moving toward you, when they get to you the starting point has been getting closer and closer so they appear to be shorter distances apart. i. fo = fs v/(v – vs) d. Reverse the signs if they are moving away from each other. i. If only one is moving, towards = higher f, away = lower f e. TOP OBSERVER MOVES, BOTTOM SOURCE MOVES [o comes before s on the alphabet so it is on top] f. Red shift (universe is expanding) – because the source of light is moving away from the observer, the frequency is shifted down (longer wavelength) so we see it light shift red toward the upper end of the visible spectrum VI. Ultrasound a. Sound waves will diffract out until they hit a surface, at which point they will reflect b. Source emits ultrasound, which will hit an object and reflect and give information to the detector. Other waves will go around until they hit another surface.
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 F 1 = A 1 /A 2 * F 2. If A 1 < A 2 then F 2 > F 1. 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 (F 1 d 1 = F 2 d 2 ) then d 2 < d 1. 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 m^3 /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: A 1 v 1 = A 2 v 2 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:
- Fluid is incompressible
- Negligible viscosity
- Laminar (non-turbulent) flow
- Steady flow rate iii. Equation: P + ½ρv^2 + ρgh = constant or P 1 + ½ρv 12 + ρgh 1 = P 2 + ½ρv 22 + ρgh 2 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 h 1 = h 2 , 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) ∝ r^4 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?
- Obstruction (rocks/plaques in arteries)
- 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
- 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/L 0 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/L 0 i. Think of it in terms of Hooke’s Law. F = kx, F = Y(A/L 0 )ΔL. Since Y is constant, Area is constant, L 0 is constant
Electricity and Magnetism I. Electric Force a. Electric Force: F = kq 1 q 2 /r^2 i. k = 9 x 10^9 Nm^2 /C^2 ii. Force is attractive if charges are opposite, repulsive if charges are same b. q – charge is quantized and comes in discrete packets i. elementary unit of charge (of one proton or electron) = 1.6 x 10-19^ C c. Conductors vs. Insulators – electrons are free to move in conductors while do not generally flow in insulators. Conductors – metals. Insulators – nonmetals like glass or plastic II. Electric Fields a. Electric fields are lines that help you visualize what a charge would feel from another charge. All charge sources will create electric fields. These field lines are vectors that show you what would happen if you put an imaginary positive test charge in the field. Closer lines denote a strong field while farther lines show weakening of the field b. F = qE or E = F/q i. E = kQ/r^2 where Q = source charge c. III. Electric Potential
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
