Physics (MCAT)

displacement

d, vector

slope is velocity

velocity

v = d/t, vector

slope is accel, area under curve is displacement

acceleration

a = v/t, vector

area under curve is velocity

g = 10 m/s^2

kinematics

d = (1/2)(vo + vf)/t, missing a
v = vo + at, missing d
d = vot + (1/2)at^2, missing vf
d = vft - (1/2)at^2, missing vo
vf^2 = vo^2 + 2ad, missing t

vector addition

place the vectors tip to tip, addition of the vectors is the new shortcut vector

break into component x and y then add up separately

vector magnitude = sqrt(x^2 + y^2)

projectile motion

constant a, constant vx, vy = 0 at top

break into component vx and vy

trig circle

sin(0) = 0, sin(30) = .5, sin(45) = .7, sin(60) = .86, sin(90) = 1

90 = (0, 1)
60 = (1/2, sqrt(3)/2)
45 = (sqrt(2)/2, sqrt(2)/2)
30 = (sqrt(3)/2, 1/2)
0 = (1,0)

Newton's laws

1. no force means no acceleration, concept of inertia
2. F_net = ma, concerned with 1 object
3. objects exert the same force on each other in opposite directions, concerned with 2 objects

units: N = kg*m/s^2

solving force problems:

1. draw force diagram on the object
2. break into x and y components, F_net = ma for both
3. solve equations

frictional force

kinetic friction- F = ukN, while moving/slipping static friction- F < usN, either rolling or not moving

gravitational force

F = Gm1m2/r^2

mass is scalar (measures inertia)
weight is vector (measures gravitational force)

uniform circular motion

turning

F = mv^2/r

a = v^2/r

force diagram on the object moving involves centripetal force, usually normal force (someone running will need to contact ground at an angle to turn)

at the top of a loop, the absolute minimum velocity would involve no normal force, so centripetal force equals gravitational force

simple harmonic motion

spring:
F = -kx
U = 1/2 k x^2
KE = 1/2 m v^2
angular frequency (w) = sqrt(k/m)
frequency (f) = 1/T = w/2pi

pendulum:
F = -mgsin(theta)
U = mgh
KE = 1/2 m v^2
angular frequency (w) = sqrt(g/L)
frequency (f) = 1/T = w/2pi

max U at ends of swing
max KE at middle of swing

torque

torque = rFsin(theta)

units are N*m

the greater the distance from the pivot point, the greater the torque

work

constant force acting on object over a distance

W = F d cos(theta)

units: J = N*m

force perpendicular to displacement does no work

area under a PV curve is work done by the system

when volume expands, work is done by the system
when volume compresses, work is done on the system

power

P = work/time = dKE/time

units: watts = J/t, J = ft*lb

energy

KE = (1/2) m v^2
PE = m g h

energy is conserved, unless outside forces act on the system

W = dK E

momentum

vector quantity

p = mv

units: kg*m/s

elastic collision- conservation of momentum, conservation of kinetic energy

inelastic collision- conservation of momentum, but kinetic energy is lost

impulse = change in momentum (dp) = Ft

thermal expansion

linear expansion: dL = a L dT

volume expansion: dV = b V dT

heat transfer

conduction- direct transfer of energy via molecular collisions

convection- transfer of energy via physical motion of heated gas or liquid

radiation- transfer of energy via EM waves

specific heat: q = m c dT
q is heat gained, does not apply during phase change

units of q: J or calories
units of c: J/g*C

heat of transformation: q = m L
units of L: J/g

first law of thermodynamics

dU = q - W
potential energy = heat gained - work done by system

no heat transfer: dU = -W
no change in volume: dU = q
no change in energy: q = W

elastic solids

shear modulus: S = (F/A) / (x/h)
young's modulus: Y = (F/A) / (dL/L)

fluid properties

density: p = m/v
units: kg/m^3

weight of fluid: F_g = pVg
units: N

buoyant force (equals weight of displaced fluid)
Fb = pfluid V_displaced g
units: N

archimedes principle:
object weight in water = object weight in air - weight of water displaced
weight of water displaced = pfluid Vdisplaced g
object weight in air = pobject Vdisplaced g

pressure: P = F/A
gauge pressure: P = pgh
absolute pressure: P = P_atm + pgh
units: pascal (P) = N/m^2

specific gravity

density of object/density of water

used in archimedes principle, where volume of object equals volume of water displaced

poiseuille's law

flow rate = ΔP(πr^4 / 8Lη)
Q = dP/R

flow rate depends on pressure difference, and radius, length, and viscosity of liquid

increased resistance can be caused by increased length, increased viscosity, or decreased radius

hydrostatic pressure

P_s = ρgy

y = distance belows the surface

shape and surface area of water does not matter

bernoulli's equation

bernoulli's equation

P + ρgh + 1/2 ρv^2 = constant

continuity equation (volume flow is constant)

vA = constant

pascal's principle (pressure is transmitted to every surface)

V = Ad = constant
P = F/A = constant
W = Fd = constant

venturi effect (constriction in pipe causes increased velocity because of continuity equation, thus causes decreased pressure because of bernoulli's equation)

P + 1/2 ρv^2 = constant

pitot tube

determine velocity of air by measuring pressure differential, one side has stagnant air so cross out KE

P1 = P2 + 1/2 ρv^2

gatorade bottle thing

KE of water coming out = PE of water at surface

due to bernoulli's equation

pressure difference

dP = pg dh

from bernoulli's equation

electrostatics

electrostatic force between two charges (coulomb's law)

F = (k q1 q2) / r^2

units of F = N
units of k: (N m^2) / C^2

electric field (field points from positive to negative charges)

E = F/q = kq/r^2
E = V/d

units of E: N/C, V/m

electric potential energy (work required to move from infinity to this point in the field)

U = q dV = qEd = k q1 q2 / r

electric potential (work required to move positive test charge to this point in the field)

V = U/q = kq/r
V = Ed (potential difference between two plates)

units of V: volt = J/C

magnetism

magnetic field (due to straight wire)
B= u I / 2 pi r
units: Tesla (T) = N s / m C

magnetic field (due to wire loop)
B = u I / 2 r

field strength is related to current over radius of loop

RHR

magnetic force (experienced by moving charge)
F = qvBsin(theta)

magnetic force (experienced by current)
F = ILBsin(theta)

force is 0 when charges move parallel/antiparallel to magnetic field

RHR

circuit properties

current: I = Q/t
units: amp (A) = C/s

voltage: V = IR
units: V = A*ohm
resistance is a property of the wire, so only V and I can realistically be changed

resistance: R = pL/A
R is directly proportional to T
units: ohm

capacitance: Q = VC
C = keA/d
U = 1/2 C V^2 (potential energy stored by capacitor)
units: farad (F) = C/V

power: P = IV
units: watt (W) = AV

power

power: P = IV
units: watt (W) = AV

power delivered = V^2/R
power lost (as heat) = I^2 R

high voltage, low current maximized power transmission!

internal resistance

R = internal resistance

V_actual = V - IR

circuits

voltage in series:
V = V1 + V2

voltage in parallel
V = V1 = V2

current in series:
I = I1 = I2

current in parallel:
I = I1 + I2

resistors in series:
R = R1 + R2

resistors in parallel:
1/R = 1/R1 + 1/R2

capacitors in series:
1/C = 1/C1 +1/C2

capacitors in parallel:
C = C1 + C2

speed of sound

343 m/s

v = sqrt(K/p)

depends on:

1. K = bulk modulus of medium (why sound is faster in solids)
2. p = density (why sound is faster in a dense solid)
3. temperature of medium (why sound is faster in hot air)

decibels

every 10 is a factor of 10

human range of hearing

20 Hz to 20 kHz
ultrasound is greater than 20 kHz

timbre

tone or character of the sound

doppler effect

4 variables:
frequency of sound, source
frequency of sound, observed
speed of sound
speed of source

if source is moving towards you, frequency observed is higher

if source is moving away from you, frequency observed is lower

used in Doppler ultrasound to calculate speed of blood or fetus, better than standard ultrasound

linear relationship between speed and frequency

stars have red shift (decreased frequency) since they're moving away from us

doppler approximation

df = frequency*relative velocity/speed of light

if wave reflects off object (police radar thing), double the doppler effect

index of refraction

n = speed of light/speed in new medium

light slows down in a medium with greater index of refraction (light is slower in glass than air)

sound is faster in glass than air

glass

frequency does not change between mediums

helium is a lie

refraction

n = c/v

n1sin(theta1) = n2sin(theta2)

always measure angles from the normal lines

when light enters higher index medium, refracts towards normal line

when light enters a lower index medium, refracts away from normal line

total internal reflection- enters lower index medium and refracts a full 90 degrees away from normal, how fiber optics work

index of refraction

w

mirrors

f > 0 means concave mirror

object beyond f means image is real (in front of mirror), inverted

object inside f means image is virtual (behind mirror), upright, enlarged

f < 0 means convex mirror

all images are virtual (behind mirror), upright, smaller

thin lens equations

1/do + 1/di = 1/f

do/di = ho/hi

height of image/object is proportional to distance of image/object

use to calculate magnification factor

lens

f > 0 means converging lens

object beyond f means image is real, inverted

object inside f means image is virtual, upright, enlarged

f < 0 means diverging lens

all images are virtual, upright, smaller

positive negative

do (object distance) is always positive

di (image distance) is positive for real images, negative for virtual images

f (focal length) is positive for concave mirrors, converging lens

m (magnification) is positive for upright image

magnification

m = -i/o
m < 1 means image is reduced m > 1 means image is enlarged

negative m means inverted image
positive m means upright image

diopter

refractive power unit (D = 1/m)

reciprocal of focal length of a lens

smaller focal length means more powerful lens

aberration

spherical aberration:

assumption of sin(theta) = theta

light at top of lens is refracted more, shorter focal length

so not all rays go through focal length, inherent property with spherical lens

chromatic aberration:

dispersion says greater wavelengths have smaller index of refraction

red refracts less than blue

human eye

ciliary muscles adjust lens to focus image on retina

requires diverging lens (concave)
corrects myopia, nearsightedness, focal length of eye is too short

requires converging lens (convex)
corrects hyperopia, farsightedness, focal length of eye is too long

image on retina is upside down, brain flips it

efficiency

work output/work input

standard units

kg
m
s

photoelectric effect

work function is energy to remove an electron

input energy is greater than output energy

KE = hf - W
W = work function

photon

E = hf
units of E: J
units of h = J s
units of f: 1/s

photon absorbed by electron raises its energy level

electron

electrons can be transferred, protons cannot